I will present a formulation of scattering amplitudes in the simplest colored, cubic, scalar theory - Tr \phi^3 - as an integral over the space of curves on Riemann surfaces, valid at all loop orders and at all orders in the topological 't Hooft expansion. This so-called "curve integral" has the main advantage of describing amplitudes as a unique object, rather than as a sum over Feynman diagrams, allowing to study phenomena which are hidden graph-by-graph and suggesting powerful techniques for the numerical evaluation of amplitudes. Furthermore, the singularity structure of the propagators of Tr \phi^3 theory is shared by any colored theory, thus suggesting the generalization of the formalism to more realistic theories by insertion of appropriate numerators in the curve integral. At the heart of the formalism is a simple counting problem associated to curves on surfaces, which surprisingly provides a combinatorial origin for the physics of scattering amplitudes. The talk is based on 2309.15913, 2311.09284 and 2402.06719.

Cosmological Correlators are one of the physical quantities that are of interest to cosmologists and are also of theoretical interest as they are related to CFT correlators via the AdS/CFT correspondence. These differ from the S-matrix as they are correlation functions computed on a given time slice. In this talk, I will review some progress in computing these in momentum space and also describe its relation to the S-matrix.

The study of scattering amplitudes in AdS space is under rapid development. The 4-point tree amplitudes of all (half-)maximally SUSY theories and all KK modes are known, and partial results are available at loop level. However, higher-point data remains limited. In this talk based on [2312.15484], I will discuss higher-point “supergluon” tree amplitudes in AdS_5 super-Yang-Mills theory / N=2 sCFT_4, which presents a simpler scenario compared to the more familiar AdS_5 supergravity / N=4 sCFT_4. Specifically, I will describe some properties of supergluon amplitudes in Mellin space and provide a constructive algorithm to compute the amplitude to all multiplicities. I will end by observing some similarity with flat-space amplitudes which hints at a possible set of Feynman rules.

Carrollian holography aims to express gravity in asymptotically flat space-time in terms of a dual Carrollian CFT living at null infinity. In this talk, I will review some aspects of Carrollian holography and argue that this approach is naturally related to the AdS/CFT correspondence via a flat limit procedure. I will then introduce the notion of Carrollian amplitude, which allows to encode massless scattering amplitudes into boundary correlators, and explain its connection to celestial amplitudes. Finally, I will present recent results concerning Carrollian OPEs and deduce how soft symmetries act at null infinity. This talk will be mainly based on https://arxiv.org/abs/2312.10138.

I will revisit a generalised crossing equation that follows from harmonic analysis on the conformal group, and is of particular interest for the cosmological bootstrap programme. Then I present an exact solution to this equation, for dimensions two or higher, in terms of 6j symbols of the Euclidean conformal group, and discuss its relevance. The presentation is based on https://arxiv.org/abs/2305.08939.

The double copy is a remarkable relationship between gauge theory and gravity that has been explored in a number of contexts, most notably scattering amplitudes and classical solutions. The convolutional double copy provides a straightforward method to bridge the two theories via a precise map for the fields and symmetries at the linearised level. This method has been thoroughly investigated in flat space, offering a comprehensive dictionary both with and without fixing the gauge degrees of freedom. In this paper, we extend this to curved space with an (anti) de Sitter background metric. We work in the temporal gauge, and employ a modified convolution that involves the Mellin transformation in the time direction. As an example, we show that the point-like charge in gauge theory double copies to the (dS-) Schwarzschild black hole solution.

Positivity bounds have wide applications from QCD to quantum gravity. However, very little is understood in the cosmological settings. In this talk, we will pursue an alternative route and place similar bounds using the classical, statistical nature of the superhorizon modes. I will describe how our approach yields constraints on the anomalous dimension of quantum fields in de Sitter (dS), as well as correlation functions in quasi-dS spacetime. Some of the results complement previous works using standard positivity arguments. This is based on 2310.02490 with Daniel Baumann, Daniel Green, and Yiwen Huang.

The scattering of gluons and gravitons in trivial backgrounds is endowed with many surprising infrared features which have interesting conformal interpretations on the two-dimensional celestial sphere. I will review the basic ideas of celestial holography and show how the infinite-dimensional chiral soft algebras (S algebra and $w_{1+\infty}$ algebra) were found in trivial backgrounds. Then, I will show in the presence of a nonvanishing cosmological constant, Strominger's infinite-dimensional $w_{1+\infty}$ algebra of soft graviton symmetries is modified in a simple way. The deformed algebra contains a subalgebra generating SO(1,4) or SO(2,3) symmetry groups of dS4 or AdS4, depending on the sign of the cosmological constant.

I will present a formulation of scattering amplitudes in the simplest colored, cubic, scalar theory - Tr \phi^3 - as an integral over the space of curves on Riemann surfaces, valid at all loop orders and at all orders in the topological 't Hooft expansion. This so-called "curve integral" has the main advantage of describing amplitudes as a unique object, rather than as a sum over Feynman diagrams, allowing to study phenomena which are hidden graph-by-graph and suggesting powerful techniques for the numerical evaluation of amplitudes. Furthermore, the singularity structure of the propagators of Tr \phi^3 theory is shared by any colored theory, thus suggesting the generalization of the formalism to more realistic theories by insertion of appropriate numerators in the curve integral. At the heart of the formalism is a simple counting problem associated to curves on surfaces, which surprisingly provides a combinatorial origin for the physics of scattering amplitudes.

We study integrated correlators of four superconformal primaries O_p with arbitrary charges p in N =4 super Yang-Mills theory (SYM). The ⟨O_2 O_2 O_p O_p⟩ integrated correlators can be computed by supersymmetric localisation, whereas correlators with more general charges are currently not accessible from this method and in general contain complicated multi-zeta values. Nevertheless we observe that if one sums over the contributions from all different channels in a given correlator, then all the multi-zeta values (and products of zeta’s) cancel leaving only ζ(2ℓ+1) at ℓ-loops. We then propose an exact expression of such integrated correlators in the planar limit, valid for arbitrary ’t Hooft coupling, which matches known strong coupling results. As an application, our result is used to determine certain 7-loop Feynman integral periods and fix previously unknown coefficients in the correlators at strong coupling.

TBA

IIntroduced in the mid-50s, crossing symmetry in interacting quantum field theory suggests that particles and anti-particles traveling back in time are indistinguishable. This perspective has significant practical implications for perturbative computations and for the S-matrix bootstrap. To prove this property rigorously, one needs to show that on-shell observables across different channels are boundary values of the same analytic function. For the simplest cases of 2-to-2 and 2-to-3 scattering, the known non-perturbative proofs rely heavily on physical principles like (micro)causality, locality, and unitarity, but also on a significant amount of several variables complex analysis, which makes their extension to arbitrary multiplicity quite challenging. In this talk, we review recent progress regarding the implications of crossing symmetry in quantum field theory, assuming analyticity. The story begins by asking what can be measured asymptotically in quantum field theory? Among the answers to this question are conventional (time-ordered) scattering amplitudes, but also a whole compendium of inclusive measurements, such as expectation values of gravitational radiation and out-of-time-ordered correlators. We show that these asymptotic observables can be related to one another through new versions of crossing symmetry, and propose generalized crossing relations together with the corresponding paths of analytic continuation. Throughout the talk, we show how to apply crossing between different observables in practice, both at tree- and loop-level. (Based on: 2308.02125 and 2310.12199)

In this talk, I will discuss some recent developments in the computation of four-point functions of half-BPS operators in a certain 4d \mathcal{N}=2 SCFT with flavour group SO(8) at large N and large ‘t Hooft coupling ë, dual to scattering of gluons at genus one on a AdS_5 X S^3 background. At tree level and in the field-theory limit, the theory is known to enjoy an 8d hidden conformal symmetry. Although the symmetry turns out to be broken by 1/N and 1/ë corrections, I will argue that it is still possible to identify an 8d dimensional organizing principle which has precise implications for the structure of one-loop Mellin amplitudes. The talk is based on https://arxiv.org/pdf/2309.15506.pdf.

I will present a constructive method to compute the Virasoro-Shapiro amplitude on AdS5xS5, order by order in AdS curvature corrections. A simple toy model for strings on AdS indicates that at order k the answer takes the form of a genus zero world-sheet integral involving single-valued multiple polylogarithms of weight 3k. The coefficients in an ansatz in terms of these functions are then fixed by Regge boundedness of the amplitude, which is imposed via a dispersion relation in the holographically dual CFT. We explicitly constructed the first two curvature corrections. Our final answer reproduces all CFT data available from integrability and all localisation results, to this order, and produces a wealth of new CFT data for planar N=4 SYM theory at strong coupling.

The on-shell approach to scattering amplitudes relies heavily on locality, gauge invariance and unitarity to fix the amplitude. In this talk, we derive new expressions for tree-level graviton amplitudes in N=8 supergravity via recursion relations. To do this, we use knowledge about the zeros of graviton amplitudes in collinear kinematics. We contextualize the expansion in the language of global residue theorems and identify canonical building blocks or G-invariants. Finally, we comment on a possible geometric origin of these G-invariants, analogous to the R-invariants in N=4 supersymmetric Yang-Mills.

As a basic observable in asymptotically flat space-time, the S-matrix provides a natural quantity to compute in any holographically dual description and has been shown, when recast in a basis of boost eigenstates, to share properties with CFT correlation functions. In this talk we will discuss the computation of scattering amplitudes in non-trivial, asymptotically flat backgrounds and their electromagnetic analogues. I will discuss the extension of the CFT description to these non-trivial backgrounds, including shockwave geometries, Schwarzschild and Kerr, and the description of the S-matrix as two-dimensional correlators.

Almost everything we know about integrability in spin chains was developed for Hamiltonians with nearest-neighbour interactions. Many of the integrable Hamiltonians that play a role in N=4 Super Yang-Mills and AdS/CFT , however, possess long-range interactions. A better understanding of this type of spin chain would therefore be very helpful in this context. In this talk, I will discuss some properties of these systems and recent progress in constructing and understanding them.

“Celestial CFT” is a formalism which attempts to recast quantum gravity in (d+2)-dimensional asymptotically flat spacetimes in terms of a d-dimensional Euclidean CFT residing at the conformal boundary. I will discuss certain universal aspects of this correspondence. As in Anti-de Sitter space, bulk gravitons produce a boundary stress tensor, and bulk gluons furnish boundary conserved currents. I will also show that continuous spaces of vacua in the bulk map directly onto the conformal manifold of the boundary CFT. This correspondence provides a new perspective on the role of the BMS group in flat space holography, and offers a new interpretation of the antisymmetric double-soft gluon theorem in terms of the curvature of an infinite-dimensional vacuum manifold. If time permits, I will also discuss the Celestial CFT duals of integrable Quantum Field Theories, which provide a useful testing ground for the correspondence.

In this talk, I shall consider the convolutional double copy for BRST and anti-BRST covariant formulations of gravitational and gauge theories. I shall give a general off-shell BRST and anti-BRST covariant formulation of linearised $\mathcal{N}=0$ supergravity using superspace methods and use this covariance to obtain an off-shell convolution map between fields in $\mathcal{N}=0$ supergravity and linearised Yang-Mills. I shall then demonstrate the validity of this map for the Schwarzschild black hole and the ten-dimensional black string solution as two concrete examples.

We derive soft theorems for theories in which time symmetries -- symmetries that involve the transformation of time, an example of which are Lorentz boosts -- are spontaneously broken. The soft theorems involve unequal-time correlation functions with the insertion of a soft Goldstone in the far past. Explicit checks are provided for several examples, including the effective theory of a relativistic superfluid and the effective field theory of inflation. We discuss how in certain cases these unequal-time identities capture information at the level of observables that cannot be seen purely in terms of equal-time correlators of the field alone. We also discuss when it is possible to phrase these soft theorems as identities involving equal-time correlators.

Scattering amplitudes of elementary particles exhibit a fascinating simplicity, which is entirely obscured in textbook Feynman-diagram computations. While these quantities find their primary application to collider physics, describing the dynamics of the tiniest particles in the universe, they also characterise the interactions among some of its heaviest objects, such as black holes. Violent collisions among black holes occur where tremendous amounts of energy are emitted, in the form of gravitational waves. 100 years after having been predicted by Einstein, their extraordinary direct detection in 2015 opened a fascinating window of observation of our universe at extreme energies never probed before, and it is now crucial to develop novel efficient methods for highly needed high-precision predictions. Thanks to their inherent simplicity, amplitudes are ideally suited to this task. I will begin by reviewing the computation of a very familiar quantity Newton's potential, from scattering amplitudes and unitarity. I will then explain how to compute directly observable quantities such as the scattering angle for light or for gravitons passing by a heavy mass such as a black hole, and how to incorporate emission of gravitational waves. These computations are further simplified thanks to a remarkable, yet still mysterious connection between scattering amplitudes of gluons (in Yang-Mills theory) and those of gravitons (in Einstein's General relativity), known as the "double copy", whereby the latter amplitudes can be expressed, schematically, as sums of squares of the former -- a property that cannot be possibly guessed by simply staring at the Lagrangians of the two theories. I will conclude by discussing the prospects of performing computations in Einstein gravity to higher orders in Newton's constant using a new, gauge-invariant version of the double copy, and as an example I will briefly discuss the computation of the scattering angle for classical black hole scattering to third post-Minkowskian order (or O(G^3) in Newton's constant G).

I will discuss generalizations of shift symmetries, galileon symmetries, and extended galileon symmetries to (A)dS space and to higher spin. These symmetries are present for fields with particular masses, and are related to partially massless symmetries. I will discuss some of the properties of non-linear extensions of these symmetries and their invariant interactions, including the existence of a unique ghost-free theory in (A)dS space that is an (A)dS extension of the special Galileon, and will speculate on possible generalizations to interacting massive higher-spin particles.

In the self-dual sector for Yang-Mills and gravity, I will show how to construct an extended phase space at null infinity, to all orders in the radial expansion. This formalises the symmetry origin of the infrared behaviour of these theories to all subleading orders. As a corollary, I will also derive a double copy mapping a subset of YM gauge transformations to a subset of diffeomorphisms to all orders in the transformation parameters.

In this talk I will discuss universal statistical properties of spectra in 4d conformal supersymmetric Yang-Mills (SYM) theories and show how they give insight into the nature of the underlying model. We will see integrability manifest itself in the planar spectra of certain SYM theories, while spectra in more generic SYM theories can be described by random matrix theory, indicating the quantum-chaotic nature of the corresponding model. In the case of the imaginary-beta deformation of N=4 SYM theory, this provides a weak-coupling analogue of the chaotic dynamics seen for classical strings in the dual background, and here we can also connect to the classical notion of chaos by studying the semi-classical Landau-Lifshitz limit.

We begin by reviewing the colour-kinematics duality of (super) Yang-Mills theory and its double copy into (super)gravity. We then show that off-shell colour-kinematics duality can be made manifest in the Yang—Mills Batalin—Vilkovisky action, up to Jacobian counterterms. The latter implies a departure from what is normally understood by colour-kinematics duality in that the counterterms generically break it. However, this notion of CK duality is very natural and, most importantly, implies the validity of the double copy to all orders in perturbations theory. Perturbatively, at least, gravity is the square of Yang—Mills! We then describe generalisations to the non-linear sigma model and super Yang-Mills theory, where Sen’s formalism for self-dual field strengths emerges automatically. We conclude by discussing the mathematical underpinnings of these observations in terms of Homotopy algebras. Figuratively, colour-kinematics duality is a symmetry of Yang—Mills in the same sense that a mug is a donut.

We study four-point functions of scalars, conserved currents, and stress tensors in a conformal field theory, generated by a local contact term in the bulk dual description, in two different causal configurations. The first of these is the standard Regge configuration in which the chaos bound applies. The second is the `causally scattering configuration' in which the correlator develops a bulk point singularity. We find an expression for the coefficient of the bulk point singularity in terms of the bulk S matrix of the bulk dual metric, gauge fields and scalars, and use it to determine the Regge scaling of the correlator on the causally scattering sheet in terms of the Regge growth of this S matrix. We then demonstrate that the Regge scaling on this sheet is governed by the same power as in the standard Regge configuration, and so is constrained by the chaos bound, which turns out to be violated unless the bulk flat space S matrix grows no faster than s^2 in the Regge limit. It follows that in the context of the AdS/CFT correspondence, the chaos bound applied to the boundary field theory implies that the S matrices of the dual bulk scalars, gauge fields, and gravitons obey the Classical Regge Growth (CRG) conjecture. We also will discuss recent progress in extending our analysis to bulk AdS exchange diagrams and AdS loops.

I will discuss the fundamentals of quantum field theory on a rigid de Sitter space. First, I will show that the perturbative expansion of late-time correlation functions to all orders can be equivalently generated by a non-unitary Lagrangian on a Euclidean AdS geometry. This finding systematizes recent findings in the literature on the relation between dS and AdS Feynman diagrams, as well as allows us to establish basic properties of these correlators, which comprise a Euclidean CFT. Second, I use this to infer the analytic structure of the spectral density that captures the conformal partial wave expansion of a late-time four-point function, to derive an OPE expansion, and to constrain the operator spectrum. Third, I will prove that unitarity of the de Sitter theory manifests itself as the positivity of the spectral density. This statement does not rely on the use of Euclidean AdS Lagrangians and holds non-perturbatively.

We discuss the structure of Regge trajectories of 6d N=(2,0) SCFTs combining analyticity in spin with supersymmetry. Focusing on the four-point function of supermultiplet we show how "analyticity in spin" holds for all spins greater than -3. Through the Lorentzian inversion formula we then describe an iterative procedure to "bootstrap" this four-point function starting from protected data, and compare the results with the numerical bootstrap bounds. This procedure works best at large but finite central charge, where non-protected contributions are suppressed by the inversion formula.

In joint work with Andreas Rosén we prove a general Weitzenböck identity for arbitrary pairs of constant coefficients homogeneous first order PDE operators on domains for fields that satisfy natural boundary conditions. This identity gives rise to a generalization of the Levi form for the classical d-bar complex in several complex variables. Under the assumption that one of the operators is cocancelling, a concept introduced by J. Van Schaftingen in his work on endpoint Sobolev estimates, and an additional algebraic condition we prove a generalized Morrey inequality. We derive from this a weighted Sobolev inequality as well as giving new proof of the equivalence of Morrey inequalities and subelliptic estimates. Using the theory of J. Kohn and L. Nirenberg this in particular implies solvability for the generalized Neumann-dbar problem on generalized strongly pseudoconvex domains for constant rank operators satisfying our conditions.

Data assimilation by nudging is done by adding a linear feedback term involving spatially coarse time series data to drive a model solution to the one observed. Typically this requires observations that are distributed throughout the spatial domain. This talk addresses using data on only a subdomain which is moving with time for the 2D Navier-Stokes equations. We prove that if the resolution of data on the subdomain is fine enough, and the subdomain moves fast enough, then the nudged solution converges to the observed solution at an exponential rate. We then present numerical evidence demonstrating the efficacy of this approach.

Empirical studies suggest that for vast tracts of land in the tropics, closed-canopy forests and savanna are alternative stable states, a proposition with far-reaching implications in the context of ongoing climate change. Consequently, numerous mathematical models, both spatially implicit and explicit, have been proposed to capture the mechanistic basis of this bistability and quantify the stability of these ecosystems. I will present analysis of a spatially extended version of the Staver-Levinforest-savanna model and highlight some open mathematical problems related to the dynamics generated by these nonlocal PDEs. On a homogeneous domain, relevant to smaller spatial scales, we uncover various types of pattern-forming bifurcations in the presence of resource limitation. On larger (continental) spatial scales, heterogeneity plays a significant role in determining observed vegetative cover. Incorporating domain heterogeneity is a pressing mathematical challenge and leads to interesting phenomena such as front-pinning, complex waves of invasion, and extensive multi-stability.

The Brascamp-Lieb inequalities are a generalisation of the Hölder, Loomis-Whitney and Young convolution inequalities, and have found many applications in harmonic analysis and elsewhere. In this talk we present an “adjoint” version of these inequalities which may be viewed as an L^p version of the entropic Brascamp-Lieb inequalities of Carlen and Cordero-Erausquin. As an application we establish some lower bounds on a range of tomographic transforms, such as the classical X-ray and Radon transforms. This is joint work with Terence Tao.

The derivation of kinetic equations from particle system is a well-established programme; a particularly well know-example is the justification of the non-linear Boltzmann equation as a scaling limit of hard sphere dynamics. I will present recent results for the considerably easier problem of ballistic annihilation where we can obtain error bounds which are much smaller than the standard Gronwall estimates.

We discuss the well-posedness theory for the Landau equation with rough, slowly decaying initial data. In particular, our method is able to treat initial data in the critical weighted $L^{3/2}$ space, where the coefficients are unbounded. This is joint work with William Golding and Maria Gualdani.

I will discuss the extension of the Aw-Rascle model to multi-dimensional case and present our recent results on that model concerning: existence and uniqueness of solutions, and their asymptotic limits in the “hard congestion regime”.

In my talk I will present a new approach to the spectral theory of systems of PDEs on closed manifolds, developed in a series of recent papers by Dmitri Vassiliev (UCL) and myself, based on the use of pseudodifferential projections. After discussing the general theory, I will turn to the (non-elliptic) operator curl, and explain how our techniques offer a new pathway to the study of spectral asymmetry.

The mathematical modelling of cancer has been increasingly applying fluid-dynamics concepts to describe the mechanical properties of tissue growth. The biomechanical pressure plays a central role in these models, both as the driving force of cell movement and as an inhibitor of cell proliferation. In this talk, I will present how it is possible to build a bridge between models that have different pressure-velocity or pressure-density relations. In particular, I will focus on the inviscid limit from a Brinkman model to a porous medium-type model, and the incompressible limit that links the latter to a Hele-Shaw free boundary problem with density constraint.

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.

In a recent work, R. Carmona, Q. Cormier and M. Soner proposed a mean field game based on the classical Kuramoto model, originally motivated by systems of chemical and biological oscillators. Such MFG model exhibits several stationary equilibria, and the question of their ability to capture long time limits of dynamic equilibria is largely open. I will discuss in the talk how to show that, up to translations, there are two possible stationary equilibria only - the incoherent and the synchronised one - provided that the interaction parameter is large enough. Finally, I will present some local stability properties of the synchronised equilibrium. Based on a joint work with A. Cesaroni.

This talk is devoted to the study of the homogenization problem for the linear Boltzmann equation in energy. Two approaches are considered. The first one is based on the two-scale convergence theory, which allows to prove the existence of a memory term in the structure of the homogenized equation. Because of this term, the semigroup property of the starting problem is lost in the limit. However, the semigroup structure of the limit equation can be preserved by working in a new framework based on an extended phase-space.

We construct global strong solutions to the 1D Boltzmann equation with angular cutoff, for generic large data on the line and for periodic boundary conditions, expanding on the work of Biryuk, Craig, and Panferov to obtain quantitative growth estimates. We use an angular averaging estimate and an improved ODE estimate to construct a global well-posedness theory for finite energy data. We show $L^\infty$ dissipation of the density on the real line and obtain an exponential bound on the $L^\infty$ growth of the density for periodic boundary conditions. Additionally, we show propagation of derivatives and moments, thus constructing a global solution theory for 1D Boltzmann in the Schwartz class.

The seminar will be focused on the analysis of fourth-order aggregation-diffusion equations using an optimal transport approach. These models have been recently obtained as approximation of nonlocal systems of PDEs describing cell-cell adhesion, which is a crucial mechanisms regulating collective cell migration during tissue development, homeostasis and repair. In a recent work, we use the 2-Wasserstein gradient flow structure of such equations to give sharp conditions for global in time existence of weak solutions, in any dimension and for general initial data. The energy involved presents two competing effects: the Dirichlet energy and the power-law internal energy. Homogeneity of the functionals reveals critical regimes that we analyse. In addition, we study a system of two Cahn-Hilliard-type equations exhibiting an analogous gradient flow structure. This is based on a joint work with J. A. Carrillo, C. Falcó, and A. Fernández-Jiménez in Oxford.

Instead of solving the eigenvalue problem $Tf=zf$ for a linear operator $T$ and eigenvalue $z$, we can use a multiplier $B$ and instead solve the linear pencil problem $BTf=zBf$. This leads us to study the numerical range and essential numerical range of linear pencils. The essential numerical range is used to describe the set of spectral pollution when approximating theeigenvalue problem by projection and truncation methods. By taking intersection over various multipliers, we get sharp enclosures. We apply the results to various differential operators. This talk is based on joint work with Marco Marletta (Cardiff).

If the solutions to a continuity equation suitably dissipate a certain energy along their flow, then it is possible to interpret such curves as paths of steepest descent on the space of probability measures endowed with the Wasserstein space, a so-called `Wasserstein gradient flow’.

Over the past 30 years, this Wasserstein gradient flow interpretation has been explored extensively and to great success, allowing for new results regarding many important PDEs including Fokker-Planck, Keller-Segel and Porous media equations.

One of the key motivating factors in the Wasserstein gradient flow interpretation is that, for many systems, several important properties of the PDE (such as existence, uniqueness, and convergence) may be established, not by studying the equation itself, but instead, by studying the properties of an energy functional which is known to dissipate its solutions.

In this talk, I describe the importance of convexity in establishing some key properties of a gradient flow in the classical Euclidean setting and then show how this framework has been abstracted in various ways to the Wasserstein space in order to establish results for various PDEs. Furthermore, I introduce the notion of convexity along acceleration-free curves and describe how this links `geodesic’ convexity on the Wasserstein space to convexity on Hilbert spaces of $L^2$ random variables.

Systems that involve many elements, be it a gas of particles or a herd of animals, are ubiquitous in our day to day lives. While such systems are of great interest to us, their investigation is hindered by the amount of (usually coupled) equations that are needed to be solved in order to understand them. The mesoscopic approach, dating back to the late 19th century, tries to simplify our dealing with such systems by finding an equation that describes the evolution of a limiting average element of said system. While widely used, the question of the validity of such equations remains an issue. One prime example is the proof of the validity of the Boltzmann equation - a problem so profound that it was included in Hilbert's famous 23 problems. In his 1956 work, Mark Kac provided a probabilistic justification to Boltzmann's equation by considering an average model of dilute gas (i.e. an evolution equation for the probability density of the ensemble) and introducing the notion of (molecular) chaos - an asymptotic correlation relation that refers to the fact that due to the rarity of the gas, any given group of particles become more and more independent as the number of particles in the system increases. Kac’s work has achieved more than its original goal and has planted the seed from which a new method, the so-called mean field limit approach, arose. The mean field limit approach attempts to find the behaviour of a limiting average element in a given system by using two ingredients: an average model of the system and an asymptotic correlation relation that expresses the emerging phenomena we expect to get as the number of elements goes to infinity. Mean field limits of average models have permeated to fields beyond mathematical physics in recent decades. Examples include models that pertain to biological, chemical, and even societal phenomena. However, to date we only use chaoticity as our asymptotic correlation relation. This doesn’t seem reasonable in models that revolve around biological and societal phenomena. Such model, Choose the Leader model, was recently constructed and was shown to break the notion of chaoticity. In our talk we will introduce Kac’s model and the notions of chaos and mean field limits. We will discuss the problem of having chaos as the sole asymptotic correlation relation and define a new asymptotic relation of order. We will show that this is the right relation for the Choose the Leader model and highlight the importance of appropriate scaling in its investigation.

This talk will discuss the rigidity of equality cases for perimeter inequality under Schwarz symmetrisation. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the Schwarz symmetric set. First, we will discuss the existing results in literature related to the corresponding rigidity problem. Then, we will present sufficient and necessary conditions for rigidity under this framework. Our analysis will be based on the properties of the corresponding distribution function and on a careful study of the transformations that can be applied to the symmetric set, without creating any perimeter contribution.

The PDE formulation of Mean Field Games (MFG) is described by nonlinear systems in which a Hamilton—Jacobi--Bellman (HJB) equation and a Kolmogorov—Fokker--Planck (KFP) equation are coupled. The advective term of the KFP equation involves a partial derivative of the Hamiltonian that is often assumed to be continuous. However, in many cases of practical interest, the underlying optimal control problem of the MFG may give rise to bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we present results on the analysis and numerical approximation of second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the partial derivative of the Hamiltonian in terms of subdifferentials of convex functions. We prove the existence of unique weak solutions to MFG PDIs under a monotonicity condition similar to one that has been considered previously by Lasry & Lions. Moreover, we introduce a monotone finite element discretization of the weak formulation of MFG PDIs and present theorems on the strong convergence of the approximations to the value function in the $H^1$-norm and the strong convergence of the approximations to the density function in $L^q$-norms. We conclude the talk with some numerical experiments involving non-smooth solutions.

[Joint with Probability] This talk will be about the convergence problem in mean field control (MFC), i.e. the challenge of rigorously justifying the convergence of certain "symmetric" \(N\)-particle control problems towards their mean field counterparts. On the one hand, this convergence problem is already well-understood from a qualitative perspective, thanks to powerful probabilistic techniques based on compactness. On the other hand, quantitative results (i.e. rates of convergence) are more difficult to obtain, in large part because the value function of the mean field problem (which is also the solution to a certain Hamilton-Jacobi equation on the Wasserstein space) may fail to be \(C^1\), even if all the data is smooth. After giving an overview of the convergence problem, I will discuss the results of two recent joint works with Cardaliaguet, Daudin, Delarue, and Souganidis, in which we use some ideas from the theory of viscosity solutions to overcome this lack of regularity and obtain rates of convergence of the $N$-particle value functions towards the value function of the corresponding MFC problem.

We obtain some new inequalities for the Newtonian capacity of compact, convex sets in $\mathbb{R}^d, d \geq 3$. Joint work with A. Malchiodi and D. Bucur.

In this talk we will discuss linear operators defined on a Banach or Hilbert space $X$ with a gap in their spectrum. Natural questions are if such gaps lead to a decomposition of the underlying space $X$ and if the gaps are stable under relatively bounded perturbations. This talk is based on joint works with C. Wyss and with J. Moreno.

We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form $A := -\partial_x + W(x)$, with $W(x) \geq 0$, defined in $L^2(\mathbb R)$. We provide a sharp estimate for the norm of its resolvent operator as the spectral parameter diverges to $+\infty$. Furthermore, we describe the C_0-semigroup generated by $-A$ and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schrödinger and damped wave operators and also the optimality of abstract resolvent bounds based on Carleman-type estimates.

The seminar is based on a joint work with A. Arnal (QUB, Belfast).

For more than a century, people have been trying to understand the precise connection between the properties of solutions of an ellipitc PDE and the geometric properties of the set where the equation is given. The scenarios are particularly interesting when the coefficients of the equation are non-smooth and the set is rough. One big breakthrough in these investigations is an equivalence between absolute continuity of the harmonic measure and rectifiability of the set. Unfortunately, this equivalence relies crucially on some topological assumptions on the set and shatters in the case of higher co-dimension (e.g. a curve in 3-dimensional space). Recently, together with J. Feneuil, we have found a unified characterization of rectifiability of a set of any codimension in terms of some Carleson estimate for the Green function. This result is built on a series of earlier works on the Green function and a smooth distance, which I will also discuss in the talk.

The GENERIC framework (General Equation for Non-Equilibrium Reversible Irreversible Coupling) [Öttinger, H. C. Beyond equilibrium thermodynamics. John Wiley & Sons, 2005] provides a systematic method to derive thermodynamically consistent evolution equations. It was originally introduced in the context of complex fluids, and has been employed to a plethora of applications in physics and engineering over the last two decades, such as to anisotropic inelastic solids, viscoplastic solids, and thermoelastic dissipative materials. However, the mathematics of this framework is still underdeveloped. In this talk, I will present some research on the variational structure of the GENERIC and discuss its connections to hypocoercivity and large deviations.

Due to unforeseen circumstances, this seminar had to be cancelled. We hope to have Matteo Capoferri visiting us soon enough to give their talk.

Vlasov-Poisson type systems are well known as kinetic models for plasma. The precise structure of the model differs according to which species of particle it describes, with the `classical’ version of the system describing the electrons in a plasma. The model for ions, however, includes an additional exponential nonlinearity in the equation for the electrostatic potential, which creates several mathematical difficulties. For this reason, the theory of the ionic system has so far not been as fully explored as the theory for the electron equation.

A plasma has a characteristic scale, the Debye length, which describes the scale of electrostatic interaction within the plasma. In real plasmas this length is typically very small, and in physics applications frequently assumed to be very close to zero. This motivates the study of the limiting behaviour of Vlasov-Poisson type systems as the Debye length tends to zero relative to the observation scale—known as the ‘quasi-neutral’ limit. In the case of the ionic model, the formal limit is the kinetic isothermal Euler system; however, this limit is highly non-trivial to justify rigorously and known to be false in general without very strong regularity conditions and/or structural conditions.

I will present a recent work, carried out in collaboration with Mikaela Iacobelli, in which we prove the quasi-neutral limit for the ionic Vlasov-Poisson system for a class of rough \(L^\infty\) data: that is, data that may be expressed as perturbations of an analytic function, small in the sense of Monge-Kantorovich distances. The smallness of the perturbation that we require is much less restrictive than in the previously known results.

We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called "binormal curvature flow". Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings, and of a helical filament, associated to a translating-rotating helix. In this talk I will consider the case of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila (U. of Bath), M. del Pino (U. of Bath) and J. Wei (U. of British Columbia).

The seminar will be held over Zoom. The details of the meeting are below Join Zoom Meeting https://durhamuniversity.zoom.us/j/96177289696?pwd=blE1UDRhSDRucUFpZXl6ME1ydXFKdz09 Meeting ID: 961 7728 9696 Passcode: 462324

The Liouville quantum gravity (LQG) surface, formally defined as a 2-dimensional Riemannian manifold with conformal factor being the exponentiation of a Gaussian free field, is closely related to random planar geometry as well as scaling limits of models from statistical mechanics. In this talk, I shall explain what the Liouville Brownian motion is and discuss certain short-time asymptotics of Liouville heat kernel which may be reinterpreted as the Weyl's law for the eigenvalues associated to the (random) Laplace-Beltrami operator on LQG surfaces. This is a joint work with Nathanael Berestycki.

In this talk we introduce a new class of kinetic models, which overcome the standard assumption in kinetic transport theory that collision processes happen instantaneously. On the level of the underlining stochastic processes this results in replacing the jump-process, which are defining the collisions, with continuous stochastic processes.

We investigate a kinetic model with non-instantaneous binary alignment collisions between particles. The collisions are described by a transport process in the joint state space of a pair of particles, where the states of the particles approach their midpoint. For two spatially homogeneous models with deterministic or stochastic collision times existence and uniqueness of solutions, the long time behavior, and the instantaneous limit are considered, where the latter leads to standard kinetic models of Boltzmann type.

Reference: L. Kanzler, C. Schmeiser, V. Tora, Two kinetic models for non-instantaneous binary alignment collisions

The seminar will be held over Zoom. The details of the meeting are below Join Zoom Meeting https://durhamuniversity.zoom.us/j/96177289696?pwd=blE1UDRhSDRucUFpZXl6ME1ydXFKdz09 Meeting ID: 961 7728 9696 Passcode: 462324

Lagrangian averaging plays an important role in the analysis of wave--mean-flow interactions and other multiscale fluid phenomena. The numerical computation of Lagrangian means, e.g. from simulation data, is however challenging. Typical implementations require tracking a large number of particles to construct Lagrangian time series which are then averaged. This has drawbacks that include large memory demands, particle clustering and complications of parallelisation. We develop a novel approach in which the Lagrangian means of various fields (including particle positions) are computed by solving PDEs that are integrated over successive averaging time intervals. We propose two strategies, distinguished by their spatial independent variables, which lead to two sets of PDEs. The first uses end-of-interval particle positions; the second directly uses the Lagrangian mean positions. The PDEs can be discretised in a variety of ways, e.g. using the same discretisation as that employed for the governing dynamical equations, and solved on-the-fly to minimise the memory footprint.

In industry, heterogeneous catalysts are widely used to enable faster large-scale production by increasing the rates of certain chemical reactions. We consider a system of parabolic PDEs in moving domains modeling mass transfer in heterogeneous catalysis with a Robin boundary condition on the interface. The behavior of such systems becomes increasingly complex as the number of catalyst particles increases. This motivates the search for a homogenized model that would describe the asymptotic behavior of the solutions to the problem and emergent properties in the limit of infinitely many particles. We transform the moving domain problem into a problem in a fixed domain by constructing a diffeomorphism out of the known solid particle velocities. We prove that solutions exist in any finite time and show that these solutions two-scale converge to solutions of a PDE/ODE system. We provide examples of solid velocities for which our result applies and discuss future research directions.\\ The seminar will be held over Zoom. The details of the meeting are below Join Zoom Meeting https://durhamuniversity.zoom.us/j/96177289696?pwd=blE1UDRhSDRucUFpZXl6ME1ydXFKdz09 Meeting ID: 961 7728 9696 Passcode: 462324

I will present an application of optimal transport theory to simplified models of large-scale rotational flows (weather). The semi-geostrophic equation is used by researchers at the Met Office to diagnose problems in simulations of more complicated weather models. It has also attracted a lot of attention in the applied analysis community, e.g., Alessio Figalli's work on the semi-geostrophic equation is listed in his Fields Medal citation. In this talk I will discuss the semi-geostrophic equation in geostrophic coordinates (SG), which is a nonlocal transport equation, where the transport velocity is defined via an optimal transport problem. Using recent results from semi-discrete optimal transport theory, we give a new proof of the existence of weak solutions of the SG equations. The proof is constructive and leads to an efficient numerical method. I will conclude talk by showing some simulations of weather fronts. This is joint work with Charlie Egan, Théo Lavier and Beatrice Pelloni (Heriot-Watt University and the Maxwell Institute for Mathematical Sciences), Mark Wilkinson (Nottingham Trent University), Steven Roper (University of Glasgow), Colin Cotter (Imperial College London) and Mike Cullen (Met Office - retired).

In this talk, we will present two kinetic models that are used to describe the evolution of charged particles in plasmas: the Vlasov-Poisson system and the Vlasov-Poisson system with massless electrons. These systems model respectively the evolution of electrons, and ions in a plasma. We will discuss the well-posedness of these systems, the stability of solutions, and their behavior under singular limits. Finally, we will introduce a new class of Wasserstein-type distances specifically designed to tackle stability questions for kinetic equations.

The seminar will be held over Zoom. The details of the meeting are below Join Zoom Meeting https://durhamuniversity.zoom.us/j/96177289696?pwd=blE1UDRhSDRucUFpZXl6ME1ydXFKdz09 Meeting ID: 961 7728 9696 Passcode: 462324

Aggregation effects appear in many applications such as in the study of flocking behaviour of swarms or in the modelling of granula media, which makes it interesting to study this phenomena also in mean-field settings. In 2018, Chen, G\"ottlich and Knapp already showed that the diffusion-aggregation model $$\partial_t u - \sigma \Delta u + \Delta u^2 = 0$$ can be derived from a system of interacting particles by using a classical mean-field limit approach. Their result shows convergence in expectation for each particle of the moderately interacting mean-field system to a limiting particle equation connected to the diffusion-aggregation system. Hence, the title of this talk could have also been \textit{Why is it sometimes useful to reprove already existing convergence results in different norms?}, but this was obviously too long. However, in this seminar talk I will answer exactly this question and explain the motivation why we reproved the mean-field limit of Chen, G\"ottlich and Knapp in $L^2$-norm and what benefits we got from this result. For a better understanding of the differences between the two convergence results, the talk will also include a short introduction to moderately interacting particle systems in mean-field regimes. This is joint work with Li Chen and Ansgar J\"ungel.

The seminar will be held over Zoom. The details of the meeting are below

Join Zoom Meeting https://durhamuniversity.zoom.us/j/96177289696?pwd=blE1UDRhSDRucUFpZXl6ME1ydXFKdz09

Meeting ID: 961 7728 9696 Passcode: 462324

A classical problem in the theory of shape optimization is to find a shape with minimal (linear) elastic compliance for a given amount of mass and prescribed external forces. It is an intriguing question with a long history, going back to Michell's seminal 1904 work on trusses, to determine what happens in the limit of vanishing mass. Contrary to all previous approaches, which utilize a soft mass constraint by introducing a Lagrange multiplier, we here consider the hard mass constraint. Our results establish the convergence of approximately optimal shapes of (exact) size tending to zero, to a limit generalized shape represented by a (possibly diffuse) probability measure. This limit generalized shape is a minimizer of the limit compliance, which involves a new integrand, namely the one conjectured by Bouchitte in 2001 and predicted heuristically before in works of Allaire-Kohn (80's) and Kohn-Strang (90's). This integrand gives the energy of the limit generalized shape understood as a fine oscillation of (optimal) lower-dimensional structures. Its appearance is surprising since the integrand in the original compliance is just a quadratic form and the non-convexity of the problem is not immediately obvious. I will also present connections to the theory of Michell trusses and show how our results can be interpreted as a rigorous justification of that theory on the level of functionals in both two and three dimensions. This is joint work with J.F. Babadjian (Paris-Saclay) and F. Iurlano (Paris-Sorbonne).

Motivated by some physical and biological models, in this talk we consider a class of degenerate parabolic equations. Our analysis is based on gradient flows in the space of probability measures equipped with the distance arising in the Monge-Kantorovich optimal transport problem. The associated internal energy functionals in general fail to be differentiable, therefore classical results do not apply in our setting. We will study the combination of both linear and porous medium type diffusions and we show the existence and uniqueness of the solutions in the sense of distributions in suitable Sobolev spaces. Our notion of solution allows us to give a fine characterization of the emerging ‘critical regions’, observed previously in numerical experiments. A link to a three phase free boundary problem will also be pointed out. It is possible to consider singular limits of our PDEs in a suitable way, to recover further degenerate models from the literature. These results have been obtained in collaboration with Dohyun Kwon (Madison).

This talk is based on a joint work with Angeliki Menegaki (IHES). I will talk about kinetic equations posed on bounded domains where the boundary conditions encode heat entering the system from the walls. We expect these open systems have non-equilibrium steady states (non Gibb's states). I will discuss our paper showing existence and some properties in the case of the BGK equation defined on an interval with a hot and cold wall. I will also discuss the challenges and prospects for analysis of these steady states.

Morrey's inequality quantifies the continuity of functions whose derivatives have high enough integrability. We study the functions for which Morrey's inequality is saturated. We will explain how various qualitative properties of these extremal functions can be deduced from the partial differential equation they satisfy.

We consider the behaviour of a passive tracer (scalar) \(\theta(x,t)\) evolving under \[\partial_t\theta + u\cdot\nabla\theta = \Delta\theta + g\] where \(u(x,t)\) is a random velocity field having the hypothesized Kolmogorov-Obukhov (in 3d) or Kraichnan-Batchelor (in 2d) energy spectrum. There are several regimes, depending on some norms of \(u\) and its high-wavenumber cutoff. We will review several recent results on this problem.

In this second talk, we will consider the questions that we have raised at the end of the previous one, present some recent results as well as the general ideas that structure the proofs.

This will naturally lead us to speak about how to assoiate a manifold to a graph with boundary, and to discuss a process called discretization, which allows to associate a graph with boundary to a manifold.

Moreover, we will state some propositions that spectrally relate these graphs and maifolds, for these links are in the heart of the proofs.

Fix the temperature to be one inside a bounded domain $D \subset \mathbb{R}^n$ and zero outside of $D$ at time $t=0$. For $t>0$ small, one would expect that the amount of heat that remains in $D$, called the heat content, should depend upon the geometry of $D$. Motivated by this idea, we shall explore the case where $D \subset \mathbb{R}^2$ is a bounded polygonal domain contained inside another polygonal domain $\Omega \subset \mathbb{R}^2$ and study the heat content of $D$ when Neumann boundary conditions are imposed on $\partial \Omega$. We will discuss how to obtain an explicit small-time asymptotic formula for the heat content of $D$ in this case which explicitly depend on the geometry of $D$ relative to $\Omega$. This talk is based on joint work with Katie Gittins.

A Mean Field Game is a differential game (in the sense of game theory) where instead of a finite number of players we have a continuous distribution of (infinitely) many players, however we make the simplifying assumption that all players are identical.

In this talk we consider the existence and uniqueness of Nash Equilibrium in Mean Field Games. We show why the study of Nash Equilibrium naturally leads to the study of a Hamilton-Jacobi equation over the space of measures called the master equation, whose solutions give rise to Nash Equilibrium for our game.

We will see that there are two natural types of noise that one can impose in a Mean Field Game, individual noise and common noise, which correspond to cases where the noise of each player is independent and identical respectively. Individual noise has a regularizing effect that is utilized in most well-posedness results for the master equation.

We explore well-posedness for the master equation in the case without individual noise, under a monotonicity condition.

The goal of this presentation is to get familiar with the discrete Steklov problem. We will recall what is the Steklov problem on a smooth compact Riemannian manifold with boundary, before presenting the concept of graphs with boundary and defining the Steklov problem in this discrete case. We will then introduce the special kind of graphs with boundary that we will study, which are what we call subgraphs of infinite Cayley graphs, by giving definitions and examples. We will then give some recent results about them, which naturally give rise to interesting questions, that we will formulate.

The large deviations of interacting diffusions have been a field of interest since the seminal work of D. A. Dawson and J. Gärtner in 1987. The large deviation principle not only provides the almost sure convergence of these interacting diffusions to their corresponding mean-field limits—so-called McKean-Vlasov equations—but also provides a variational formulation for the limiting distribution. In this talk, I will give a brief introduction to the basic concepts of large-deviation theory, particularly Sanov’s theorem and Varadhan’s integral lemma. I will then discuss how these concepts can be used to prove large-deviations principles for interacting diffusions, mentioning the technical difficulties in applying standard theory and illuminating the need for extensions of the standard theory. The last part of the talk will focus on insights into an extension that is applicable for interacting diffusions with singular interacting kernels.

In mean field game theory, the Master Equation is a PDE whose classical solutions may tell us all we need to know about the Nash equilibrium among large numbers of agents. A challenging aspect of the equation is that one of its independent variables is a measure (the mean field), which is infinite dimensional. Since the pioneering work of Cardaliaguet, Delarue, Lasry, and Lions (2019), there have been several results on the well-posedness of the Master Equation, but lots of work remains to be done. In this talk I will focus on a recent contribution (joint work with Ronnie Sircar) in which we prove the existence of classical solutions to the Master Equation for a mean field game of controls with Dirichlet boundary conditions on a half-line, which is a model for exhaustible resource production.

Variational problems with a convexity constraint arise in different scientific disciplines such as Newton's problem of minimal resistance in Physics, the Rochet-Choné model of monopolist's problem in Economics, and wrinkling patterns in floating elastic shells in Elasticity. The convexity constraint renders serious challenges in numerically computing minimizers of these variational problems, and calls for robust approximation schemes. In this talk, we will discuss how these minimizers can be approximated by solutions of singular, fourth order Abreu equations that arise in extremal metrics in complex geometry. Our tools involve sharp regularity estimates for the linearized Monge-Ampere equations in divergence form.

We review Turing-type instabilities and their analysis via classical linear instability analysis. From here we discuss several cases where this approach must be modified to account for complex properties of the medium. Firstly we describe the case of a heterogeneous domain, where a pre-pattern or other spatial heterogeneity influences the reaction-diffusion dynamics. Under the approximation of a sufficiently smooth heterogeneity relative to small diffusion coefficients, we can use WKB asymptotics to show a localisation of the classical Turing conditions. In the opposite regime, when the medium is composed of two distinct stratified layers, we can also make some progress under suitable assumptions. In both scenarios we find much richer dispersion relations, as well as numerous open questions suitable for a range of mathematical techniques and new ideas. Throughout we relate our modelling and analysis back to biological questions of form and function, and suggest further areas of exploration not present in even these more complicated models.

A natural classification of random walks is the one into recurrent and transient ones. This is equivalent to the non-/validity of the Hardy inequality for the energy functional associated with the Laplace operators on the graph. The latter is an abstract inequality between functionals and can be generalised further. The corresponding theory is known as criticality theory.

In this talk, we introduce quasi-linear Schrödinger operators on graphs and show many equivalent statements of a Hardy inequality to hold true. If the time permits, we also discuss the optimality of the corresponding Hardy weight.

The talk is based on work in progress.

In this seminar we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporates the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.

We will consider a non-convex model of single-crystal elasto-plasticity, where the non-convexity arises through the imposition of a hard "single-plane" side condition on the plastic deformation. This well-posed variational model arises as the relaxation of a more fundamental "single-slip" model in which at most one slip system can be activated at each spatial point. The relaxation procedure is motivated by the desire for efficient, oscillation-free simulation of single-crystal plasticity, but it is not immediately obvious how to implement the strict side-condition (that at most one slip-plane may be activated at each point) numerically. Our approach to this problem is to regularize the side-condition by introducing a large, but finite, cross-hardening penalty into the plastic energy. The regularized model is then amenable to implementation with standard finite-element methods, and, with the aid of div-curl arguments, one can show that it Gamma-converges to the single-plane model for large penalization.

First, we discuss optimal spectral enclosures for discrete Laplacians on the line and the half-line perturbed by complex summable potentials. Second, we present related results on a spectral stability of discrete Schrödinger operators on the half-line with small complex potentials. The talk is based on joint projects with O. O. Ibrogimov, D. Krejčiřı́k, and A. Laptev.

Bacterial chemotaxis describes the ability of single-cell organisms to respond to chemical signals. In the case where the bacterial response to these chemical signals is sharp, the corresponding chemotaxis model for bacterial self-organization exhibits a discontinuous advection speed. This is a key challenge for analysis. We propose a new approach to circumvent the discontinuity issue following a perturbativ approach, where the shape of the cellular profile is clearly separated from its global motion. As a result, we obtain exponential relaxation to equilibrium with an explicit rate. This is joint work with Vincent Calvez (Université Claude Bernard Lyon 1, France).

In this talk we analyse the long-time behaviour of solutions to prototypical transport-relaxation systems of BGK-type. We present a hypocoercivity method that modifies the standard norm based on Lyapunov matrix inequalities. The method provides optimal decay rates for constant relaxation rates, where the equation can be Fourier decomposed into finite ODEs. Expressing the Lyapunov functional through pseudo-differential operators allows us to go beyond the ODE approach and prove explicit decay rates for spatially non-homogeneous relaxation. We further discuss how the two-velocity example connects the presented method to other hypocoercivity methods. This is joint work with Anton Arnold, Amit Einav and Beatrice Signorello.

We are concerned with finding Fokker-Planck equations in whole space with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrarily close to its infimum. This infimum is 1, corresponding to the high-rotational limit in the Fokker-Planck drift.

Such an ìoptimalî Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. The proof is based on the recent result that the $L^2$-propagator norms of the Fokker-Planck equation and of its drift-ODE coincide for all time.

Finally we give an outlook onto using Fokker-Planck equations with t-dependent coefficients.

This talk is based on joint work with Beatrice Signorello.

References:

* A. Arnold, B. Signorello: Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium, preprint 2021.

* A. Arnold, C. Schmeiser, B. Signorello. Sharp decay estimates and $L^2$-propagator norm for Fokker-Planck equations with linear drift, to appear in Comm. Math. Sci., 2022.

In this talk we develop the theory of a metric, which we call the $\nu$-based Wasserstein metric and denote by $W_\nu$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure $\nu$ and is relevant in particular for the case when $\nu$ is singular with respect to $m$-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $\nu$-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $\nu$; we also characterize it in terms of integrations of classical Wasserstein distance between the conditional probabilities when measures are disintegrated with respect to optimal transport to $\nu$, and through limits of certain multi-marginal optimal transport problems. As we vary the base measure $\nu$, the $\nu$-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance (obtained when $\nu$ is a Dirac mass) and a metric associated with the uniquely defined generalized geodesics obtained when $\nu$ is sufficiently regular (eg, absolutely continuous with respect to Lebesgue). When $\nu$ concentrates on a lower dimensional submanifold of $\mathbb{R}^m$, we prove that the variational problem in the definition of the $\nu$-based Wasserstein distance has a unique solution. We establish geodesic convexity of the usual class of functionals, and of the set of source measures $\mu$ such that optimal transport between $\mu$ and $\nu$ satisfies a strengthening of the generalized nestedness condition introduced in McCann&Pass 2020. If time permitted we also present an applications of the ideas introduced: we also use the multi-marginal formulation to characterize solutions to the multi-marginal problem by an ordinary differential equation, yielding a new numerical method for it.

We use entropy methods to show that the heat equation with Dirichlet boundary conditions on the complement of a compact set in R^d shows a self-similar behaviour much like the usual heat equation on R^d, once we account for the loss of mass due to the boundary. Giving good lower bounds for the fundamental solution on these sets is surprisingly a relatively recent result, and we find some improvements using some advances in logarithmic Sobolev inequalities.

By using optimal mass transport theory we prove a sharp isoperimetric inequality in CD(0,N) metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature. The equality cases are also discussed, establishing various rigidities. Talk based on a joint work with Z. Balogh (Universitat Bern).

We will consider a non-convex model of single-crystal elasto-plasticity, where the non-convexity arises through the imposition of a hard "single-plane" side condition on the plastic deformation. This well-posed variational model arises as the relaxation of a more fundamental "single-slip" model in which at most one slip system can be activated at each spatial point. The relaxation procedure is motivated by the desire for efficient, oscillation-free simulation of single-crystal plasticity, but it is not immediately obvious how to implement the strict side-condition (that at most one slip-plane may be activated at each point) numerically. Our approach to this problem is to regularize the side-condition by introducing a large, but finite, cross-hardening penalty into the plastic energy. The regularized model is then amenable to implementation with standard finite-element methods, and, with the aid of div-curl arguments, one can show that it Gamma-converges to the single-plane model for large penalization.

Everybody enjoys the famous properties of the heat equation like the maximum principle. Except systems of parabolic equations such as systems of reaction-diffusion equation. We present some recent progresses on the existence of classical/weak/renormalised global in time solutions as well as general results on the convergence to a chemical equilibrium state.

Collective dynamics has stimulated intense mathematical research in the last decade. Many different models have been proposed but most of them rely on describing agents as point particles in position-velocity space. We propose a model where the particles carry more complex geometric structure. Specifically, the particles are rigid bodies whose attitude (or body orientation) is described by an orthonormal frame. Particles tend to align their frame with those of their neighbours. In this talk we will review recent results on this model which are issued from collaborations with Antoine Diez, Amic Frouvelle, Sara Merino-Aceituno, Mingye Na and Ariane Trescases.

Reaction-Diffusion systems appear naturally in many biological and chemical phenomena with underlying chemical reactions. One extremely successful method to investigate such systems, and in particular their long time behaviour, is to explore natural functionals of these systems usually known as entropies by means of the so-called Entropy method. Such an approach is commonly used to investigate reversible reaction-diffusion systems where one is guaranteed a strictly positive equilibrium state. It can’t be directly applied, however, in situations where some of the substances that are involved in the reactions get destroyed over time - a common feature in irreversible systems. In our talk, based on recent work with Marcel Braukhoff and Bao Quoc Tang, we will present a new approach to circumvent the problems that arise when dealing with decaying substances by considering "cut off" entropies which, when combined with a decreasing mass term, give a new entropy-like functional for which we can apply the ideas of the Entropy method. We will use this approach in the setting of a well known irreversible enzyme system to not find explicit rates of convergence of the aforementioned entropy, but also for the $L^\infty$ norm of the associated concentrations.

NB: This seminar will take place in person in MCS3070 (please ignore zoom link).

Given a complete non-compact Riemannian manifold (M, g) with certain curvature restrictions, in this talk we introduce an expansion condition concerning a group of isometries G of (M, g) that characterizes the coerciveness of G in the sense of Skrzypczak and Tintarev (Arch Math 101(3): 259–268, 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey).

We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments.

Nonlocal energies are continuum models for large systems of particles with long-range interactions. Under the assumption that the interaction potential is radially symmetric, several authors have investigated qualitative properties of energy minimisers. But what can be said in the case of anisotropic kernels? Motivated by the example of dislocation interactions in materials science, we pushed the methods developed for nonlocal energies beyond the case of radially symmetric potentials, and discovered surprising connections with random matrices, fluid dynamics, and Calderón-Zygmund operators.

In this talk, we will consider scalar first-order minimization problems $\int f(x_0,u,\nabla u)$ with mass constraint $\int u = m$. We will show that suitable rescalings of such functionals $\Gamma$-converge towards a concave $H$-mass functional over the space of positive measures, under mild assumptions on the Lagrangian. From this result we may recover the concentration phenomenon of Cahn-Hilliard fluids into droplets, and obtain variational approximations of general branched transport models by elliptic energies. This is joint work with Antonin Monteil.

Eigenvalue problems are often used to model the stationary states of physical systems. For instance, eigenfunctions of the Laplacian in a planar domain, with a Dirichlet boundary condition, describe the small oscillations of a membrane whose boundary is fixed: the membrane remains motionless where the eigenfunction vanishes. A natural question has emerged while studying this topic: in how many subdomains does the zero-set of an eigenfunction divide the original domain?

I will recall some tools used to tackle this problem and some classical results, focusing on the asymptotic upper bound discovered by Pleijel in 1956. I will then describe the extension of Pleijel's theorem to a large class of Schrödinger operators. This talk is based on joint work with Philippe Charron.

In this talk, we will consider scalar first-order minimization problems $\int f(x_0,u,\nabla u)$ with mass constraint $\int u = m$. We will show that suitable rescalings of such functionals $\Gamma$-converge towards a concave $H$-mass functional over the space of positive measures, under mild assumptions on the Lagrangian. From this result we may recover the concentration phenomenon of Cahn-Hilliard fluids into droplets, and obtain variational approximations of general branched transport models by elliptic energies. This is joint work with Antonin Monteil.

We consider scalar transport equations involving nonlocal interaction terms and different kinds of mobility and we present how to obtain weak solutions (in some regimes even entropy solutions) as many particle limit of a suitable nonlocal version of the deterministic follow-the-leader scheme, which can be interpreted as the discrete Lagrangian approximation of the target pde. We discuss both the cases of linear and nonlinear mobilities as well as how the evolution is affected when a diffusive term is taken into account. The content of this talk is based on several works obtained in collaborations with S. Daneri, M. Di Francesco, S. Fagioli, E. Runa and F. Stra.

Casimir interactions are forces between objects such as perfect conductors. There are three mainstreams of calculating Casimir forces in the physics literature. The first one, due to Casimir, is known as the spectral zeta regularisation of the vacuum energy. The second one is calculated via a surface integral of the renormalised stress energy tensor of the electromagnetic field. The third one is using a Fredholm determinant constructed from layer potential operators. We prove that they are equivalent in our recent work. In this talk, we will briefly introduce these three methods and discuss some main tools we used to prove the equivalence of the methods. This is a joint work with Alex Strohmaier.

We study long time dynamics of combustive processes in random media, modeled by reaction-diffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large space-time scale dynamics of solutions to these equations is almost surely governed by a homogeneous Hamilton-Jacobi equation. While this was previously shown in one dimension as well as for radially symmetric reactions in several dimensions, we prove this phenomenon in the general non-isotropic multidimensional setting. We also prove the rate of convergence when the initial data is close to an indicator function of a convex set.

This talk is based on the joint work with Andrej Zlatos.

We study the asymptotical dynamics of the lake equations in case of vanishing or emerging of an island. We derive an asymptotic lake-type equation for both scenarios. In the first case, the asymptotic dynamics displays an additional Dirac mass in the vorticity. The main mathematical difficulty is that the equations are singular when the water depth vanishes. We provide new uniform estimates in weighted spaces for the related stream functions in order to obtain a compactness result.

This is joint work with Lars Eric Hientzsch and Christophe Lacave (Université Grenoble Alpes).

The theory of Mean Field Games is devoted to thestudy of Nash equilibria in (differential) games involving a population of infinitely many identical players. In the PDE framework, a MFG system consists of coupled Hamilton-Jacobi and Fokker-Planck equations, which characterize equilibria of the population. I will discuss in the talk some results on the existence and non-existence of these equilibria, in a setting where players are strongly encouraged to aggregate. The results have been obtained in collaboration with D. Ghilli (LUISS).

In this talk, we aim to present recent results aboutthe rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. The hydrodynamic system that we obtain is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms and, to our knowledge, it is the first hydrodynamic system that properly describes rapid granular flows consistent with the kinetic formulation in physical dimension d=3. For that purpose, one of the main mathematical difficulty is to understand the relation between the restitution coefficient, which quantifies the energy loss at the microscopic level, and the Knudsen number. This is achieved by identifying the correct nearly elastic regime to capture nontrivial hydrodynamic behavior. The talk is based on a joint work with Ricardo Alonso (Texas A&M University at Qatar) and Isabelle Tristani (ENS Paris,Université PSL).

The Poisson summation formula tells us that the Dirac comb (the sum of delta-functions on integer points) is invariant the Fourier transform, which rotates by pi/2 in phase space. We discuss what results when rotating by other angles, as an application of classical tools for affine transformations on phase space (known as the Heisenberg / metaplectic / Jacobi groups).

The study of p-harmonic functions on Riemannian manifolds has invoked the interest of mathematicians and physicists for nearly two centuries. Applications within physics can for example be found in continuum mechanics, elasticity theory, as well as two-dimensional hydrodynamics problems involving Stokes flows of incompressible Newtonianfluids.

In my talk I will focus on the construction of explicit p-harmonic functions on rank-one Lie groups of Iwasawa type. This is joint work with Sigmundur Gudmundsson and Marko Sobak.

We discuss the rigidity of measurable subsets in the Euclidean space such that the Lebesgue measure of their intersection with a ball of radius r, centred at any point in the essential boundary, is constant. Based on a joint work with Dorin Bucur.

Phase transitions are present in a wide array of systems ranging from traffic to machine learning algorithms. In this talk, we will relate the concept of phase transitions to the convexity properties of the associated thermodynamic energy. Motivated by noisy stochastic gradient descent in supervised learning, we will consider the problem of understanding the thermodynamic limit of exchangeable weakly interacting diffusions (AKA propagation of chaos) from an energetic perspective. The strategy will be to exploit the 2-Wasserstein gradient flow structure associated with the thermodynamic energy in the infinite particle setting. Using this perspective, we will show how phase transitions affect the homogenization limit or the stability of the log-Sobolev inequality.

I will give an introduction to the phenomenon of “revivals”, or“dispersive quantisation”. Although first reported experimentally in 1835 by Talbot, this phenomenon was only studied in the ’90’s. In particular, in the context of PDEs, it was studied for the periodic free space Schroedinger equation by Berry and al, and it was then rediscovered for the Airy equation by Peter Olver in 2010. Since then, a sizeable literature has examined revivals for the periodic problem for linear dispersive equations with polynomial dispersion relation. What I will discuss in this talk is further occurrences of this phenomenon for different boundary conditions, a novel form of revivals for more general dispersion relations and nonlocal equations such as the linearised Benjamin-Ono equation, and nonlinear (integrable) generalisations. This work is joint with Lyonell Boulton, George Farmakis, Peter Olver and David Smith.

The physical size of particles plays an important role in many applications in physics and the life sciences. However, the problem of including such finite size effects in mean-field models consistently is mostly open. In this talk I will discuss different microscopic models for mixtures of hard-spheres as well as their formal limiting PDEs. These continuum equations often have similar properties, such as degenerate mobilities, cross-diffusion terms or a lack of an underlying gradient flow (GF) structure. The lack of a full GF structure is often caused by the approximations made when passing to the limit. But the resulting systems often have an asymptotic gradient flow (AGF) structure, that is a gradient flow structure up to a certain order. This structure can be used to study the behavior of solutions close to equilibrium. However, different entropy-mobility pairs can be found to describe such equations (up to a certain order), some of them satisfying physical principles, others not. I will discuss the behavior of solutions to these possible entropy-mobility pairs and illustrate their difference with numerical simulations.

I will consider an optimal control problem for a large population of interacting agents with deterministic dynamics, aggregating potential and constraints on reciprocal distances, in dimension 1 and I will discuss existence and qualitative properties of periodic in time optimal trajectories, with particular interest on the compactness of the solutions' support and on the saturation of the distance constraint. Moreover, I will show the consistency of the mean-field optimal control problem with density constraints with the corresponding underlying finite agent one and deduce some qualitative results for the time periodic equilibria of the limit problem. Joint work with Marco Cirant (Padova).

In this talk, I will present stochastic material models described by strain-energy densities where the parameters are characterised by probability distributions at a continuum level. To answer important questions, such as “what is the influence of probabilistic parameters on predicted mechanical responses?” and “what are the possible equilibrium states and how does their stability depend on the material constitutive law?”, I will focus on likely instabilities in nematic liquid crystal elastomers. I will discuss the soft elasticity phenomenon where, upon stretching at constant temperature, the homogeneous state becomes unstable and alternating shear stripes develop at very low stress, and also some classical effects inherited from the underlying polymeric network, such as necking, cavitation, and shell inflation instabilities. These fundamental problems are important in their own right and may stimulate related mechanical testing of nematic materials.

We will discuss a derivation of the incompressible Euler equations from a N-particle system with repulsive Coulomb interaction.

The proof is based on the study of a suitable discrete modulated energy functional.

This is a joint work with Mikaela Iacobelli (ETH Zurich).

Given a bounded open set $\Omega$ in $ \mathbb R^2$ and a regular partition of $\Omega$ by $k$ open sets $D_j$, and assume that:

* This is an equipartition $\mathfrak l_k:= \lambda(D_j)$(for all $j$) where $\lambda(D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$

* It has the pair compatibility condition, i.e. for any pair of neighbors in the partition $D_i,D_j$, there is a linear combination of the ground states in $D_i$ and $D_j$ which is an eigenfunction of the Dirichlet problem in $\text{int}(\overline{D_i\cup D_j})$.

Typical examples are nodal partitions and spectral minimal partitions. The aim is to extend the indices and Dirichlet-to-Neumann like operators introduced by Berkolaiko--Cox--Marzuola in the nodal case to this more general situation. Like in the analysis of minimal partitions, this will involve in particular the analysis of suitable Aharonov-Bohm operators.

This talk is a small journey through sub-Riemannian geometries. We will focus on the α-Grushin plane to illustrate key features of the theory. Geodesics can be obtained from the Hamiltonian point of view: they can be expressed with special trigonometric functions. Finally, we will investigate different notions of curvature and see how well they are fitted for these types of singular spaces.

The talk will be about compressible and incompressible nonlinear elasticity variational problems. Our contribution is to provide a convex relaxation for a class of non convex problem, together with sufficient conditions guaranteeing its tightness. Our relaxation is based on a notion of Dirichlet energy for measure valued mappings which is interesting in itself, and the proof of tightness relies on convex analysis and the study of a dual problem.

This is joint work with Nassif Ghoussoub, Young-Heon Kim and Aaron Palmer (UBC):https://arxiv.org/abs/2004.10287

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In this talk we consider a class of finite horizon first order mean field games. The main focus here will be to provide a simple proof of convergence of symmetric N-player Nash equilibria in distributed open-loop strategies to solutions of the mean field game in Lagrangian form. The talk is based on a joint work with Markus Fischer (University of Padua).

The theory of metric measure spaces verifying the Riemannian-Curvature-Dimension condition RCD(K,N) has attracted a lot of interest in the last years.

They can be thought as a non smooth counterpart of the class of Riemannian manifolds with Ricci curvature bounded from below by K and dimension bounded from above by N.

So far we have reached a good understanding of their structure up to negligible sets and it seems natural to push the study further, up to codimension one.

In this talk I will outline some recent developments about the structure of boundaries of sets of finite perimeter obtained in joint works with Ambrosio, Brue' and Pasqualetto, where we extended De Giorgi's celebrated theorem to this framework. The results are expected to be useful to improve our knowledge on the fine structure of these spaces and on their global shape.

In this talk I will consider the spectral resolution of the Laplacian operator on a manifold and discuss the question of how spectral projectors can concentrate on a given subset of the manifold. In particular we will consider two cases : compact manifolds with or without boundary in which the purely discrete spectrum leads to finite combinations of eigenfunctions and the unbounded case without boundaries, in which the spectrum contains a continous part. In both cases we give quantitative estimates for the localisation of the spectral projection in terms of the highest frequnecy involved, which are essentially optimal. We also try to refine the uncertainty principle in this situation so as to consider the smallest possible localisation.

This is based on joint work with G. Lebeau (Nice) and N. Burq (Orsay).

It has been discovered that the lower bound of Ricci curvature $\kappa$ of a Riemannian manifold can be characterized by the displacement $\kappa$-convexity (in the Optimal transport sense) of the Boltzmann-Shannon entropy. Via this characterization, Sturm ('˜06) and Lott-Villani ('˜09) defined the well-known notion of 'Ricci curvature' for a more general class of metric measure spaces. Inspired by the previous work, Erbar and Maas ('˜11) gave the modified definition of this Ricci curvature for discrete Markov chains, and they also described this curvature in terms of Bochner's inequality and gradient estimate with respect to the heat semigroup (in the spirit of Bakry-Émery).

After discussing the history of this entropic Ricci curvature, I will briefly talk about my work on how to apply the gradient estimate to obtain an upper bound of the diameter of the underlying graph of the Markov chains with positive Ricci curvature.

The mean field opinion model treats the evolution of opinions as Markov Chain with a mean field interaction. The models exhibits a wide array of behaviours depending on the time and space scales. The purpose of this talk is to introduce the model and explain its properties. This is a joint work with Inês Armendáriz, Monia Capanna and Pablo Ferrari.

Let $\Omega$ be a planar, bounded, connected, open set with Lipschitz boundary. Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either a Dirichlet, Neumann or Robin boundary condition. We are interested in the number of nodal domains of $u$.

If an eigenfunction $u$ associated with the $k$--th eigenvalue has exactly $k$ nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we call the corresponding eigenvalue Courant-sharp.

The Courant-sharp Dirichlet, respectively Neumann, eigenvalues of the square are known due to Pleijel, B\'erard--Helffer, respectively Helffer--Persson-Sundqvist.

We discuss whether the Robin eigenvalues of the square are Courant-sharp.

This is based on joint work with B. Helffer (Université de Nantes).

The Fokker-Planck equation plays an important role when one considers problems that involve white noise. As such, it has a long and illustrious history with many applications in statistical physics, plasma physics, stochastic analysis and mathematical finances. Recent studies have focused on the case where the diffusive part of the equation is degenerate, an issue that can impact the long time behaviour of the solution to this equation. This difficulty, however, can be corrected by the drift mechanism in the system, as long as it manages to 'mix' the diffusive and non-diffusive directions. When this happens, a simple and natural equilibrium emerges. The strong connection between the Fokker-Planck equation and the world of statistical physics also fosters the tool to investigate the aforementioned convergence: The notion of an entropy for the system. One common methodology to investigate the long time behaviour under an entropy is the so-called entropy method, where one searches for a geometric functional inequality between the entropy and its formal production under the evolution flow. This methodology, however, is problematic when degeneracy appears. In the case where the diffusion and drift parts of the equation are constant, Anton and Erb have introduced in 2014 a 'modified' production functional (motivated by notions of hypocoercivity), which have managed to yield the sharp convergence rate for a large family of entropies, when the diffusion matrix is degenerate or not - as long as the drift matrix wasn't defective. The problems the defectiveness of the drift bring become apparent when one examines the standard technique to obtain the desired inequality of the entropy method, the so-called Bakry-Èmry method. In this technique one uses the entropy method again - but on the production functional. In this talk we shall consider a different approach to the problem, which yields the sharp rate of convergence to equilibrium for a family of natural entropies to the system. We will circumvent the entropy method by carefully exploring the spectral properties of the Fokker-Planck operator in an appropriate Hilbert setting, resulting in the desired convergence for one particular entropy, which will then be cascaded to all other entropies by the use a newly found non-symmetric hypercontractivity result.

This talk is based on a joint work with Anton Arnold and Tobias Wöhrer.

this talk will not be taking place on this date as the speaker is unable to travel. new date tba.

It is a classical problem of inverse spectral geometry to find geometric quantities that can be determined from the spectrum of an elliptic differential operator. For example, it follows from the work of Girouard, Parnovski, Polterovich and Sher that spectral asymptotics for the Dirichlet-to-Neumann map of the Laplacian determine the number and lengths of the boundary components of a surface, but not any more. In this talk, I will explain how we can recover more geometric information from a surface if we consider instead the Dirichlet-to-Neumann maps associated with Schrödinger operators. I will also explain how this has application to the inverse scattering problem, and therefore to non-destructive testing. This is joint work with Simon St-Amant (Université de Montréal).

NOTE: exceptionally, this talk will take place in CG93.

We provide examples of initial data which saturate the enhanced diffusion rates proved for general shear flows which are Hölder regular or Lipschitz continuous with critical points, and for regular circular flows, establishing the sharpness of those results. The proof makes use of a probabilistic interpretation of the dissipation of solutions to advection diffusion equations.

It has been known for several decades that, despite being a PDE, the 2d Navier-Stokes equations are effectively governed by a finite number of degrees of freedom. We will discuss how these are related to the complexity of the flow and present some results obtained by former Durham PhD students.

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We will look at the classical problem of minimizing the Dirichlet energy of a map $u :\Omega\subset\mathbb{R}^2\to N$ valued into a compact Riemannian manifold $N$ and subjected to a Dirichlet boundary condition $u=\gamma$ on $\partial\Omega$. It is well known that if $\gamma$ has a non-trivial homotopy class in $N$, then there are no maps in the critical Sobolev space $H^1(\Omega,N)$ such that $u=\gamma$ on $\partial\Omega$. To overcome this obstruction, a way is to rather consider a relaxed version of the Dirichlet energy leading to singular harmonic maps with a finite number of topological singularities in $\Omega$. This was done in the 90's in a pioneering work by Bethuel-Brezis-Helein in the case $N=\mathbb{S}^1$, related to the Ginzburg-Landau theory. In general, we will see that minimizing the energy leads at main order to a non-trivial combinatorial problem which consists in finding the energetically best topological decomposition of the boundary map $\gamma$ into minimizing geodesics in $N$. Moreover, we will introduce a renormalized energy whose minimizers correspond to the optimal positions of the singularities in $\Omega$.

We consider Schroedinger operators on the whole Euclidean space or on the half-space, subject to real Robin boundary conditions. I will present the construction of a non-real potential that decays at infinity so that the corresponding Schroedinger operator has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum. This proves that the Lieb-Thirring inequalities, crucial in quantum mechanics for the proof of stability of matter, do no longer hold in the non-selfadjoint case.

We extend a result of Davies and Nath [1] on the location of eigenvalues of Schrödinger operators $-\Delta+V$ with slowly decaying complex-valued potentials to higher dimensions. We also discuss examples related to the Laptev--Safronov conjecture [2], which stipulates that the absolute value of any complex eigenvalue can be bounded in terms of the $L^q$ norm of $V$, for a certain range of exponents $q$. The talk is based on [3].

[1] Davies, E. B. and Nath, J. Schrödinger operators with slowly decaying potentials J. Comput. Appl. Math., 2002, 148, 1-28

[2] Laptev, A. and Safronov, O. Eigenvalue estimates for Schrödinger operators with complex potentials Comm. Math. Phys., 2009, 292, 29-54

[3] Cuenin, J.-C. Improved eigenvalue bounds for Schr\"odinger operators with slowly decaying potentials arXiv e-prints, 2019, arXiv:1904.03954

We present a probabilistic analysis of the long-time behaviour of the nonlocal, diffusive equa- tions with a gradient flow structure in 2-Wasserstein metric, namely, the Mean-Field Langevin Dynamics (MFLD). Our work is motivated by a desire to provide a theoretical underpinning for the convergence of stochastic gradient type algorithms widely used for non-convex learning tasks such as training of deep neural networks. The key insight is that the certain class of the finite dimensional non-convex problems becomes convex when lifted to infinite dimensional space of measures. We leverage this observation and show that the corresponding energy functional defined on the space of probability measures has a unique minimiser which can be characterised by a first order condition using the notion of linear functional de- rivative. Next, we show that the flow of marginal laws induced by the MFLD converges to the stationary distribution which is exactly the minimiser of the energy functional. We show that this convergence is exponential under conditions that are satisfied for highly regularised learning tasks. At the heart of our analysis is a pathwise perspective on Otto calculus used in gradient flow literature which is of independent interest. Our proof of convergence to stationary probability measure is novel and it relies on a generalisation of LaSalle's invariance principle. Importantly we do not assume that interaction potential of MFLD is of convolution type nor that has any particular symmetric structure. This is critical for applications. Finally, we show that the error between finite dimensional optimisation problem and its infinite dimensional limit is of order one over the number of parameters.

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoglu-Otto and Laux-Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space and in particular we construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedoglu-Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. The talk is based on a recent joint work with Matt Jacobs (UCLA) and Inwon Kim (UCLA).

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoglu-Otto and Laux-Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space and in particular we construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedoglu-Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. The talk is based on a recent joint work with Matt Jacobs (UCLA) and Inwon Kim (UCLA).

I will present the results from my recent paper* (with Jan Sieber) and show intuitively how forest dynamics and bistability at the landscape scale emerge from the microscale reaction rules. There will be plenty of cellular automaton simulation videos. I will also show how the forest and fire automaton relates to epidemic models and how one can derive a coarse-grained description without relying on mean-field approximations. *https://www.pnas.org/doi/10.1073/pnas.2211853120

The term 'submesoscale' refers to ocean features with a horizontal timescale of 100m - 10km. These features consist of a range of Eddie's, fronts and waves which exist in a regime where the typical non-dimensional parameters of the system are all order 1. I will explain the importance and challenges of modelling submesoscale motion and present some of my work on ocean fronts and frontal instabilities.

In tissues as diverse as amphibian skin and the human airway, the cilia that propel fluid are grouped in sparsely distributed multiciliated cells (MCCs). I will discuss fluid transport in this “mosaic” architecture, with emphasis on the trade-offs that may have been responsible for its evolutionary selection. Live imaging of MCCs in embryos of the frog Xenopus laevis shows that cilia bundles behave as active vortices that produce a flow field accurately represented by a local force applied to the fluid. A coarse-grained model that self-consistently couples bundles to the ambient flow reveals that hydrodynamic interactions between MCCs limit their rate of work so that they best shear the tissue at a finite but low area coverage, a result that mirrors findings for other sparse distributions such as cell receptors and leaf stomata. I will conclude by discussing the implications of our findings in the context of tissue development.

Data-driven methods play a crucial role in helping scientists discover governing equations from data, leading to a deeper understanding of natural mechanisms. These equations are often partial differential equations (PDEs), containing linear and nonlinear partial derivative terms. We propose an extension to the ARGOS framework, developing a sparse regression algorithm based on the multi-step adaptive lasso to automatically identify PDEs with limited prior knowledge. The proposed framework automates calculating partial derivatives, constructing a candidate library, and estimating a sparse model. We rigorously evaluate the performance of our method by identifying canonical PDEs under various noise levels and sample sizes, highlighting its robustness in handling noisy and non-uniformly distributed data. We also assess the algorithm's effectiveness with pure random noise to simulate scenarios where data quality is compromised. Our findings demonstrate the effectiveness and reliability of the new approach in identifying the underlying PDEs from data.

In this talk I will present past and recent results on the mathematical development of a theory for multi-dimensional localised patterns which are solutions of continuum models that have a spatially localised region of cellular pattern embedded in a quiescent state. These patterns occur in fluid experiments, vegetation patches, buckling of cylinders and biological pattern formation. I will present the range of analytical and numerical techniques that have been developed (in particular related to the recent SIAM 2024 T. Brooke Benjamin Prize) and show that this is an open field with many interesting and fruitful avenues to explore.

Superfluids, such as those formed by ultra-cold atomic Bose-Einstein Condensates (BECs), have incredible properties, such as the ability to flow without viscous effects, and the fact that vorticity is quantized.

Although the problem of superfluid flow past a potential barrier is a well-studied problem in BECs, fewer studies have considered the case of superfluid flow through a disordered potential. We consider the case of a superfluid in a channel with multiple point-like barriers, randomly placed to form a disordered potential. We identify the relationship between the relative position of two point-like barriers, and the critical velocity for vortex nucleation of this arrangement, before considering a system with many obstacles. We then study how the flow of a superfluid in a point-like disordered potential is arrested through the nucleation of vortices and the breakdown of superfluidity. We then consider the vortex decay rate as the width of the barriers and show that vortex pinning becomes an important effect.

We then turn our attention to a two-component BEC in the immiscible limit. In such a system, if vortices are formed in a ``majority’’ component, atoms in the ``minority’’ component will fill the vortex cores, modifying the vortex profile. We show that a variational approach can be employed to approximate the vortex profile for a range of atom numbers in the in-filling component, and that these solutions are stable to small perturbations. We then consider the dynamics of these in-filled vortices.

How do mobile organisms situate themselves in space? This is a fundamental question in both ecology and cell biology but, since space use is an emergent feature of movement processes operating on small spatio-temporal scales, it requires a mathematical approach to answer. In recent years, increasing empirical research has shown that non-locality is a key aspect of movement processes, whilst mathematical models have demonstrated its importance for understanding emergent space use patterns. In this talk, I will describe a broad class of models for modelling the space use of interacting populations, whereby directed movement is in the form of non-local advection. I will detail various methods for ascertaining pattern formation properties of these models, fundamental for answering the question of how organisms situate themselves in space, and describe some of the rich variety of patterns that emerge.

Biofilm formation impacts many fields, from medical technologies (e.g. catheter design) to infrastructure development (e.g. water supply pipes) due to contamination and infection risks. For the case of motile micro-swimmers, the early stages of biofilm behaviour are dependent on the physical properties of swimmers and their flow environments as these affect the likelihood of surface interactions and surface colonisation. In our work, we highlight the effect of boundary conditions on the bulk flow distributions, such as through the development of boundary layers or secondary peaks of cell accumulation in bulk-flow swimmer dynamics. For the case of a dilute swimmer suspension in 2D channel flow, we compare distributions (in physical and orientation space) obtained from individual-based stochastic models with those from continuum models, and identify mathematically sensible continuum boundary conditions for different physical scenarios (i.e. specular reflection, uniform random reflection and absorbing boundaries). We identify the dependence of the spread of preferred cell orientations on the interplay between rotation driven by sheared flows and rotational diffusion. We further highlight the effects of swimmer geometries, fluid shear, and the full history of bulk-flow dynamics on the orientation distributions of micro-swimmer wall incidence.

Androgen deprivation therapy’s ability to reduce tumour growth represents a milestone in prostate cancer treatment. Nonetheless, most patients eventually become refractory and develop castration-resistant prostate cancer (CRPC). Second-generation drugs and their combination have been recently approved for the treatment of CRPC. However, cases of tumour resistance to these new drugs have now been reported. In the last few years, many mathematical models have been proposed to describe the dynamics of prostate cancer under treatment. So far, one of the significant challenges has been the development of mathematical models that could represent experiments under in vivo conditions (experiments on individuals) and, therefore, be suitable for clinical applications while being mathematically tractable. In this talk, I will present a comprehensive study of the phenomena of castration and drug resistance in prostate cancer. I will show how, through the integration of experimental data, statistical analysis and mathematical and computational approaches, it was possible to gain insights into the reasons behind resistance and potential therapeutic strategies to overcome it. Two models will be proposed: a nonlinear distributed-delay dynamical system that explores neuroendocrine transdifferentiation in human prostate cancer in vivo under androgen deprivation therapy [1] and a hybrid cellular automaton with stochastic elements to represent multiple drug therapies [2]. The analytical and numerical study of the first dynamical system showed how the choice of the delay distribution is critical in defining the system’s dynamics and determining the conditions for the onset of oscillations following a Hopf bifurcation. On the other hand, through the computational analysis of the hybrid model, it was possible to investigate the spatial behaviour of tumour cells, the effectiveness of multiple drug therapies on prostate cancer growth, and to identify the best drug combination strategies and treatment schedules to achieve the extinction of cancer cells and avoid metastasis formation. The model revealed that combination and alternating therapies can delay the onset of drug resistance and, in suitable scenarios, can eliminate the disease. The presented mathematical systems incorporate phenomena previously reported in the literature [3,4,5] and verified in the laboratory, such as cell phenotype switching due to drug resistance acquisition and the micro-environment dynamics’ effect on the tumour cells’ necrosis and apoptosis. References [1] Turner, L., Burbanks, A., & Cerasuolo, M. (2021). PCa dynamics with neuroendocrine differentiation and distributed delay. Mathematical Biosciences and Engineering, 18(6), 8577–8602. [2] Burbanks A, Cerasuolo M, Ronca R, Turner L, 2023. A hybrid spatiotemporal model of PCa dynamics and insights into optimal therapeutic strategies. Mathematical Biosciences, 355, 108940. [3] J. Baez, Y. Kuang, Mathematical models of androgen resistance in prostate cancer patients under intermittent androgen suppression therapy, Appl. Sci., 6 (2016), 352. [4] A. Anderson, 2005. A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion, Math. Med. Biol. 22, 163-186. [5] M. Cerasuolo, F. Maccarinelli, D. Coltrini, A. Mahmoud, V. Marolda, G. Ghedini, S. Rezzola, A. Giacomini, L. Triggiani, M. Kostrzewa, R. Verde, D. Paris, D. Melck, M. Presta, A. Ligresti, R. Ronca, 2020. Modeling acquired resistance to the second-generation androgen receptor antagonist enzalutamide in the tramp model of prostate cancer, Cancer Res. 80 (7), 1564–157.

Following a chemical weapons attack, it is crucial for public health that the toxic chemical agent is located and cleaned up. One particular issue is when the agent has been absorbed into porous building materials such as concrete, brick or plasterboard. The government bodies with responsibility for this remediation, Defra and DSTL, must ensure they have protocols in place for effective and efficient decontamination. This talk will follow a story of how mathematicians have worked with Defra and DSTL to model the transport and chemical decontamination reaction between the toxic agent and an applied cleanser within a porous material, beginning with a Study Group and including some of my recent work developing asymptotic methods to model multi-scale free-boundary reaction fronts. I will also discuss some benefits and challenges I’ve experienced working with industrial partners more broadly.

Organoids are three-dimensional multicellular tissue constructs used in applications such as drug testing and personalised medicine. We are working with the biotechnology company Cellesce, who develop bioprocessing systems for the expansion of organoids at scale. Part of their technology includes a bioreactor, in which organoids are embedded within a layer of hydrogel and a flow of culture media across the hydrogel is utilised to enhance nutrient delivery to, and facilitate waste removal from, the organoids. A complete understanding of the system requires spatial and temporal information regarding the relationship between flow and the resulting metabolite concentrations throughout the bioreactor. However, it is impractical to obtain these data empirically, as the highly-controlled environment of the bioreactor poses difficulties for online real-time monitoring of the system. Mathematical modelling can be used to improve the yield of organoids grown within the bioreactor, by predicting the metabolite concentrations during culture for different operating conditions. However, since millions of discrete organoids are grown simultaneously, modelling the mass transport and organoid growth is computationally infeasible in this multiply connected three-dimensional problem involving many moving boundaries of organoid-hydrogel interface. We present a general mathematical model for the transport of nutrient and waste metabolite to and from organoids growing within the hydrogel. We use an asymptotic (multiscale) approach to systematically determine the correct system of effective equations that govern the macroscale mass transport within the hydrogel. The homogenised hydrogel model is coupled to the behaviour within the culture media region, and we explore this model for different culture conditions for the bioreactor. Our results show how the operating protocol influences the metabolite transport within the bioreactor, highlighting the importance of the role of flow in the bioreactor in enhancing metabolite transport, and consequently improving organoid growth.

Physics-informed neural networks (PINNs) provide a novel approach for numerical simulations, tackling challenges of discretization and enabling seamless integration of noisy data and physical models (e.g., partial differential equations). This presentation highlights new opportunities that are enabled through physics-informed machine learning. We will discuss the results of our recent studies where we apply PINNs for coronal magnetic field simulations of solar active regions, which are essential to understand the genesis and initiation of solar eruptions and to predict the occurrence of high-energy events from our Sun. We optimize our network to match observations of the photospheric magnetic field vector at the bottom-boundary, while simultaneously satisfying the force-free and divergence-free equations in the entire simulation volume. We demonstrate that our method can account for noisy data and deviates from the physical model where the force-free magnetic field assumption cannot be satisfied. We simulate the evolution of the active region (AR) NOAA 11158 over 5 continuous days of observations at full cadence of 12 min. The derived evolution of the free magnetic energy and helicity in the active region, demonstrates that our model captures flare signatures, and that the depletion of free magnetic energy spatially aligns with the observed EUV emission. The total computation time requires about 10 hours, which presents the first method that can perform realistic coronal magnetic field extrapolations in quasi real-time, and allows for advanced space weather monitoring. Physics-informed neural networks can flexibly combine multiple data sources in a single simulation. We demonstrate this by utilizing multi-height magnetic field measurements, where the additional chromospheric field information leads to a more realistic approximation of the solar coronal magnetic field. We conclude with an outlook on our ongoing work, where we extend this approach to MHD simulations and perform global magnetic field simulations.

The density stratified, rotating, ocean permits an energetic field of internal gravity waves. Lee waves are one variety of such waves, generated when steady currents interact with rough seafloor topography. These waves are an important sink of energy and momentum from the mean flow, and their effect must therefore be parameterised in global climate models. Typically, linear theory is used to estimate the energy flux into lee waves and to construct global maps of the resulting energy dissipation and mixing. In this talk, I will introduce and discuss linear theory for lee wave generation, extending to the more realistic case of propagation through spatially variable flows, and reflection at the ocean surface.

The patterning of biological tissues is essential for the formation of the organs that make up our bodies. It relies on self-organisation, which emerges from dynamic, iterative interactions between components from molecular to cellular to tissue levels. Polarisation is one of the most basic levels of cell and tissue scale patterning. In developing epithelial tissues, planar polarisation is vital for coordinated cell behaviours during morphogenesis. Alongside experimental approaches, mathematical modelling offers a useful tool with which to unravel the underlying mechanisms. I will describe our recent and ongoing efforts to model the planar polarised behaviours of cells in developing epithelial tissues, how these models have given new mechanistic insights into various aspects of embryonic development, and the mathematical and computational challenges associated with this work.

Large (macro) bodies, such as whales and bumblebees, move about using thin membranes (fins and wings etc.). Very small (micro) bodies, such as spermatozoa, use slender filaments for movement. At the macroscale, locomotion is achieved by imparting momentum to the surrounding fluid through inertial forces. At the microscale, such a strategy would be foiled by large viscous drag forces; hence, locomotion is achieved by exploiting drag forces. At some lengthscale, there is a shift from using thin membranes to using hairs to move. In this talk, we will explore the hydrodynamic basis of locomotion in this "mesoscale" realm, with the common dandelion fruit as our tour guide.

The concept of a diluted soliton gas was introduced in 1971 by V.E. Zakharov as an infinite collection of weakly interacting solitons. More recently, this concept has been extended to dense gases, in which solitons interact strongly and continuously. In the last few years, this field of studies has attracted a rapidly growing interest from both theoretical and experimental points of view. One of the main reasons for this change is the development of numerical algorithms, which made it possible for the first time to simulate wave dynamics of a dense soliton gas. With the help of these algorithms, it has been demonstrated recently that soliton gas dynamics underlies some fundamental nonlinear wave phenomena, such as the spontaneous modulation instability and formation of rogue waves. In my talk, I am going to review recent results and outline perspectives for the current studies of solitonic models, demonstrating that these models might be the key to solving some of the oldest and most pertinent problems in the nonlinear waves theory.

Filamentous material may exhibit structure dependent material properties and function that depends on their entanglement. Even though intuitively entanglement is often understood in terms of knotting or linking, many of the filamentous systems in the natural world are not mathematical knots or links. In this talk we will introduce a novel framework in knot theory that can characterize the complexity of simple curves in 3-space in general. In particular, it will be shown how the Jones polynomial, a traditional topological invariant in knot theory, is a special case of a general Jones polynomial that applies to both open and closed curves in 3-space. Similarly, Vassiliev measures will be generalized to characterize the knotting of open and closed curves. When applied to open curves, these are continuous functions of the curve coordinates instead of topological invariants. We will apply our methods to polymeric systems and show that the topological entanglement captured by these mathematical methods indeed captures polymer entanglement effects in polymer melts and solutions. These novel topological metrics apply also to proteins and we will show that these enable us to create a new framework for understanding protein folding, which is validated by experimental data. These methods thus not only open a new mathematical direction in knot theory, but can also help us understand polymer and biopolymer function and material properties in many contexts with the goal of their prediction and design.

Many time-dependent systems in finance, industry and natural sciences are characterised by two or more distinct time scales. The prominent examples are the geophysical flows that consist of fast waves and slowly moving vortices. The numerical time integration of these systems is challenging as the fast oscillations require very small timesteps. In this talk, I review the “phase averaging” method for the time integration of these system and then present my complementary idea to improve this method. To alleviate the stiffness of multi-timescale systems the nonlinear terms in the equation are averaged over fast oscillations, which leads to an integral in the averaged equation. This integral is then approximated by a quadrature sum where each term can be evaluated in parallel to reduce the overall computational time. My complementary idea is based on layers of correction with different averaging windows which are smartly stacked up such that they can be run in parallel. In a few numerical examples, I show how this allows us to take ridiculously long timesteps (which are sometimes even larger than the period of fast oscillations) to capture the slow dynamics that incorporates the oscillations feedback.

Vegetation patterns are a ubiquitous feature of semi-arid regions and are a prime example of a self-organisation principle in ecology. In this talk, I present bifurcation analyses of two PDE models to (i) investigate the effects of nonlocal seed dispersal, and (ii) identify a mechanism that enables species coexistence despite competition for a limiting resource.

Quasi-static modelling of the Sun’s corona using the magneto-frictional approximation makes predictions of how its large-scale magnetic structure might vary over the solar cycle. Unlike the traditional potential field source surface model, this approach is able to probe the effects of low coronal electric currents and the corresponding free magnetic energy. In particular, ongoing footpoint shearing and flux cancellation lead to the concentration of free energy within closed-field regions, naturally forming sheared arcades and magnetic flux ropes. I will show model results for Cycle 24 driven by magnetogram data from SDO/HMI, with the aim of comparing to earlier results from a decade ago for Cycle 23. In particular, we will consider how solar flux ropes and their eruptions vary over the solar cycle, at least according to this model.

We explore the dynamics of a compressible fluid bubble surrounded by an incompressible fluid of infinite extent in three-dimensions, constructing bubble solutions with finite time blowup under this framework when the equation of state relating pressure and volume is soft (e.g., with volume singularities that are locally weaker than that in the Boyle-Mariotte law), resulting in a finite time blowup of the surrounding incompressible fluid, as well. We focus on two families of solutions, corresponding to a soft polytropic process (with the bubble decreasing in size until eventual collapse, resulting in velocity and pressure blowup) and a cavitation equation of state (with the bubble expanding until it reaches a critical cavitation volume, at which pressure blows up to negative infinity, indicating a vacuum). Interestingly, the kinetic energy of these solutions remains bounded up to the finite blowup time, making these solutions more physically plausible than those developing infinite energy. For all cases considered, we construct exact solutions for specific parameter sets, as well as analytical and numerical solutions which show the robustness of the qualitative blowup behaviors for more generic parameter sets. Our approach suggests novel -- and perhaps physical -- routes to the finite time blowup of fluid equations.

Magnetic flux ropes are coherent bundles of twisted magnetic field lines that wind about a common axis. These structures, formed at the Sun, evolve through footpoint motions, magnetic reconnection, and other processes. The degree of twist in the flux rope helps us to understand their evolution and potential eruption. This twist can be difficult to compute so it is often approximated such that the geometry of the flux rope is neglected. However, despite the relative simplicity of their computation, the results of these approximations require careful analysis to ensure proper understanding. Here we present different definitions of twist and our recently released magnetic field analysis tools (MAFIAT) Python package which is designed to investigate them. Furthermore, we discuss its research applications and our latest developmental features.

Averaging over fast time scales or short spatial scales is a key ingredient in the modelling of complex fluid flows. It has long been realised that Lagrangian averaging, in which the average is carried out at fixed particle label, has advantages over the standard Eulerian averaging, carried out a fixed spatial position. Mainly this is because Lagrangian averaging preserves material conservation laws such as the conservation of vorticity or potential vorticity. The theory of generalised Lagrangian mean (GLM) provides a complete description of the dynamics of Lagrangian mean flows. It is however rather daunting. I will show how adopting a geometric approach helps clarify the theory. The approach shows in particular that most results are independent of the specific (and somewhat problematic) definition of the mean flow used in GLM. I will also discuss recent progress leading to efficient numerical methods for the computation of Lagrangian means.

We describe and model experimental results on the dynamics of a ”ludion” - a neutrally buoyant body – immersed in a layer of stably stratified salt water. By oscillating a piston inside a cylinder communicating with a vessel containing the stratified layer of salt water, it is easy to periodically vary the hydrostatic pressure of the fluid. The ludion or Cartesian diver, initially positioned at its equilibrium height and free to move horizontally, can then oscillate vertically when forced by the pressure oscillations. Depending on the ratio of the forcing frequency to the Brunt- Väisälä frequency N of the stratified fluid, the ludion can emit its own internal gravity waves that we measure by Particle Image Velocimetry. Our experimental results describe first the resonance of the vertical motions of the ludion when excited at different frequencies. A theoretical oscillator model is then derived taking into account added mass and added friction coefficients (Voisin, 2007) and its predictions are compared to the experimental data. Then, for the larger oscillation amplitudes, we observe and describe a bifurcation towards free horizontal swimming. For forcing frequencies close to N , chaotic trajectories are recorded. The statistical analysis of this dynamics is in progress but already suggests the existence of an underlying nonequilibrium thermodynamics with even possible ”condensations” of ludions in pairs or in triplets when several ludions are introduced in the container. Finally, we also observed that the ludion can interact with its own internal gravity wave field and possibly become trapped in the container gravity eigenmodes. However, it seems that, contrary to the surface waves associated with Couder walkers (Couder et al., 2005) the internal waves are not the principal cause of the horizontal swimming. This does not however, exclude possible hydrodynamic quantum analogies to be explored in the future (Bush, 2015). For instance, we observe that ludion trajectories in circular cylindrical corral seems to possess some preferred radii showing that this work on quantum analogy is very promising.

Magnetic helicity quantifies the geometrical complexity of a magnetic field as it measures the twist and writhe of individual magnetic field lines and the intertwining of pairs of field lines. It plays an important role in the study of magnetized plasmas since it is conserved in ideal magneto-hydrodynamics. A means to identify the spatial locations where magnetic helicity is more important, such as magnetic flux ropes, is provided by field line helicity. In astrophysical conditions, this is better expressed by relative field line helicity (RFLH). The first study of the photospheric morphology of RFLH in a solar active region (AR), the famous AR 11158, revealed that RFLH can associate the large decrease in the value of helicity during a strong flare of the AR with the magnetic structure that later erupted. After reviewing these results, we examine the evolution of the morphology of RFLH in a sample of solar ARs that exhibit M-class flares and above. We analyze the role that various RFLH-deduced parameters play in indicating upcoming eruptive events in the AR and report on their possible predictive capabilities.

Stochastic individual-based modelling approaches allow for the investigation of complex biological systems, for example cell populations that exhibit single cell dynamics. These models generally include rules that each cell follows independently of other cells in the population. From these models one can derive the continuum limits from the underlying random walk, providing a deterministic PDE description of these processes to allow for mathematical analysis. We build upon previous work to develop models for the growth of cell populations where individual cell movement via chemotaxis plays a role. Through both analysis and numerical simulations of the models we investigate the role of chemotaxis and phenotypic trade-offs in emergence of complex spatial patterns of population growth.

The fluid dynamical mechanisms responsible for tidal dissipation in giant planets and stars remain poorly understood. One key mechanism is the interaction between tidal flows and turbulent convection. This is thought to act as an effective viscosity in damping the equilibrium tide, but the efficiency of this mechanism is still a matter of much debate.

Using hydrodynamical simulations we investigate the dissipation of the equilibrium tide as a result of its interaction with convection. We model the large-scale tidal flow as an oscillatory background shear flow inside a small patch of convection zone. We simulate Rayleigh-Bénard convection in this Cartesian model and explore how the effective viscosity of the turbulence depends on the tidal (shear) frequency.

We will present the results from our simulations to determine the effective viscosity, and its dependence on the tidal frequency. The main results are: a new scaling law for the frequency dependence of the effective viscosity which has not previously been observed in simulations or predicted by theory and occurs for frequencies smaller than those in the fast tides regime; negative effective viscosities, which can be thought of as tidal anti-dissipation (or inverse tides), are possible in this system.

This is joint work with Rod Cross and Andrew Wade.

The development of a complex functional multicellular organism from a single cell involves tightly regulated and coordinated cell behaviours coupled through short- and long-range biochemical and mechanical signals. To truly comprehend this complexity, alongside experimental approaches we need mathematical and computational models, which can link observations to mechanisms in a quantitative, predictive, and experimentally verifiable way. In this talk I will describe our recent efforts to model aspects of embryonic development, focusing in particular on the planar polarised behaviours of cells in epithelial tissues, and discuss the mathematical and computational challenges associated with this work. I will also highlight some of our work to improve the reproducibility and re-use of such models through the ongoing development of Chaste (https://github.com/chaste), an open-source C++ library for multiscale modelling of biological tissues and cell populations.

Turbulent convection occurs in the cores of stars whose masses are at least 1.1 times greater than the Sun. This convection generates gravity waves in the adjacent stable region, and signatures of these waves can be seen at the stellar surface and used to constrain models of the stellar interior. "Standard" stellar evolution models without excess mixing beyond the boundary of the convection zone fail to reproduce many observations of these stars. In this seminar, I will present a review of the observations of excess mixing in the cores of massive stars. I will then discuss from a hydrodynamical perspective the different forms of convective boundary mixing which can occur in stars or models of stars. I will present the results of simulations which demonstrate "convective penetration," in which there is a turbulent well-mixed region which is typically assumed to be stable. I will also present a separate set of new simulations of 3D, spherical, compressible core convection and show that the wave generation and propagation in these simulations matches the theory developed for 2D Boussinesq simulations.

Longstanding among the open questions in solar physics is the coronal heating problem. Above the surface, at a few thousand degrees, the temperature of the Sun's atmosphere rises to many millions in the corona. Why this is so, and the mechanisms that contribute to this heating, remain unclear.

Self-organized criticality is a physical paradigm that has been applied to many natural systems, including neuroscience, tectonics, and financial markets. Such systems are conjectured to exist in minimally stable states, in which they are subject to perturbation by an external driving, leading to redistributive avalanches of varying size. From small, local disturbances, these can propagate through chain reactions of like events, becoming very large: 'avalanches'.

Since the 1990s, this concept has been applied to the coronal heating problem. Convective motions drive the magnetic field, and, when sufficiently stressed, its energy is released through highly localized magnetic reconnection. Each such event contributes heating, and has the capacity to start a chain of other events. While these MHD avalanches have been shown to be feasible among straightened models of coronal loops, their viability is to be determined in models that reflect the true geometric curvature of coronal loops above the solar surface.

Using three-dimensional MHD simulations, avalanches are verified within this more realistic geometry. Similarly to the simpler case, an ideal MHD kink-mode instability occurs, but in a modified manner, preferentially aligned with the curvature in the field. Instability spreads over a volume far larger than that of the original flux tubes, causing widespread heating. Substantial amounts of energy are released, contributing to coronal heating, in a series of nanoflare-type events. Overwhelmingly, shocks, jets, and related processes dominate over Ohmic dissipation. Narrow and intense heating occurs, throughout the loop, without obvious spatio-temporal preference.

Prospects for developing and further analysing this model, including with the full treatment of thermodynamic terms, are discussed.

Bacteria represent the major component of the world’s biomass. A number of these bacteria are motile and swim with the use of flagellar filaments, which are slender helical appendages attached to a cell body by a flexible hook. Low Reynolds number hydrodynamics is the key for flagella to generate propulsion at a microscale [1]. In this talk I will discuss two topics related to swimming of a model bacterium Escherichia coli (E. coli).

E. coli has many flagellar filaments that are wrapped in a bundle and rotate in a counterclockwise fashion (if viewed from behind the cell) during the so-called ‘runs’, wherein the cell moves steadily forward. In between runs, the cell undergoes quick ‘tumble’ events, during which at least one flagellum reverses its rotation direction and separates from the bundle, resulting in erratic motion in place. Alternating between runs and tumbles allows cells to sample space by stochastically changing their propulsion direction after each tumble. In the first part of the talk, I will discuss how cells reorient during tumble and the mechanical forces at play and show the predominant role of hydrodynamics in setting the reorientation angle [2].

In the second part, I will talk about hydrodynamics of bacteria near walls in visco-elastic fluids. Flagellar motility next to surfaces in such fluids is crucial for bacterial transport and biofilm formation. In Newtonian fluids, bacteria are known to accumulate near walls where they swim in circles [3,4], while experimental results from our collaborators at the Wu Lab (Chinese University of Hong Kong) show that in polymeric liquids this accumulation is significantly reduced. We use a combination of analytical and numerical models to propose that this reduction is due to a viscoelastic lift directed away from the plane wall induced by flagellar rotation. This viscoelastic lift force weakens hydrodynamic interaction between flagellated swimmers and nearby surfaces, which results in a decrease in surface accumulation for the cells.

References

[1] Lauga, Eric. "Bacterial hydrodynamics." Annual Review of Fluid Mechanics 48 (2016): 105-130.

[2] Dvoriashyna, Mariia, and Eric Lauga. "Hydrodynamics and direction change of tumbling bacteria." Plos one 16.7 (2021): e0254551.

[3] Berke, Allison P., et al. "Hydrodynamic attraction of swimming microorganisms by surfaces." Physical Review Letters 101.3 (2008): 038102.

[4] Lauga, Eric, et al. "Swimming in circles: motion of bacteria near solid boundaries." Biophysical journal 90.2 (2006): 400-412.

In this talk, we will present the results of the study of two pattern-forming models solved on a finite domain. In the first, the dynamics of the real and complex cubic-quintic Swift-Hohenberg equation over a finite disk with no-flux boundary conditions are studied. We predict the unstable modes of the trivial state using a linear stability analysis. These modes are followed via numerical continuation, revealing a great variety of spatially extended and spatially localized behaviors. Notably, we find solutions localized in the interior as well as solutions localized along the boundary or part of the boundary. Bifurcation diagrams summarizing these results and their stability properties are presented, linking the different solutions. The findings of this study are likely relevant to nonlinear optics, combustion as well as convection.

In the second model, A simple equation modeling an inextensible elastic lining subject to an imposed pressure is derived from the idealised elastic properties of the lining and the pressure. The equation aims to capture the wrinkling response of arterial endothelium to blood pressure changes. A bifurcation diagram is computed via numerical continuation. Wrinkling, buckling, folding, and mixed-mode solutions are found and organised according to system-response measures including tension, in-plane compression, maximum curvature and energy. Approximate wrinkle solutions are constructed using weakly nonlinear theory, in excellent agreement with numerics. We explain how the wavelength of the wrinkles is selected as a function of the parameters in compressed wrinkling systems and show how localized folds and mixed-mode states form in secondary bifurcations from wrinkled states.

The solar corona is shaped and mysteriously heated to millions of degrees by the Sun’s magnetic field. It has long been hypothesized that the heating results from a myriad of tiny magnetic energy outbursts called nanoflares, driven by the fundamental process of magnetic reconnection. Misaligned magnetic field lines can break and reconnect, producing nanoflares in avalanche-like processes. This theory recently received major support through the observational discovery of nanojets - very fast (100-200 km/s) and bursty (

The talk will start with a short introduction to the open-source Dedalus computational framework for solving PDEs. I'll briefly discuss some design motivations and interesting applied problems different groups are studying with the software.

However, the true point of the talk will be to describe some of the mathematical aspects underlying Dedalus' flexible framework for curvilinear coordinates. Dedalus can compute arbitrary covariant calculus for scalar, vector and tensor fields on spheres and balls in one, two, and three dimensions. I plan to discuss some modest fraction of the wonderful mathematics used at the core of the code. In particular, Dedalus uses orthogonal polynomial bases on an infinite hierarchy of nested Hilbert spaces, with multiple derivative, multiplication and embedding operators acting between them. The structures become complicated quickly, but everything boils down to some simple guiding principles that I will demonstrate using freely available properties of meek little sine and cosine functions. No prior background beyond basic calculus and linear algebra is assumed.

Shocks are a universal feature of warm plasma environments, such as the lower solar atmosphere and molecular clouds, which consist of both ionised and neutral species. Including partial ionisation leads to the existence of a finite width for shocks, where the ionised and neutral species decouple and recouple. As such, drift velocities exist within the shock that lead to frictional heating between the two species, in addition to adiabatic temperature changes across the shock. The local temperature enhancements within the shock alter the recombination and ionisation rates and hence change the composition of the plasma. We study the role of collisional ionisation and recombination in slow-mode partially ionised shocks. In particular, we incorporate the ionisation potential energy loss and analyse the consequences of having a non-conservative energy equation. A semi-analytical approach is used to determine the possible equilibrium shock jumps for a two-fluid model with ionisation, recombination, ionisation potential, and arbitrary heating. Two-fluid numerical simulations are performed using the (PIP) code. Results are compared to the magnetohydrodynamic (MHD) model and the semi-analytic solution. Accounting for ionisation, recombination, and ionisation potential significantly alters the behaviour of shocks in both substructure and post-shock regions. In particular, for a given temperature, equilibrium can only exist for specific densities due to the radiative losses needing to be balanced by the heating function. A consequence of the ionisation potential is that a compressional shock will lead to a reduction in temperature in the post-shock region, rather than the increase seen for MHD. The numerical simulations pair well with the derived analytic model for shock velocities.

In the talk we will discuss the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains where the pressure is constant on the boundary but not in the interior. The pressure is piecewise constant and the plasma current exhibits current sheets. The toroidal domains where these equilibria are shown to exist do not need to be small perturbations of an axisymmetric domain, and in fact they can have any knotted topology. The building blocks we use in our construction are analytic toroidal domains satisfying a certain nondegeneracy condition, which roughly states that there exists a force-free field that is ergodic on the surface of the domain. The proof involves three main ingredients: a gluing construction of piecewise smooth MHD equilibria, a Hamilton–Jacobi equation on the two-dimensional torus that can be understood as a nonlinear deformation of a cohomological equation (so the nondegeneracy assumption plays a major role in the corresponding analysis), and a new KAM theorem tailored for the study of divergence-free fields in three dimensions. The talk is based on joint work with A. Luque and D. Peralta-Salas.

Lithium-ion batteries have become ubiquitous over the past decade, and they are called to play even a more important role with the electrification of vehicles. In order to design better and safer batteries and to manage them more efficiently, we need models than can predict the battery behaviour accurately and fast. However, in many cases these models are still posed in an ad hoc way, which makes them hard to extend and may lead to inconsistencies. In this talk we will see some examples on how asymptotic methods can be applied to obtain simple models that can be used in battery control and parameterisation.

The Juno spacecraft is currently orbiting Jupiter. Newly available data reveal a complex magnetic field morphology. The dominant dipolar component is accompanied by strong magnetic flux patches and a narrow field belt in the equatorial region. The gravitational sounding also indicates that the surface zonal jets extends thousand kilometers below the cloud level. Those are key signatures of the intricate Jovian internal dynamics.

Jupiter's internal structure comprises an outer layer filled with a mixture of molecular hydrogen and helium where the zonal flows are thought to be driven; and an inner region where hydrogen becomes metallic and dynamo action is expected to sustain the magnetic field. Several internal structure models suggest a more complicated structure with a small intermediate region in which helium and hydrogen would become immiscible.

During this seminar, I will review the main results obtained using global 3-D numerical simulations of Jupiter in spherical geometry. I will discuss the success and the limitations of those models in reproducing the key features of the Jovian internal dynamics.

Density-stratified fluid systems such as the oceans and atmosphere undergo irreversible diascalar transport with respect to density, heat, etc., known as mixing. When turbulence is present, the effective diffusivity can be greatly magnified and the rate of mixing enhanced. To understand this process involving turbulent flows, one can consider the energetics by partitioning the potential energy into available and background parts to signify the irreversibility. Following the energetic approach, this talk reviews aspects, such as the rate and efficiency, of stratified turbulence mixing.

Eukaryotic cells exhibit a complicated rheology in response to mechanical stimuli with an elastic component being primarily due to a network of cross-linked filamentous proteins called cytoskeleton. Rational upscaling of existing detailed (but computationally expensive) discrete models into continuum is largely missing and the manner in which microscale parameters and processes manifest themselves at the macroscale is thus often unclear.

We introduce a discrete mathematical model for the mechanics of the cell cytoskeleton. The model involves an initially regular (planar) array of pre-stretched protein filaments (e.g. actin, vimentin) that exhibit tensile forces and resistance to bending. Assuming that the inter-crosslink distance is much shorter than the region of the cell under consideration, we upscale the force balance equations using discrete-to-continuum methods based on Taylor expansions to form a continuum system of governing equations and infer the corresponding macroscopic stress tensor.

We solve these discrete and continuum models numerically to analyse an imposed displacement of a bead placed in the domain and infer force-displacement curves, which show good quantitative agreement between the approaches. Furthermore, we linearise the continuum model to derive analytical approximations of the stress and strain fields in the neighbourhood of a small bead, explicitly computing the net force required to generate a given deformation as a function of key model parameters.

Future work will also incorporate nonlinear effects in polymer elasticity and the additional influence of fluid transport within the cytoskeleton.

This talk focusses on events known as coronal mass ejections. Since their discovery in the early 1970s it has been realised that these eruptions occur due to changes in the Sun’s magnetic field. The eruptions appear to be related to a certain magnetic field configuration known as a flux rope. Understanding how and where flux ropes form has unravelled some of the mysteries around coronal mass ejections and understanding their magnetic structure has not only helped explain why these eruptions occur, but also what their space weather impact might be if they are ejected toward the Earth. This talk will also discuss a new approach that uses magnetic helicity to shed new light on how eruptions might be forecast ahead of time. An exciting step forward that might be useful in the years to come for space weather forecasting.

Polymer physics successfully describes most of the polymeric materials that we encounter everyday. In spite of this, it heavily relies on the assumption that polymers do not change topology (or architecture) in time or that, if they do alter their morphology, they do so in equilibrium. This assumption spectacularly fails for DNA in vivo, which is constantly topologically re-arranged by ATP-consuming proteins within the cell nucleus. Inspired by this, here I propose to study entangled systems of DNA which can selectively alter their topology and architecture in time and may expend energy to do so. I argue that solutions of topologically active (living) polymers can display unconventional viscoelastic behaviours and can be conveniently realised using solutions of DNA functionalised by certain families of vitally important proteins.

In this talk I will present some results on the microrheology of entangled DNA undergoing topological alterations via proteins, for instance digestion by restriction enzymes and ligation by T4 ligase. I will present theory, modelling and experimental data showing that that we can harness non-equilibrium processes to design materials and complex fluids with time-varying viscoelastic behaviours.

Shocks driven by Coronal Mass Ejections (CMEs) are primary agents of space weather. They can accelerate particles to high energies and can compress the magnetosphere thus setting in motion geomagnetic storms. For many years, these shocks were studied only in-situ when they crossed over spacecraft or remotely through their radio emission spectra. Neither of these two methods provides information on the spatial structure of the shock nor on its relationship to its driver, the CME. In the last two decades, we have been able to not only image shocks with coronagraphs but also measure their properties remotely through the use of spectroscopic and image analysis methods. Thanks to instrumentation on STEREO and SOHO we can now image shocks (and waves) from the low corona, through the inner heliosphere, to Earth. Here, we review the progress made in imaging and analyzing CME-driven shocks and show that joint coronagraphic and spectrscopic observations are our best means to understand shock physics close to the Sun.

Topology has become a much-discussed part of current research in solid-state magnetism, providing an organising principle to classify field theories, and their ground states and excitations, that are now regularly realized in magnetic materials. Examples include topological excitations such as skyrmions which exist in the spin textures of an expanding range of magnetic systems, and one-dimensional spin-chain systems, where topological considerations are key in understanding their properties. Central to this story is the role of the sine-Gordon model, which was important in motivating Skyrme's work and also in understanding the properties of spin chains using field theories. From this starting point, I will review some of the states that we might expect to realize in magnetic materials, and provide two sets of examples of where and how these have been found. Firstly, I will present examples spin chains and ladders formed of molecular building blocks, where the versatility of carbon chemistry allows access to spin Luttinger liquids, and sine-Gordon and Haldane chains. Secondly I shall discuss materials that host magnetic skyrmions and related excitations, along with the prospects for finding still more of these in the future.

Curvelets are a special type of wavelet that, by design, efficiently represent highly anisotropic signal content, such as local linear and curvilinear structures, e.g. edges and filaments. In addition, analyses of data will be the most efficient and accurate if the technique is derived to accommodate the underlying geometry of the data. In this seminar, I will talk about a new-generation scale-discretised curvelet transform suitable for analysing signals of arbitrary spin defined on a sphere (Chan et al. 2016), e.g. signals observed on planetary surfaces and the celestial sky, 360-degree images taken by omnidirectional cameras, and medical imaging such as retinal visualisation. Our constructed curvelets, namely the second-generation curvelets, have many desirable properties that are lacking in first-generation constructions: they live directly on the sphere, exhibit a parabolic scaling relation, are well localised in both spatial and harmonic domains, support the exact analysis and synthesis of both scalar and spin signals, and are free of blocking artefacts. We apply our curvelet transform to some example natural spherical images and demonstrate the effectiveness of curvelets for representing directional curve-like features.

Papers: (1) Second-generation of Curvelets on the Sphere: Jennifer Y. H. Chan, Boris Leistedt, Thomas D. Kitching, and Jason D. McEwen, IEEE Trans. Signal Process., vol. 65, no. 1, pp. 5-14, 2017 (9 pages, 7 figures). DOI: 10.1109/TSP.2016.2600506 and arXiv:1511.05578. Code: http://astro-informatics.github.io/s2let/

(2) Wavelet-based Segmentation on the Sphere: Xiaohao Cai, Christopher G. R. Wallis, Jennifer Y. H. Chan, and Jason D. McEwen, Pattern Recognition, vol. 100, pp. 0031-3203, 2020 (23 pages, 8 figures). DOI: 10.1016/j.patcog.2019.107081; arXiv:1609.06500.

Travelling waves arise in a wide variety of biological applications, from the healing of wounds to the migration of populations. Such biological phenomena are often modelled mathematically via reaction-diffusion equations; however, the resulting travelling wave fronts often lack the key feature of a sharp ‘edge’. In this talk, we will examine how the incorporation of a moving boundary condition in reaction-diffusion models gives rise to a variety of sharp-fronted travelling waves for a range of wavespeeds. In particular, we will consider common reaction-diffusion models arising in biology and explore the key qualitative features of the resulting travelling wave fronts.

Some proteins are known to form open ended knots. Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. A crucial step in this direction is being able to efficiently and meaningfully characterise their entanglement. Mathematically, this is a non-trivial problem. In this talk I will present some of the techniques that can be used to do that, and some applications to knotted proteins.

One of the most commonly used models for the magnetic field in the solar corona is a Potential Field, Source-Surface (PFSS) model. These models use a known radial magnetic field distribution on the Sun's photospheric surface to compute a suitable current-free magnetic field. They are relatively simple to calculate and have become somewhat ubiquitous over the last few decades. There are however severe limitations with the model, in particular that the solar wind is not properly accounted for and the open magnetic flux is considerably underestimated compared to observations. We propose an improvement to this model which seeks to address these problems without any significant increase in computational expense.

Dynamical systems have enjoyed huge success in the analysis of systems described by ordinary differential equations, such as nonlinear oscillators, chemical reactions, electronic devices, population dynamics, etc. Usually, in the dynamical systems approach, one is concerned with the identification of the basic building blocks of the system under investigation and how they interact with each other to produce the observable dynamics, as well as how they can be manipulated to produce a desired output, in the cases where control is pursued. Examples of those building blocks are unstable equilibrium and periodic solutions, nonattracting chaotic sets and their manifolds, which are special surfaces in the phase space that basically control the dynamics, guiding solutions in preferred directions.Despite its success in those areas, many still think that the theory has limited value when applied to fully developed turbulence, like observed in solar convection, due to the infinite dimension of the phase space. In this talk, we show that this difficulty can be overcome by adopting a Lagrangian reference frame, where the phase space for each fluid particle becomes three-dimensional and the building blocks of the turbulence can be efficiently extracted by appropriate numerical tools. We reveal how finite-time Lyapunov exponents, a traditional measure of chaos, can be usedto detect attracting and repelling time-dependent manifolds that divide the fluid in regions with different behavior. These manifolds are shown to accurately mark the boundaries of granules in observational data from the photosphere. In addition, stagnation points and vortices detected as elliptical Lagrangian coherent structures complete the set of building blocks of the photospheric turbulence. Such structures are crucial for the trapping and transport of mass and energy in the solar plasma.

In this talk, I will present a theoretical framework for the topological study of time series data [1]. After setting up a basic background on topological data analysis, I will describe how ideas from dynamical systems and persistent homology can be combined to gain insights from time-varying data. Several applications to real data will be shown at the end of the talk, including anomaly detection in ECG, pattern recognition in bird songs and prediction of epilepsy seizures [2].

[1] Perea, J.A., Harer, J. Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis. Found Comput Math 15, 799–838 (2015). [2] Fernandez, X., Borghini, E., Mindlin, G., Groisman, P. Intrinsic persistent homology via density-based metric learning. (2020) Preprint: arXiv:2012.0762

In this talk, we will study knotoids in the plane or in the 2-sphere S^2. Knotoids are knotted arcs,generalizing classical knot theory and also ‘closing up’ to virtual knots. Knotoids can be viewed as line projections of spatial open curves with open ends fixed on two infinite lines. With this interpretation, they provide a topological classification for proteins. I will first introduce essential notions of knotoids and then construct invariants of knotoids.

To survive and to thrive, plants, which are sessile by nature, rely on their ability to sense multiple environmental signals, such as gravity or light, and respond to them by growing and changing their shape. To do so, the signals must be transduced down to the cellular level to create the physical deformations leading to shape changes. The challenge for a general mathematical theory of tropism is that these processes take place at very distinct scales. In this talk, I will present a multiscale theory of tropism that takes multiple stimuli and transforms them into auxin transport that drives tissue-level growth and remodelling, thus modifying the plant shape and position with respect to the stimuli. This feedback loop can be dynamically updated to understand the response to individual stimuli or the complex behavior generated by multiple stimuli such as canopy escape or pole wrapping for climbing plants

I shall present two recent pieces of work investigating how shape effects the transport of active particles in shear. Firstly we will consider the sedimentation of particles in 2D laminar flow fields of increasing complexity; and how insights from this can help explain why turbulence can enhance the sedimentation of negatively buoyant diatoms [1]. Secondly, we will consider the 3D transport of elongated active particles under the action of an aligning force (e.g. gyrotactic swimmers) in some simple flow fields; and will see how shape can influence the vertical distribution, for example changing the structure of thin layers [2].

[1] Enhanced sedimentation of elongated plankton in simple flows (2018). IMA Journal of Applied Mathematics W Clifton, RN Bearon, & MA Bees [2] Elongation enhances migration through hydrodynamic shear (in Prep), RN Bearon & WM Durham,

Global Rossby waves, interacting with mean east-west flow on the Earth's atmosphere, produce jet streams, which are responsible for causing winter storms, and cold outbreaks that we experience in midlatitudes. Rossby waves arise in thin layers within fluid regions of stars and planets. These global wave‐like patterns occur due to the variation in Coriolis forces with latitude. It has recently been discovered that the Sun has Rossby waves too. But unlike the Earth's Rossby waves, due to the presence of strong magnetic fields solar Rossby waves are magnetically modified. Therefore, the Sun's global magnetic fields and flows are also influenced by these global‐scale waves. In this talk, I will demonstrate through model-simulations how solar Rossby waves, nonlinearly interacting with differential rotation and spot-producing magnetic fields, can cause the seasonal/sub-seasonal (6-18 months) variability in solar activity, which is, in turn, the origin of space weather on intermediate time-scales. Space weather occurring on a very short time-scale (hours-to-days) and on much longer time-scale (decadal-to-millennial) have been studied extensively, but there exists a gap, namely the occurrence of space weather on the seasonal/sub-seasonal time-scale of a few weeks to several months. I will discuss how the knowledge of MHD Rossby waves can plausibly fill-in this gap, and help simulate the short-term variability in solar activity, and hence space-weather seasons. Two helpful links are: http://dx.doi.org/10.1029/2018SW002109 https://physicsworld.com/a/rossby-waves-on-the-sun-provide-a-tool-for-forecasting-space-weather

Many previous studies have shown that the magnetic precursor of a coronal mass ejection (CME) takes the form of a magnetic flux rope, and a subset observed at plasma temperatures of ~10^7 K have become known as ‘hot flux ropes’. We seek to identify the processes by which these hot flux ropes form, with a view of developing our understanding of CMEs and thereby improving space weather forecasts. Extreme-ultraviolet observations were used to identify five pre-eruptive hot flux ropes in the solar corona, and the evolution of the photospheric magnetic field was studied over several days before they erupted to investigate how they formed. Evidenced by confined solar flares in the hours and days before the CMEs, we conclude the hot flux ropes formed via magnetic reconnection in the corona, contrasting many previously-studied flux ropes that formed lower down in the solar atmosphere via magnetic cancellations. This coronal reconnection is driven by observed ‘orbiting’ motions of photospheric magnetic flux fragments around each other, which bring magnetic flux tubes together in the corona. This represents a novel trigger mechanism for solar eruptions, and should be considered when predicting solar magnetic activity.

The EU Horizon2020 project EUHFORIA 2.0 aims at developing an advanced space weather forecasting tool, combining the MHD solar wind and CME evolution model EUHFORIA. With the Solar Energetic Particle (SEP) transport and acceleration model PARADISE. We will first introduce EUHFORIA and PARADISE and then elaborate on the plans of the EUHFORIA 2.0 project which will address the geoeffectiveness of impacts and mitigation to avoid (part of the) damage, including that of extreme events, related to solar eruptions, solar wind streams, and SEPs, with particular emphasis on its application to forecast Geomagnetically Induced Currents (GICs) and radiation on geospace.

The EUHFORIA 2.0 project started on 1 December 2019, and yielded some first results. These concern alternative coronal models, the application of adaptive mesh refinement techniques in the heliospheric part of EUHFORIA, alternative flux-rope CME models, evaluation of data-assimilation based on Karman filtering for the solar wind modelling, and a feasibility study of the integration of SEP models. The novel tool will be accessible by the whole space weather community via the ESA Space Weather Service Network as it will be integrated in the Virtual Space Weather Modelling Centre (VSWMC)[3], which is part of that network.

The rate of magnetic field diffusion plays an essential role in several astrophysical processes. It has been demonstrated that the omnipresent turbulence in astrophysical media induces fast magnetic reconnection, which consequently can lead to large-scale magnetic flux diffusion at a rate independent of the plasma microphysics. This process is called ``reconnection diffusion'' (RD) and allows for the diffusion of fields which are dynamically important. The current theory describing RD is based on incompressible magnetohydrodynamic (MHD) turbulence. We have tested quantitatively the predictions of the RD theory when magnetic forces are dominant in the turbulence dynamics. We employed the PENCIL Code to perform simulations of forced MHD turbulence, extracting the values of the diffusion coefficient using the Test-Field method. Our results are consistent with the RD theory (diffusion proportional to the Alfvenic Mach number Ma to the power of 3 for Ma < 1) when turbulence approaches the incompressible limit (sonic Mach number Ms < 0.02), while for larger Ms the diffusion is faster (proportional to Ma to the power of 2). Our results generally support and expand the RD theory predictions.

Turbulent dynamos are present in many astrophysical systems. We consider dynamo action in turbulent flows that contain coherent structures. In particular we aim to assess the extent to which the growth rate of such a dynamo is controlled by the coherent stuctures in the flow, as opposed to the turbulent eddies. One approach to this problem is to apply Fourier filtering to the velocity field (Tobias & Cattaneo 2008) to identify the dominant scales of motion in the dynamo. However, localised coherent structures are not always well-represented by such filtering schemes, with information distributed across many Fourier components. An alternative approach is to use wavelets, which are better suited to describing such localised structures. We will present simulations of 2.5D dynamo action, using flows derived from 2D hydrodynamic turbulence. The flows are filtered in wavelet space, retaining only those wavelet coefficients whose magnitude exceeds a certain threshold; only a small fraction of the relevant modes must be kept in order to ensure the retention of the dominant coherent structures. We will describe the extent to which the dynamo growth rate for these filtered flows depends upon the filtering threshold, comparing our findings with comparable Fourier-based filters. Based upon this comparison of these two filtering approaches, we will discuss the extent to which a wavelet-based approach could be used to better understand astrophysical dynamos.

The solar dynamo involves the production of toroidal field by the winding up of toroidal field, and the production of poloidal field from the toroidal field. In this talk we will quantify the relevant production processes, and discuss the transport of magnetic flux which is required to move the magnetic flux from where it is generated to where it is needed for the above production processes to work. The multiple essential roles of flux emergence will be described, and clues as to the nature of flux emergence will be presented.

The flow of a thin film down an inclined plane is a canonical setup in fluid mechanics and associated technologies, with applications such as coating, where the liquid-gas interface should ideally be flat, and heat or mass transfer, where an increase of interfacial area is desirable. In each of these applications, we would like to robustly and efficiently manipulate the flow in order to drive the dynamics to a desired interfacial shape. In this talk, I will propose a feedback control methodology based on same fluid blowing and suction through the wall. The controls will be developed in the lower rungs of a hierarchy of models for a falling liquid film based on reduced-order modelling and asymptotic analysis. The goal is to develop control strategies at these more cost-effective levels of the hierarchy (both the lowest rung modelled by the Kuramoto-Sivashinsky equations, and at a 'middle' level, consisting of two long-wave models) and investigate their ability to translate across the hierarchy into real-life solutions by using direct numerical simulations of the Navier-Stokes equations, which in this context act as an in silico experimental framework. I will discuss distributed controls as well as (more realistic) point-actuated controls, their robustness to parameter uncertainties and validity across the hierarchy of models.

Apart from its about 11-year periodicity, the most striking property of the solar activity record is the notable variability of the cycle amplitudes. Nonlinear and/or stochastic mechanisms are required to modulate the cycle amplitudes. During the past decade, Babcock-Leighton (BL) mechanism is demonstrated at the essence of the solar cycle. In the seminar, I will present our series of studies on the identification and quantitative evaluation of nonlinear and stochastic mechanisms on understanding of solar cycle variability in the framework of BL-type solar dynamo.

Recent advances in solar instrumentation have revealed a rich variety of eruptive activity in striking detail. The largest such eruptions, coronal mass ejections and associated solar flares, drive the shocks and energetic particles that play a major role in the most hazardous space weather events. To predict the space weather impact of solar eruptions, we must understand three vital questions: How does energy build up in the corona? What triggers its explosive release? How is that energy transferred to nonthermal particles? To accurately model explosive activity, it is important to capture both the large-scale dynamics of the energy buildup and release and the fine-scale structure that plays a critical role in particle energization. I present recent advances in tackling these questions using high-resolution, three-dimensional magnetohydrodynamics simulations of solar eruptions. I also discuss promising avenues for future work and prospects for comparison to ground-based (DKIST, EOVSA) and space-borne (PSP, SolO) observations.

Aggregation phenomena in biology and beyond are often attributed to attraction between individuals. In this work we study how elastically tethered obstacles interacting with the swimmers impact the macroscopically created patterns. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations model, for which we assume large tether stiffness. The result is a coupled system of non-linear, non-local partial differential equations. We use linear stability analysis to predict pattern size from model parameters. Further analysis of the macroscopic equations reveal that, surprisingly, the obstacle interactions induce short-ranged swimmer aggregation, irrespective of whether obstacles and swimmers are attractive or repulsive.

Three years ago, we asked a simple question: how does large-scale organisation emerge in magnetohydrodynamics (MHD)? In this talk, I will review the progress we have made using a cocktail of novel approaches. First, I will show how a quasi-conserved quantity named the field line helicity (FLH) links to self-organisation. Then I will discuss various models which we constructed to help us understand better the evolution of topology: 1) unstirring a 2D scalar using effective magnetic relaxation, or variational methods, 2) 3D & 2D magnetic reconnection simulations for comparison, 3) 1D effective model with a focus on local reconnection. These approaches are complementary as each is designed with a specific property in mind: 1) to test if pure advection plays a role in the simplification of FLH, 2) to analyse the change of magnetic structures in plasmas, 3) to test whether self-organisation in MHD behaves as an emerging phenomena. One benefit of using these drastically different models is that we can gain deeper insight into a non-trivial physical process. Besides presenting the results, I will also discuss the limitations and setbacks, and then conclude with possible future plans.

Taffy is a type of candy made by repeated 'pulling' (stretching and folding) a mass of heated sugar. The purpose of pulling is to get air bubbles into the taffy, which gives it a nicer texture. Until the late 19th century, taffy was pulled by hand, an arduous task. The early 20th century saw an avalanche of new devices to mechanize the process. These devices have fascinating connections to the topological dynamics of surfaces, in particular with pseudo-Anosov maps. Special algebraic integers such as the Golden ratio and the lesser-known Silver ratio make an appearance, as well as more exotic numbers. We examine different designs from a mathematical perspective, and discuss their efficiency. This will be a "colloquium style" talk that should be accessible to graduate students.

Complex fluids appear in many biological and industrial settings. A key feature is that their interesting macroscale behaviour derives from their complex microscale structure. In many cases, these fluids are suspensions - some viscous fluid with particles or fibres suspended within. For example, in polymeric fluids, flexible suspended fibres lead to non-Newtonian bulk responses such as shear thinning or viscoelasticity. When entangled or connected in networks, fibres form gels and disordered solids as is the case in important biological materials such as mucus.

These systems are interesting for mathematicians because the question is raised as to which scale is 'right' to investigate these fluids.

The project I have been working on for the last few years - modelling sperm swimming through mucus - argues that large-scale simulation of microscale fibres, interacting with fluid, allows us to bridge this gap.

In this talk, I give a brief overview of suspension mechanics, then I present our progress so far in modelling fibre suspensions - the challenges behind moving to 3D, how we created large networks for sperm-like swimmers to navigate through, as well as the interesting behaviour of fibres in other configurations. I will share my (honest!) experience of using computational infrastructure, as well as presenting avenues for the future.

The magnetic field of the corona stores the energy that is released via magnetic reconnection during solar flares and coronal mass ejections (CMEs). Flares with CMEs are often described by the '˜standard' eruptive flare (CSHKP) model and this offers a conceptual framework in which to investigate the global characteristics of the energy release and transport in the context of the magnetic field configuration. The low plasma beta environment of the corona means the magnetic field plays a central role in the energy transport, and different magnetic field configurations can lead to a variety of outcomes in terms of the evolution of the energy release, the efficiency of the energy transport mechanisms and the locations where the energy is deposited. Despite the often rather good agreement between observations and the '˜standard' model, many open questions remain particularly in respect to the triggering of the energy release. In this talk I will discuss how multi-wavelength spectroscopy used in tandem with magnetic field information can help shed light on some of these open questions, and also how new facilities might provide new insights in the future.

The origin of the large-scale magnetic fields of astrophysical objects remains one of the great unsolved problems of theoretical astrophysics. From planets (including exoplanets) to stars and galaxies, astrophysical objects typically are observed to have organised magnetic fields with systematic properties. For example, the eleven-year solar cycle is a remarkable example of spatio-temporal organisation emerging from a turbulent astrophysical system. In this talk I shall show some observations of astrophysical magnetic fields, before briefly reviewing our current understanding, highlighting limitations of our current theories, and the somewhat perplexing role of magnetic helicity conservation. I shall conclude by proposing possible avenues for future research suggested by recent results of dynamos with large-scale shear flows and helicity loss.

The magnetic field of the corona stores the energy that is released via magnetic reconnection during solar flares and coronal mass ejections (CMEs). Flares with CMEs are often described by the '˜standard' eruptive flare (CSHKP) model and this offers a conceptual framework in which to investigate the global characteristics of the energy release and transport in the context of the magnetic field configuration. The low plasma beta environment of the corona means the magnetic field plays a central role in the energy transport, and different magnetic field configurations can lead to a variety of outcomes in terms of the evolution of the energy release, the efficiency of the energy transport mechanisms and the locations where the energy is deposited. Despite the often rather good agreement between observations and the '˜standard' model, many open questions remain particularly in respect to the triggering of the energy release. In this talk I will discuss how multi-wavelength spectroscopy used in tandem with magnetic field information can help shed light on some of these open questions, and also how new facilities might provide new insights in the future.

Complex magnetic fields in plasmas may eventually relax to a simple state even if the resistivity is small. Nevertheless, what predicts the end state remains unclear. Using 3D magnetohydrodynamic (MHD) models, we find that there are two stages of topological evolution. The first is a fast reconnection phase constrained by the topological degree. The next is a slow phase dominated by diffusion and shows rearrangement and reconnection of discrete flux tubes. The end state always has two flux tubes with opposite twists, just as predicted by E. N. Parker. Meanwhile, we find the topological change can also be studied in 2D effective models at a low computational cost. Interestingly, the structural change in the reduced model during the first stage is similar to the reverse process of fluid mixing. The overall reduction in complexity is consistent with an optimal unstirred state.

Alfvénic waves have long been considered a leading participant in the transfer of energy around cool, magnetised stars' atmospheres; potentially responsible for the heating of the Sun's corona and the acceleration of the solar wind. In the early 2000's, various self-consistent models of Alfvénic wave turbulence from photosphere to heliosphere were developed that supported the role of wave dissipation. However, it wasn't until a few years later that Alfvénic waves were unambiguously observed in the Sun's corona, both in spectroscopic and imaging data. Since then, we have begun to probe the properties of the Alfvénic waves; providing key constraints for the wave turbulence heating models and challenging some of the long-held paradigms that these models rely upon. Here, I will discuss how ground-based observations of the corona in the Infrared are helping to uncover the behaviour of Alfvenic waves and the implications for our understanding of energy transfer via Alfvénic waves.

Active regions are the source of the majority of magnetic flux rope ejections that become Coronal Mass Ejections (CMEs) in the outer corona. To identify in advance which active regions will produce an ejection and when this ejection will occur is key for both space weather prediction tools and future science missions such as Solar Orbiter.

The aim of this study is to develop a new technique to identify which active regions are more likely to generate magnetic flux rope ejections. Once fully developed, the new technique will aim to: (i) produce timely space weather warnings and (ii) open the way to a qualified selection of observational targets for space-borne instruments.

We use a data-driven Non-linear Force-Free Field (NLFFF) model to describe the 3D evolution of the magnetic field of a set of active regions. From the 3D magnetic field configurations and comparison with observations, we determine a metric to distinguish eruptive from non-eruptive active regions based on the Lorentz force. Furthermore, using a subset of the observed magnetograms, we run a series of simulations to test whether the time evolution of the metric can be predicted.

We find that the identified metric successfully differentiates active regions observed to produce eruptions from the non-eruptive ones in our data sample. A meaningful prediction of the metric can be made between 6 to 16 hours in advance.

Additionally, we introduce a new operational metric that may be used in a ``real-time" operational sense were three levels of warning are categorised. These categories are: high risk (red), medium risk (amber) and low risk (green) of eruption. Through considering individual cases we find that the separation into eruptive and non-eruptive active regions is more robust the longer the time series of observed magnetograms used to simulate the build up of magnetic stress and free magnetic energy within the active region.

Magnetic helicity is a fundamental quantity of magnetohydrodynamics that carries topological information about the magnetic field. By `topological information', we usually refer to the linkage of magnetic field lines. For domains that are not simply connected, however, helicity also depends on the topology of the domain. In this paper, we expand the standard definition of magnetic helicity in simply connected domains to multiply connected domains in R^3 of arbitrary topology. We also discuss how using the classic Biot-Savart operator simplifies the expression for helicity and how domain topology affects the physical interpretation of helicity.

The talk is devoted to homogenization of periodic differential operators. We study the quantitative homogenization for the solutions of the hyperbolic system with rapidly oscillating coefficients. In operator terms, we are interested in approximations of the cosine and sine operators in suitable operator norms. Approximations for the resolvent of the generator of the cosine family have been already obtained by T. A. Suslina. So, we rewrite hyperbolic equation as parabolic system and consider corresponding unitary group. For this group, we adopt the proof of the Trotter-Kato theorem by introduction of some correction term and derive hyperbolic results from elliptic ones.

Consider a mixture of M non-interacting immiscible fluids under isothermal conditions at thermal equilibrium. The configurations seen in the experiments can be described as the (local) minimizers of a Gibbs free energy introduced by Van der Waals (later rediscovered by Cahn and Hilliard). In this model, a parameter $\epsilon$ describes the typical size of the interface regions separating areas of pure phases.

A mathematical challenge of the 80s was to understand the behaviour of the model as $\epsilon \to 0$. It was proved by Modica that the Van der Waals energy converges, in the sense of Gamma-convergence, to the surface energy of the interfaces separating the stable phases, as conjectured by Gurtin some years earlier.

In this talk a variant of the above model allowing for small scale heterogeneities in the fluids is presented. In particular, the case where the scale $\epsilon$ of the small homogeneities is of the same order of the scale governing the phase transition is considered. The interaction between homogenization and the phase transitions process will lead, in the limit as $\epsilon \to 0$, to an anisotropic interfacial energy.

The talk is based on a work in collaboration with Irene Fonseca (CMU), Adrian Hagerty (CMU), and Cristina Popovici (Loyola University).

To gain a better understanding of the formation and evolution of the pre-eruptive structure of CMEs requires the direct measurement of the coronal magnetic field, which is currently very difficult. An alternative approach, such as the simulation of the photospheric magnetic field must be used to infer the pre-eruptive magnetic structure and coronal evolution prior to eruption. The evolution of the coronal magnetic field of a small sub-set of bipolar active regions is simulated by applying the magnetofrictonal relaxation technique of Mackay et al. (2011). A sequence of photospheric line-of-sight magnetograms produced by SDO/HMI are used to drive the simulation and continuously evolve the coronal magnetic field of the active regions through a series of non-linear force-free equilibria. The simulation is started during the first stages of active region emergence so that the full evolution from emergence to decay can be simulated. A comparison of the simulation results with SDO/AIA observations show that many aspects of the observed coronal evolution of the active regions can be reproduced, including the majority of eruptions associated with the regions.

I will talk about the spectrum of double layer potential operators for 3D surfaces with rough features. The existence of spectrum reflects the fact that transmission problems across the surface may be ill-posed for (complex) sign-changing coefficients. The spectrum is very sensitive to the regularity sought of solutions. For L^2 boundary data, for domains with corners and edges, the spectrum is complex and carries an associated index theory. Through an operator-theoretic symmetrisation framework, it is also possible to recover the initial self-adjoint features of the transmission problem '“ corresponding to H^{1/2} boundary data '“ in which case the spectral picture is more familiar.

Discrete Lorenz attractors are chaotic attractors, which are the discrete-time analogues of the well-known Lorenz attractors in differential equations. They are true strange attractors, i.e. they do not contain simpler regular attractors such as stable periodic orbits. In addition, this property is preserved also under small perturbations. Thus, the Lorenz attractors, discrete and continuous, represent the so-called robust chaos. In the talk I will present a list of global (homoclinic and heteroclinic) bifurcations, in which it was possible to prove the appearance of discrete Lorenz attractors in the Poincare map. In some cases in was also possible to prove the coexistence of infinitely many attractors.

Vortices are the muscles of fluid motion. In quantum fluids, such as superfluid Helium and ultracold gases, these muscles are particularly simple, having fixed core size and circulation. This, along with the absence of viscosity, makes these fluids a highly idealised system to study vortex dynamics and turbulence. Moreover, recent experimental advances now enable precise, real-time monitoring of individual quantum vortices.

Here I will discuss our work in understanding the dynamics of quantum vortices. This will range from their microscopic behaviour, such as vortex nucleation and reconnection events between two vortices, to their macroscopic domain of collective structures and quantum turbulence. Throughout, I will relate our findings to the latest experiments and analogs in classical fluids. If time permits, I will also discuss an even more exotic fluid - the quantum ferrofluid.

Consider a mixture of M non-interacting immiscible fluids under isothermal conditions at thermal equilibrium. The configurations seen in the experiments can be described as the (local) minimizers of a Gibbs free energy introduced by Van der Waals (later rediscovered by Cahn and Hilliard). In this model, a parameter $\epsilon$ describes the typical size of the interface regions separating areas of pure phases.

A mathematical challenge of the 80s was to understand the behaviour of the model as $\epsilon \to 0$. It was proved by Modica that the Van der Waals energy converges, in the sense of Gamma-convergence, to the surface energy of the interfaces separating the stable phases, as conjectured by Gurtin some years earlier.

In this talk a variant of the above model allowing for small scale heterogeneities in the fluids is presented. In particular, the case where the scale $\epsilon$ of the small homogeneities is of the same order of the scale governing the phase transition is considered. The interaction between homogenization and the phase transitions process will lead, in the limit as $\epsilon \to 0$, to an anisotropic interfacial energy.

This is a work in collaboration with Irene Fonseca (CMU), Adrian Hagerty (CMU) and Cristina Popovici (Loyola University).

We review the Kolmogorov-Kraichnan spectra of turbulence in three and two dimensions, as well as the mathematical attempts to derive them from the Navier-Stokes equations. We then consider the simpler problem of deriving analogous spectra for passive tracers governed by advection-diffusion equations. This is largely work in progress with Mike Jolly (Indiana University).

The flow between differentially rotating cylinders, so-called Taylor-Couette flow, is one of the oldest problems in classical fluid dynamics. Taking the fluid to be electrically conducting, and applying axial and/or azimuthal magnetic fields opens up a range of new possibilities, so-called magnetorotational instabilities. I will review some of this work, including theoretical/numerical results, laboratory experiments, and astrophysical applications.

Accurate prediction of solar activity calls for precise calibration of solar cycle models. Consequently we aim to find optimal parameters for models which describe the physical processes on the solar surface, which in turn act as proxies for what occurs in the interior and provide source terms for coronal models. We use a genetic algorithm to perform the optimization, and apply it to both a 1D model that inserts new magnetic flux in the form of idealized bipolar magnetic regions, and also to a 2D model that assimilates specific shapes of real active regions. The genetic algorithm searches for parameter sets that produce the best fit between observed and simulated butterfly diagrams, weighted by a latitude-dependent error structure which reflects uncertainty in observations. We also approach the problem using powerful Bayesian emulation techniques, and compare the efficiency of the two methods.

The topology of coronal magnetic fields near the open-closed magnetic flux boundary is important to the the process of interchange reconnection, whereby plasma is exchanged between open and closed flux domains. Maps of the magnetic squashing factor in coronal field models reveal the presence of the Separatrix-Web (S-Web), a network of separatrix surfaces and quasi-separatrix layers, along which interchange reconnection is highly likely. Under certain configurations, interchange reconnection within the S-Web could potentially release coronal material from the closed magnetic field regions to high-latitude regions far from the heliospheric current sheet where it is observed as slow solar wind. It has also been suggested that transport along the S-Web may be a possible cause for the observed large longitudinal spreads of some impulsive, 3He-rich solar energetic particle events. Here we demonstrate that certain features of the S-Web reveal structural aspects of the underlying magnetic field, specifically regarding the arcing bands of highly squashed magnetic flux observed at the outer boundary of global magnetic field models. In order for these S-Web arcs to terminate or intersect away from the helmet streamer apex, there must be a null spine line that maps a finite segment of the photospheric open-closed boundary up to a singular point in the open flux domain. We propose that this association between null spine lines and arc termination points may be used to identify locations in the heliosphere that are preferential for the appearance of solar energetic particles and plasma from the closed corona, with characteristics that may inform our understanding of interchange reconnection and the acceleration of the slow solar wind.

We describe a systematic approach to (linear) instability theory for steady galaxies and stars as described by the Einstein-Vlasov and Einstein-Euler system respectively. One of the several outcomes of our analysis is a statement that galaxies with high central redshifts are dynamically unstable. This is a joint work with Zhiwu Lin (GeorgiaTech) and Gerhard Rein (Bayreuth).

The so-called Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase change, for example ice passing to water, and one aims to describe the regularity of the interface separating the two phases.

In its stationary version, the Stefan problem reduces to the classical obstacle problem, which consists in finding the equilibrium position of an elastic membrane whose boundary is held fixed, and that is constrained to lie above a given obstacle.

The aim of this talk is to give an overview of the classical theory of the obstacle problem, and then discuss some very recent developments on the optimal regularity of the free boundary both in the static and the parabolic setting.

The Crab Nebula is one of the most iconic astronomical objects which was, is and will keep making a very strong impact on the development of astrophysics for some time to come. The nebula was created by one of the historic supernovae almost two thousand years ago, but it is constantly invigorated by a powerful relativistic magnetised wind produced by the Crab pulsar. The interaction of this wind with the nebula leads to some spectacularly dynamic phenomena in its inner region: the famous jet, torus, wisps and few bright peculiar knots. Whether it is the dynamics of relativistic plasma, properties of relativistic shock waves, magnetic reconnection, or mechanisms of non-thermal particle acceleration, the Crab Nebula is a unique space laboratory to study all these phenomena which are important in many other areas of high energy astrophysics. In my talk, I will focus on some of the recent advances in the astrophysics of the Crab Nebula, describe what we have learned from these and what still remains poorly understood.

It has been known for almost two decades (starting with a 2000 paper by Zhikov) that elliptic partial differential operators with periodic high-contrast coefficients, describing certain composite materials, have band gap spectrum described by Zhikov's beta-function in the homogenisation limit. The homogenised operator is of the two-scale nature, it has a macroscopic and microscopic (corresponding to the period of the composite) parts. While the stochastic homogenisation is a well established area of mathematics, the high-contrast stochastic homogenisation has hardly been addressed, and it seems that there is not a single paper studying the spectral problems in high-contrast stochastic homogenisation. We initiate the research in this direction and show that similarly to the periodic case in the stochastic high contrast setting(under some lenient conditions) the spectrum has a band gap structure characterised by a function similar to Zhikov's beta-function, we study the limit two-scale operator with the stochastic microscopic component and prove the convergence of the spectra.

Solar flares are highly explosive events which release significant quantities of energy (up to 10^32 ergs) from specific magnetic configurations in the solar atmosphere. As part of this process, flares produce unique signatures across the entire electromagnetic spectrum, from radio to ultra-violet (UV) and X-ray wavelengths, over extremely short length and timescales. Many of the observed signals are indicative of strong particle acceleration, where highly energised electron and proton populations rapidly achieve MeV/GeV energies and therefore form a significant fraction of the energy budget of each event. It is almost universally accepted that magnetic reconnection plays a fundamental role (on some level) in the acceleration of particles to such incredible energies. In this talk, I will briefly summarise a series of recent experiments where test particles are introduced into a number of 3D magnetic reconnection configurations. I will discuss the particle population response to each configuration and what these responses might infer for both simulations and observations of magnetic reconnection in the flaring solar corona.

The multi-fluid nature of neutron star interiors leads to several processes that can drive magnetic field evolution despite the high electrical conductivity of the interior. Indeed, observations of neutron stars show that a significant fraction of young neutron stars are magnetically active. This includes the highly magnetized magnetars as well as high magnetic field radio pulsars. In this talk I will give an overview of the open questions regarding the magnetic fields of neutron stars and discuss recent work to understand how fields evolve in the solid crust of the star, where the non-linear Hall drift can act on short timescales and likely sources much of the observed magnetic activity.

A classic modelling task in Solar Physics is a boundary value problem: how to reconstruct the 3D magnetic field in the Sun's atmosphere given boundary data on the Sun's surface? The new generation of magnetic field models are time dependent, but this brings new problems as boundary data for the electric field, rather than just the magnetic field, are required. In this talk, I will present recent work on inverting Faraday's law: i.e., determining the electric field from observations of only the magnetic field. I will show that L1-minimization provides an elegant solution to this seemingly ill-posed inverse problem.

The Vlasov-Poisson system is a kinetic equation that models collisionless plasma. A plasma has a characteristic scale called the Debye length, which is typically much shorter than the scale of observation. In this case the plasma is called '˜quasineutral'. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling, the formal limit of the Vlasov-Poisson system is the Kinetic Isothermal Euler system. The Vlasov-Poisson system itself can formally be derived as the limit of a system of ODEs describing the dynamics of a system of N interacting particles, as the number of particles approaches infinity. The rigorous justification of this mean field limit remains an open problem. In this talk I will present recent joint work with Mikaela Iacobelli, in which we derive the Kinetic Isothermal Euler system from a regularised particle model. Our approach uses a combined mean field and quasineutral limit.

Motivated by astrophysical relativistic jets with curved streamlines, we study the onset and the evolution of the Relativistic Centrifugal Instability (RCFI). As a first step, we study axisymmetric rotating flows, where the density and angular velocity change discontinuously at a given radius. Following the original physical argument of Lord Rayleigh, we derive the relativistic version of the Rayleigh criterion for this problem and use axially symmetric computer simulations to verify its predictions. The inclusion of a uniform axial magnetic field can suppress the centrifugal instability for low flow velocities. However, in highly relativistic flows, such a field is no longer in equilibrium because of the electric field induced, and needs to be balanced by some extra pressure. This extra pressure term, in general, destabilises the flow.

In the setting of so-called evolutionary equations invented by Rainer Picard in 2009 we identify homogenisation problems as being equivalent to a certain type of a continuity property of solution operators. Indeed, it can be shown that $G$-convergence of matrix-coefficients is equivalent to convergence of certain inverses in the weak operator topology. With this, one can show various homogenisation results for a wide class of standard linear equations in mathematical physics. Furthermore, the genericity of memory effects to arise due to the homogenisation process in the context Maxwell's equations can be explained by operator-theoretic means.

In this talk I would like to present new results concerning the existence of smooth uniform attractors for nonautonomous damped wave equation with nonlinearities of quintic growth. It is well known that to prove even wellposedness of the wave equation in 3D with fast enough growing nonlinearities the only energy estimate is not enough and some extra estimates, known as Strichartz estimates, are required. To the best of our knowledge, previously these type of estimates, in the critical quintic case, were known only for the autonomous equation. We prove that Strichartz type estimates remain valid for the quintic wave equation with nonatunomous forcing. Furthermore, it appears that the forcing can be given by a vector-valued measure with bounded total variation. Based on these estimates we introduce several classes of "nice" external forces for which we show that the quintic damped wave equation possesses smooth uniform attractors. This is joint work with Sergey Zelik.

Rheology is the study of flowing complex materials that usually have a stress response governed by the presence of some microstructure within the material.

In my talk I will discuss the rheology of long chain polymers that entangle with themselves. I will show how coarse-grained models of these entangled polymer melts are necessary in order to characterise experimental measurements and hence deduce material microstructure. Furthermore, I will talk about a the work done on a recent impact funding award in collaboration with an adhesives company, Henkel. Henkel's problem is one of formulation, whereby there are thousands of degrees of freedom to consider when making a particular adhesive. The company needs to be able to predict how a particular formulation of adhesive will flow in the process they use to stick two films together. We used the models of entangled polymers in combination with 2D finite element simulations to predict if a given material would successfully survive the industrial process, i.e., that flow instabilities would not be a significant feature of the flow.

We discuss several temporal discretisation schemes and their applications to the Navier-Stokes equations. Of particular interest is the convergence of long-time statistics in the 2d case. We will also comment briefly on the situation in 3d.

A recent class of composite materials, known as Metamaterials, have gained much attention and interest in the Mathematics and Physics community over the last decade or so. These composites can roughly be characterised as exhibiting much more pronounced physical properties than their constituent components. These responses are due to scale-interaction effects.Mathematically, such metamaterial type effects could be rigorously justified and explained due to 'partial degeneracies' in underlying multi-scale continuum models.

In this talk, we shall introduce a notion of a partial degeneracy in parameter-dependent variational systems, motivated by examples from classical and semi-classical homogenisation theory, and present an approach to study the leading-order asymptotics of such systems. The determined asymptotics of the variational system can serve as effective models for phenomena due to multi-scale interactions and are given with order-sharp error estimates in the uniform operator topology.

This is joint work with Dr Ilia Kamotski(UCL) and Prof. Valery Smyshlyaev(UCL).

The notion of fatigue is founded on the concept of degraded or tired material and is linked to the observation of damage, a consequence of the loading and unloading cycles. It is apparent that fatigue produces progressive damage involving plastic deformation, crack nucleation, creep rupture and finally rapid fracture. So, damage is the consequence of the gradual process of mechanical deterioration, that basically results in a structural component failure. The evolution of damage will be described by the coefficient of a fractional derivative, that represents the phase field, satisfying the Ginzburg-Landau equation.

We will discuss several symmetrizations (Steiner, Ehrhard, and spherical symmetrization) that are known not to increase the perimeter. We will show how it is possible to characterize those sets whose perimeter remains unchanged under symmetrization. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when those sets whose perimeter remains unchanged under symmetrization, are trivially obtained through a rigid motion of the (Steiner, Ehrhard or spherical) symmetral. We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function for a special class of sets. These results are obtained together with several collaborators (Maria Colombo, Guido De Philippis, Francesco Maggi, Matteo Perugini, Dominik Stoger).

The Navier-Stokes equations describe the motion of fluids, with the two and three dimensional cases exhibiting certain very different characteristics. Kolmogorov's 1941 theory regarding the energy cascade in 3D turbulence and its 2D enstrophy analogue by Kraichnan in 1967 suggest that the behaviour of a fluid can be described by finite degrees of freedom, despite it being described by a PDE which is, fundamentally, infinite-dimensional. To develop this idea further, the idea of determining modes was introduced by Foias et al in 1983.

The beta-plane approximation is applied to simulate the effect that the earth's rotation has on the 2D NSE, where the rotation varies linearly with the latitude. Physical arguments and numerics indicate that the flow in such a simulation will be come zonal with time. Al-Jaboori and Wirosoetisno (2011) proved that the flow becomes more zonal with stronger rotation.

In this talk, I will introduce the concepts of determining modes and the beta-plane approximation, go over developments and improvements that have been made and cover some results that we have made by combining these two ideas.

The problem of quantization of a d-dimension probability distribution by discrete probabilities with a given number of points can be stated as follows: given a probability density $\rho$, approximate it in the Wasserstein metric by a convex combination of a finite number N of Dirac masses. In a paper in collaboration with E. Caglioti and F. Golse we studied a gradient flow approach to this problem in one dimension. By embedding the problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles tends to infinity. Under some suitable regularity assumptions on the density, we prove uniform stability and quantitative convergence result for the discrete and continuous dynamics.

In this talk I'll introduce some important concepts from the fashionable field of optimal transport theory. No previous knowledge of optimal transport theory is required, and the aim of the talk is to prepare the audience for future seminars on this topic.

The solar corona, the sun's hot outer atmosphere, is a hotbed of activity driven by continual changes in the magnetic field that permeates it. In the largest events, free energy is stored in the corona in the form of filaments '“ long snake-like features with highly sheared magnetic fields. These filaments can become unstable and erupt outwards into interplanetary space as Coronal Mass Ejections (CMEs). A multitude of smaller jet-like events are also present which launch hot tapered spires of plasma up from near the surface. Although smaller, these events are much more plentiful and extend down to the finest scales resolvable by current instruments. Until recently, jets and CMEs were thought to be distinct phenomena resulting from quite different mechanisms. However, the latest observations suggest that some jets are in fact miniature CMEs. In this talk I will introduce a new model for these mini-CME-type jets based on high-resolution MHD simulations. I'll discuss how this jet model is a natural extension of a prominent CME model and how this shows the direct link between the two phenomena for the first time.

This talk is in the area of mathematical modeling of materials science. We discuss the continuum limit of two discrete models for crystalline structures evolving on a flat substrate. The first model is based on microscopic Burton-Cabrera-Frank (BCF) models for stepped surfaces, and the second one is based on atomistic Solid-on-Solid (SOS) models. We prove that the BCF model is a finite difference scheme for a continuum PDE, and describe the macroscopic long term behavior and self similar solutions. For the SOS model, we use statistical mechanics techniques to prove that, in the discrete setting, a facet (flat face of the crystal) emerges as a consequence of the model. We hope to carry this over to the continuum setting.

Observations by the Hinode satellite of quiescent prominences have revealed in great detail the dynamics of plumes rising through the prominence material. These plumes, created by the magnetic Rayleigh-Taylor instability, rise through the prominence material. In this talk I will show how the growth rate for the linear instability and the nonlinear dynamics of the rising plume in the nonlinear regime provide diagnostic tools for investigating both the plasma beta and the magnetic field direction in the prominence. These methods will be compared to observations of plumes with both Doppler velocity and magnetic field measurements (both strength and direction) by Orozco Suarez et al (2014) to confirm their validity. It is through application of these new methods that I believe we will be able to make great strides in understanding the role of the magnetic field in the small-scale dynamics of quiescent prominences without the need for complex polarization measurements.

We introduce a new class of "filtered" schemes for some first order non-linear Hamilton-Jacobi equations. The work follows recent ideas of Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423-444, 2013) and Oberman and Salvador (J. Comput. Phys., Vol 284, pp. 367-388, 2015) for steady equations. Here we mainly study the time dependent setting and focus on fully explicit schemes. Furthermore, specific corrections to the filtering idea are also needed in order to obtain high-order accuracy. The proposed schemes are not monotone but still satisfy some epsilon-monotone property. A general convergence result together with a precise error estimate of order h^{1/2} are given (h is the mesh size). The framework allows to construct finite difference discretizations that are easy to implement and high-order in the domain where the solution is smooth. A novel error estimate is also given in the case of the approximation of steady equations. Numerical tests including evolutive convex and nonconvex Hamiltonians and obstacle problems are presented to validate the approach. We show with several examples how the filter technique can be applied to stabilize an otherwise unstable high-order scheme. This is joint work with O. Bokanowski and M. Falcone.

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The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the "holes" that a minimiser may have. The class of potentials for which we prove existence of minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. Finally, using Euler-Lagrange conditions on local minimisers we give a link to classical obstacle problems in the calculus of variations.

This is a joint work with J. A. Carrillo and F. Patacchini from Imperial College London.

A common pursuit is to optimise a number of control parameters in order to suppress or avoid the appearance of an instability, e.g. for drag reduction. In recent years, however, it has become possible to use similar variational methods as a means to determining the most direct route to instability. This involves the construction of a variational problem where the `parameter'-space is now huge, being the entire space of possible velocity fields. This field, if corresponding to a flow, must be subject to the constraints of boundary conditions and the governing equations.

The natural application to shear flows has enabled us to identify the minimal flow structures that lead to instability, i.e. to the transition to turbulence. For the dynamo instability one seeks a velocity field that leads magnetic energy growth, and it has been possible to put a lower bound on the `power' of the driving flow.

Whilst powerful at first sight, the variational method suffers a number of difficulties that the settings above highlight.

Pringle, C.T., Willis, A.P., and Kerswell, R.R., Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos, J. Fluid Mech., 702, 415-443 (2012).

Willis, A.P., Optimization of the magnetic dynamo, Phys. Rev. Lett. 109, 251101 (2012)

It is well known that the plastic, or permanent, deformation of a metal is caused by the movement of curve-like defects in its crystal lattice. These defects are called dislocations. What is not known is how to use this microscale information to make theoretical predictions at the continuum scale. A mathematical procedure that has proved to be very successful for the micro-to-macro upscaling of equilibrium problems in materials science is Gamma-convergence.

Macroscopic plasticity, however, is heavily dependent on dynamic properties of the dislocation curves. Motivated by this, M.G. Mora, M.A. Peletier and I recently upscaled a time-dependent system of discrete, interacting dislocations by combining Gamma-convergence methods with the theory of rate-independent systems. In the continuum limit we obtained an evolution law for the dislocation density. In this talk I will present this result and discuss its limitations and further extensions towards more realistic and complex systems.

We consider weak stationary solutions to the incompressible Euler equations and show that the analogue of the h-principle obtained by De Lellis and Szekelyhidi for time-dependent weak solutions continues to hold. The key difference arises in dimension d = 2, where it turns out that the relaxation is strictly smaller than what one obtains in the time-dependent case. This is joint work with Laszlo Szekelyhidi Jr.

In this talk I will present a work in collaboration with M. Bonnivard (Paris-Diderot) and F. Santambrogio (Orsay) where we propose an approximation for the Steiner problem (i.e. finding a connected set of minimal length joining a prescribed given set) by a family of elliptic type functionals, in the way of Modica-Mortola or Ambrosio-Tortorelli. The novelty in this work is a term of new type which is able to take care of the connectivity constraint.

In this talk, we will consider travelling waves in discrete systems. There are two models that will be discussed, i.e. coupled oscillators modelling a one-dimensional array of metamaterials and a discrete nonlinear Schrodinger equation modelling arrays of waveguides. In the first part of the talk, we will present the existence and uniqueness of periodic and asymptotic travelling waves in the first system. In the second part of the talk, we will discuss the second equation and informally study the existence of homoclinic orbits to saddle-points of the corresponding advance-delay equation, which are commonly referred to as embedded solitons.

This talk will introduce stochastic differential equations from scratch. Starting with Brownian motion I will introduce stochastic differential equations (SDEs) and a new numerical method to integrate the Stratonovich form. From there I will introduce stochastic partial erential equations (SPDEs) and how to compute travelling waves for SPDEs. Finally I plan to discuss a technique that freezes the wave and stops it from moving.

A moving vessel carrying a fluid can give rise to a wide range of complex and beautiful fluid motions. On the other hand, the motion of the interior fluid induces forces and moments on the vessel which can lead to unintended vessel motions and these motions could even lead to a destabilization of the vessel dynamics. One of the simplest experiments which demonstrates the fluid-vessel interaction is Cooker's pendulous sloshing experiment. This experiment consists of vessel with a rectangular cross-section which is partially filled with an inviscid, incompressible fluid and is suspended by two cables. The centre of mass of the system is allowed to rotate in a vertical plane, while the tank bed remains horizontal. This generates an irrotational fluid motion in the vessel.

In this talk we will show that this experimental setup contains an internal 1:1 resonance where the anti-symmetric fluid sloshing modes, which induce the vessel motion, have exactly the same frequency as the symmetric sloshing modes which occur in a stationary vessel. We also show that an exchange of energy between the vessel dynamics and the fluid motion can occur close to the 1:1 resonance when nonlinearity is considered.

We present a discrete model of cell motility in one dimension which incorporates the effects of volume filling and cell-to-cell adhesion. The continuum limit of the model is a nonlinear diffusion equation for the cell density, such that the diffusivity can turn negative if the adhesion coefficient is large. The consequent ill-posedness explains the pattern-forming behaviour observed in simulations of the underlying discrete model. The relationship between the discrete and continuum models is explored mathematically and numerically, and a Stefan-problem formulation of the continuum limit is proposed. Finally, we factor chemotaxis into the model, and show simulations of singular aggregation patterns arising from small initial data.

We study numerically phase separation in a binary fluid subject to an applied shear flow in two dimensions, with full hydrodynamics. For systems with inertia, we reproduce the nonequilibrium steady states reported previously by Stansell et al. [Phys Rev Lett 96 085701 (2006)]. The domain coarsening that would occur in zero shear is arrested by the applied shear flow, which restores a finite domain size set by the inverse shear rate. For inertialess systems, in contrast, we find no evidence of nonequilibrium steady states free of finite size effects: coarsening persists indefinitely until the typical domain size attains the system size, as in zero shear. We present an analytical argument that supports this observation, and that furthermore provides a possible explanation for a hitherto puzzling property of the nonequilibrium steady states with inertia. [SMF Phys Rev E 77 021504 (2008)]

Mimetic Finite Difference (MFD) methods are relatively new numerical techniques that have already been applied to the solution of problems in continuum mechanics, electromagnetics, gas dynamics and linear diffusion. They may be classified as standing in between Mixed Finite Element Methods and Finite Volumes. The idea behind this new discretisation technique is to define discrete operators by imposing that the essential properties of the underlying differential operators are preserved. For instance, when applied to linear diffusion problems written in mixed form, the discrete (mimetic) differential operators are defined imposing the Green's formula with respect to some discrete scalar products. In this way, conservation laws and solution symmetries are embedded in the method. Another crucial property of MFD is that very general polyhedral mesh elements can be handled, allowing for non-convex, degenerate polyhedrons, and even polyhedrons with curved faces. The flexibility in the mesh design gives an obvious advantage in the treatment of complex solution domains and heterogeneous materials. Moreover, allowing non-matching, non-convex mixed types of elements facilitates adaptive mesh refinement, particularly in the coarsening phase, making it a completely local process. The talk will overview the definition and features of MFD concentrating on linear diffusion problems. We shall demonstrate through extensive numerical examples the flexibility of the method, and present our recent analysis on the method's superconvergence properties and their use in a-posteriori error estimation. Finally, we shall present the first a priori analysis of the method applied to the solution of steady convection-diffusion problems.

Many surfaces in nature and industry develop coatings of unwanted material as a result of micro-organisms colonising them to form biofilms, or the conditions at the surface promoting reaction to form fouling layers that degrade the performance of the equipment. The economic cost and environmental impact of fouling and of cleaning (removing) can be substantial. There is therefore a need to understand the mechanisms by which these layers develop and decay, but experimental studies are often complicated by the fact that layers generated in a liquid environment are often highly porous and collapse when dried, or deform when contacted by a measurement stylus.

Our group has developed a simple technique to measure the thickness of soft fouling layers which allows us to locate the surface of the layer and thereby measure its thickness in situ and in real time. We can currently achieve measurement accuracies of +/- 10 micron which allows us to study the growth or swelling of biofilms, protein gels, milk-based foulants and polymer films. This fluid dynamic guaging technique exploits the flow characteristics of a siphon nozzle as it approaches a surface to locate the interface without touching it: it therefore mimics the operation of an atomic force microscope, albeit at micron length scales.

We have developed the technique (and stretched the analogy with atomic force microscopy) by using computational fluid dynamics simulations of the creeping and laminar flows of Newtonian fluids involved to calculate the flow field and thus estimate the stresses imposed on the surface. This allows us to determine the yield characteristics of the soft layers and other aspects of their microstructure.

The technique also works well when the bulk fluid is moving: we can track the development and destruction of fouling layers in ducts and thereby simulate real flow conditions. Our simulation work in this area has identified several challenges, which will be outlined is this presentation alongside some of the potential applications of the technique.

Results from two-dimensional direct numerical simulations of the governing equations that model incompressible fluid flow over a flat plate are presented. The Navier-Stokes equations are cast in a novel velocity-vorticity formulation (see Davies and Carpenter (2001)) and discretized with a mixed pseudospectral and compact finite-difference scheme in space, and a three-level backward-difference scheme in time.

A method to determine the envelope of a wavepacket (from numerical data) was developed. Based on the usual Hilbert Transform, new stages were incorporated to ensure a smooth envelope was found when the wavepacket was asymmetric.

The early transitional stages of the Blasius flow (flow over a flat plate with zero streamwise pressure gradient) are investigated with particular regard to a weakly nonlinear effect called wave-envelope steepening. Blasius flow is linearly unstable and so-called Tollmien-Schlichting modes develop. As nonlinearities become significant, the envelope of the wavepacket starts to develop differently at its leading and trailing edges. Numerical results presented here show that the envelope becomes steeper at the leading edge than it is at the trailing edge.

The effect of a non-zero streamwise pressure gradient on wave-envelope steepening is investigated by using Falkner-Skan profiles in place of the Blasius profile.

Natural transition is triggered by randomly-modulated waves. A disturbance with a randomly-modulated envelope was modelled and its effect on wave-envelope steepening was studied.

The higher-order Ginzburg-Landau equation was used to model the evolution of an envelope of a wavepacket disturbance. These results gave good qualitative comparison with the direct numerical simulations.

Finally, in preparation for developing a three-dimensional nonlinear version of the code, the discretization of one of the governing equations (the Poisson equation) was extended to three dimensions. Results from this new three-dimensional version of the Poisson solver show good agreement with those from an iterative solver, and also demonstrate the robustness of the numerical scheme.

I will address the usage of the elliptic reconstruction technique (ERT) in a posteriori error analysis for fully discrete schemes for evolution partial differential equations with particular focus on parabolic equations. A posteriori error estimates are effective tools in error control and adaptivity and a mathematical rigorous derivation justifies and improves their use in practical implementations.

The flexibility of the ERT allows a virtually indiscriminate use of various parabolic PDE techniques such as energy methods, duality methods and heat-kernel estimates, as opposed to direct approaches which leave less maneuver room. Thanks to the ERT, parabolic stability techniques can be combined with different elliptic a posteriori analysis techniques, such as residual or recovery estimators, to derive a posteriori error bounds. The method unifies and simplifies most previously known analysis, and it provides previously unknown error bounds (e.g., pointwise norm error bounds for the heat equation). [Results with Ch. Makridakis and A. Demlow.]

A further feature of the ERT, which I would like to highlight, of the ERT is its simplifying power. It allows to derive estimates where the analysis would be dautingly complicated. As an example, I will illustrate its use in the context of non-conforming methods, with a special eye on discontinuous Galerkin methods [with E. Georgoulis] and "ZZ" recovery-type estimators [with T. Pryer].

In this talk, I will outline a general recent methodology for solving boundary value problems for linear and integrable nonlinear PDEs in two variables. I will focus on the case of of evolution problems (hence one variable always models time). I will discuss the case of classical boundary value problems posed on a finite or semi-infinite interval, as well as the case when such a problem is posed in a time-dependent domain. In both cases, I will illustrate how this methodology yields new fast strategies for evaluating the solution numerically. This work is in collaboration with A.S. Fokas and my student S. Vetra.

"The Variational Particle-Mesh method is a general method for discretising fluid equations which can be derived from a variational principle. The method encodes the relationship between Lagrangian and Eulerian fluid mechanics. I will discuss results for the N-dimensional Camassa-Holm equation where it is possible to show that the method has a discrete particle relabelling symmetry which leads to a set of conservation laws that are related to Kelvin's circulation theorem. If there is time I will also discuss a new application to medical imaging. "

"I will discuss two topics involving the efficient evaluations of theta functions connected with algebraic curves and integrable systems. These are of interest on both practical and theoretical grounds. One is the use of the Richelot transformation to evaluate genus two hyperelliptic integrals, a generalization of the Algebraic-Geometric Mean of Gauss. The other is the study of reducible period matrices, when the algebraic curve is a cover of one or more of lower genus. In this case the higher genus theta function can be written as a sum of products of lower genus (often g=1) theta functions. "

"We explore some recent topics of interest in both linear and nonlinear wave propagation. We do so in the context of problems from continuum mechanics which involve shear (or transverse) waves, compressional (or longitudinal) waves, and kinematic waves. Specifically, the following three topics will be considered: Instant steady-state and Stokes' second problem of dipolar fluids; Nonlinear acoustics in Darcy-type porous media: The transition from acceleration to shock waves; and Growth and decay of shock waves in a traffic flow model with relaxation. Employing both analytical and numerical techniques, we carry out this investigation with the purpose of gaining a better understanding of, and deeper insight into, the physical phenomena represented in the mathematical models. (Work supported by ONR/NRL funding.)"

"Geotechnical engineering is concerned with the response of existing rock and soil features to new constructions, and the behaviour of the structures themselves. This includes tunnels, retaining walls and foundations. Two problems usually require solutions: determination of movements during and after construction and assessment of overall stability. The former problem is often tackled using conventional finite element techniques. Geomaterials (e.g. soil and rock) are, however, very poorly modelled with linear elasticity and complex non-linear elasto-plastic models, devised especially for geomaterials, are necessary. In addition, modelling often includes many loading stages and the simulation of processes such as tunnel lining and drainage. These factors togethre with the size of many of the models makes finite element modelling in geotechnics unlike most other areas of civil engineering and much more challenging. This talk will highlight some of these issues using examples from my research over the past few years. In addition I will also discuss the modelling of collapse situations in geotechnics, using finite element limit analysis techniques."

"Multigrid methods are among the fastest algorithms for the solution of linear systems of equations Ax=b. For many problems the computational efforts for the multigrid solution of the linear system are of the same complexity as the multiplication of a vector with the matrix A. This talk deals with multigrid algorithms for structured linear systems. Particular focus is put on Toeplitz matrices, i.e. matrices with entries constant along diagonals. For the case of Toeplitz systems generated by nonnegative functions with a finite number of zeros of finite order new multigrid algorithms are proposed and efficiently implemented. It is pointed out why these algorithms are computationally superior to existing approaches. Imaging applications are the most important practical source of Toeplitz systems: I will focus on Fredholm integral equations of the first kind arising from image deblurring. For the resulting discretization matrices a multigrid algorithm employing a natural coarse grid operator is implemented which improves on an existing approach by R.Chan, T.Chan and J.Wan. Finally, it will be explained how the new method can be viewed in the context of established multigrid approaches for Fredholm integral equations of the second kind. "

"Two different approaches to mathematical modelling in renal research, done in collaboration with the Renal Unit at Glasgow's Western Infirmary, will be discussed. The first approach uses a discrete event simulation that has been developed to predict the benefits to be gained by correcting for potentially remediable cardiovascular risks. The model is based on a retrospective study of renal graft patients from the West of Scotland. The second approach discusses mathematical models for progressive renal disease. To date, the literature on such models is scarce. However, one such model exists, developed using a simple dynamical systems approach. This is discussed and compared with my own model, which I have recently been developing."

" DIGM occurs when a thin polycrystalline specimen made of one metal (the solvent) is put in the vapour of another metal (the solute). The solute diffuses into the specimen along grain boundaries and sets up elastic stresses which cause the grain boundary to move. The shape of the moving grain boundary is determined by a differential equation which balances this elastic force against the forces due to the curvature of the boundary and the resistance to its motion. Meanwhile, the shapes of the surfaces of the two grains also satisfy differential equations, because solvent atoms diffuse along the surface and their chemical potential depends on its curvature. The DE's for the grain boundary and the two grain surfaces are coupled by conditions at the triple junction where they meet. I will show some cases where the resulting mathematical problem can be solved to determine such things as the speed of a steadily moving grain boundary."

"A fully practical finite element approximation of an Allen-Cahn/Cahn-Hilliard system with a degenerate mobility and a logarithmic free energy is considered. The system arises in the modelling of phase separation and ordering in binary alloys. In addition to showing well-posedness and stability bounds for the approximation, convergence in one space dimension is proved. Finally some numerical simulations will be presented. "

"This talk focuses on the dynamics of scroll waves in three-dimensional reaction-diffusion systems with excitable reaction kinetics. Results will be presented from three different approaches to understanding the dynamics of these waves: direct numerical simulations, bifurcation analyses, and singular perturbation methods."

"The seminar consists of two parts: (a) A versatile Liapunov functional appropriate to nonlinear diffusion is discussed. (b) A class of (cross-sectional) velocity - acceleration relations is discussed for steady flow of an incompressible continuum. "

" The pulmonary airways are lined with a thin liquid film which affects many aspects of the lung's mechanical behaviour. In the smaller airways, surface tension induces a large pressure jump across the film's highly curved air-liquid interfaces. The resulting compression of the elastic airway walls can lead to significant wall deformations and causes a strong interaction between fluid and solid mechanics. This talk will provide an overview of recently developed computational models of such problems and will illustrate their applications to airway closure and reopening. "

"This talk will discuss Grid Computing with particular reference to its use in Applied Mathematics. After a brief introduction to the main ideas behind the Grid, the Netsolve system developed at the University of Tennessee, Knoxville, will be used as an example of an interface to remote computing resources. The Triana system developed at Cardiff University will also be described. This will lead on to a discussion of Application Service Providers as a model for accessing remote computational resources. The importance of incorporating existing ?legacy? software into component-based distributed computing frameworks will be stressed, and the use of the Java-C Automatic Wrapper tool (JACAW) for doing this will be presented. Finally, the potential for using these types of tools and systems to develop high-level problem-solving environments will be assessed."

"the 16th in a series of annual lectures on the History and Philosophy of Mathematics given in memory of Sir Edward Collingwood, FRS"

" We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operator. We show that the leading term in the spectral asymptotics is closely related to a certain convex polygon, and that the spectrum does not determine the operator up to similarity. Two elliptic systems which only differ in their boundary conditions may have entirely different spectral asymptotics."

"We consider attractors of parabolic equations on large and unbounded domains. We define the epsilon-entropy which is a measure of the complexity of the attractor, similar to a dimension per unit volume. We show how to get bounds on the epsilon-entropy for both PDEs in unbounded domains and their numerical approximations."

"Liquid crystalline phases typically occur between conventional crystal and liquid phases. Many of the properties they exhibit are intermediate between liquids (e.g. flow properties) and crystalline materials (e.g. anisotropic properties). The coupling between these give rise to many interesting physical phenomena. In particular, electrooptic and magnetooptic effects can occur. Here, changes to an applied electric or magnetic field produce a change in molecular ordering and consequently induce a change in the optical response of the liquid crystal. Such responses have been harnessed in several well-known devices, including the twisted nematic display (TND) (common in everything from mobile phones to laptop computer displays), adaptive optic devices for telescopes, and switchable windows. This talk describes work carried out into the study of liquid crystalline phases using computer simulation methods. Two well-known techniques have been employed, molecular dynamics and Metropolis Monte Carlo. These methods have been used to simulate the phase behaviour of a range of simple models. The talk will show that simple single-site anisotropic models are sufficient to demonstrate the presence of liquid crystalline phases and that these models can be used to develop methods for predicting key material properties. More sophisticated models, which represent the molecular structure of individual molecules, can also be employed. Although, expensive in terms of computer time, these models can be used to provide an insight into how changes in chemical structure influence phase behaviour and material properties. It is shown that such models can provide a path to "Molecular Engineering", whereby in the future, it may be possible to design molecules that have the desired physical properties, starting only from knowledge of their molecular structure and their interactions. "

A method for quantifying the solution error in compact finite-difference schemes has been used to design optimal and near-optimal schemes for progressively more complicated decay/advection/diffusion problems. The new schemes achieve given levels of accuracy with larger time steps and wider grid spacing. The required computational resources can be as small as 0.001 of traditional schemes.

"Elliptic problems with gradient constraints arise naturally in the elastoplastic torsion of a bar. These problems are well understood in the framework of variational inequalities and may correspond to quasi-variational inequalities if the gradient bound depends on the solution itself. While the stationary problems may be solved by fixed point techniques, once the continuous dependence on the convex set is shown, the corresponding evolutionary problems are much more delicate to analyse, as those arising in some superconductivity models. For instance, in the Bean critical state problem, the gradient of the average magnetic field is constrained by a critical value. When this value is constant, the mathematical problem consists of a parabolic variational inequality which can be analysed and computed developing standard functional and numerical methods. In the physical variant in which that threshold may be a function of the magnetic field, the corresponding model is an evolutionary quasi-variational inequality. We survey a few known results for the variational problems and some of its variants, we describe the recent existence result for the quasi-variational inequality, obtained in a joint work with Lisa Santos."

"It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretization. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this work, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretizations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions that are oscillatory when the mesh Peclet number is large. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Peclet number and boundary conditions of the problem. When streamline upwinding is included in the discretization, we show precisely how the amount of upwinding included in the discrete operator affects solution oscillations and accuracy when boundary layers are present. Joint work with Alison Ramage of The University of Strathclyde."

"In many applications it is desirable with high aspect ratio elements because of anisotropic behaviour of the solution. This is typically the case in problems involving for instance boundary layers. However, standard a posteriori error estimators for finite element methods do not work well for such elements. They typically over estimate the error. This means that they may lead to refinements in ""wrong"" areas if they are used for adapting the grid, and they may suggest an over refined grid to get the error down to the desired level. Different error estimators and possible remedies for the problems stated above will be discussed."

"We discuss the computation and numerical continuation of connections to periodic orbits, implementing in Auto97 (Doedel et al) projection boundary conditions proposed by Beyn 1993. As an illustrative example we consider the Baer--Rinzel model of a single neuron which models active spines (with Hodgkin--Huxley dynamics) linked by a diffusive cable. The model is known to support various travelling waves: solitary waves, periodic waves, multi-bump waves. Physically, heteroclinic connections between periodic solutions correspond to waves connecting periodic spike trains. We examine propagation failure as parameters such as resistance and spine density vary."

Stochastic delay differential equations (SDDEs) generalise deterministic delay differential equations as well as stochastic ordinary differential equations. Applications of SDDEs may be found for example in physiological systems and finance. In this talk I will give an overview over analytical and numerical methods and some concepts of the relevant stability analysis.

The talk will give an overview of some of the work being done at Strathclyde on adaptive methods. It will focus on Hermite collocation solution of near-singular problems using coordinate transformations based on adaptivity. A coarse grid is generated by an adaptive finite difference method and this grid is used to construct a coordinate transformation that is based on monotonic cubic spline approximation. An uneven grid is then generated by means of the coordinate transformation and the differential problem is solved on this grid using Hermite collocation. Numerical results are presented for steady and unsteady problems in 1D and for steady problems in 2D.

"We describe a parallel algorithm for the finite element solution of a general class of elliptic partial differential equations in three dimensions based upon a weakly overlapping domain decomposition preconditioner. The approach, which extends previous work for two-dimensional problems, is briefly described and analysed, and numerical evidence is provided to demonstrate the potential of our proposed parallel algorithm. "

"The general N-body problem (N interacting point masses or rigid bodies) is an ubiquitous component of modern chemical, physical and engineering research. For simulation of long time-interval dynamics, improved stability is often obtained by using a ""geometric integrator,"" for example a symplectic or time-reversible method. In this talk, I will describe new geometric integration methods for rigid body and particle systems subject to molecular forces, hard-walls and Coulombic collisions. "

"I will review a convergence theory of numerical approximations of stochastic differential equations. The theory concerns the approximation of long time(ergodic) properties of the underlying model. A motivating example is used throughout the talk, so called Dissipative Particle Dynamics, which is used inindustry to study phase formations in polymer mixtures. The convergence theory has been applied more generally to parabolic PDEs and impulsed ODEs. "

"Observations of cooperative behaviour among unrelated individuals (e.g. food sharing, grooming, predator inspection) are of great interest to evolutionary biologists, not least because such altruism appears open to abuse (receiving help without giving it). The classical Prisoners' Dilemma game, in which players repeatedly choose to cooperate (C) or defect (D), remains the central tool for identifying the types of cooperative strategy that might evolve in the natural world, but it is often criticised by empiricists. For instance, in the real world individuals can vary their investments in partners, so the option of co-operation is rarely all or nothing. Similarly, defection is rarely more than the passive strategy of not cooperating. In this talk, I present a simple model of biological trade which is consistent with the Prisoners' Dilemma model, but which allows individuals to vary their cooperative investments. In reformulating the model in this way, it is clear that whole new ways of cheating are possible, such undercutting your partner's investment. Despite these potentially erosive forces, I show that cooperation can still thrive under these conditions, and that it is likely to do so by strategies which both build up trust in partners and which react quantitatively to partners that short-change them.In the final section of my talk I present the results of more recent analyses in which I demonstrate: (i) that cooperative interactions can still occur despite wide variation in the frequency of needing help, and (ii) that individuals who can't (rather than won't!) reciprocate are likely to play an important role in enhancing the evolutionary stability of cooperation."

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It was established by Koike how to use the commuting actions of U(n) and S_k x S_l on tensor powers of C^n and it's dual space (C^n)* to explicitly construct subspaces that are isomorphic to certain irreducible representations of U(n). I will discuss a similar construction, using the commuting actions of S_n and S_k on the k'th tensor power of C^n to construct subspaces that are isomorphic to certain irreducible representations of S_n x S_k. Time permitting, I will also discuss some applications of this work towards the study of the expected character of random permutations for 'stable' irreducible representations of S_n.

In recent work Le, Le Hung, Levin and Morra have proved a generalisation of Breuil’s Lattice conjecture in dimension three. This involved showing that lattices inside representations of GL3(Fp) coming from both a global and a local construction coincide. Motivated by this we consider the following graph. For an irreducible representation ô of a group G over a finite extension K of Qp we define a graph on the OK-lattices inside ô whose edges encapsulate the relationship between lattices in terms of irreducible modular representations of G (or Serre weights in the context of the paper by Le et al.).

In this talk, I will demonstrate how one can apply the theory of graduated orders and their lattices, established by Zassenhaus and Plesken, to understand the lattice graphs of residually multiplicity free representation over suitably large fields in terms of a matrix called an exponent matrix. Furthermore I will explain how I have been able to show that one can determine the exponent matrices for suitably generic representation go GL3(Fp) allowing us to construct their lattice graphs.

The central value of the L-function of an elliptic curve has been the object of extensive studies in the last 50 years. Associated with such a curve we wish to understand also families of twists of it, leading to the study of twisted L-functions.

On the other hand, modular symbols have been a useful tool to study the space of holomorphic cusp forms of weight 2, and the homology of modular curves. They have been the object of extensive investigations by many mathematicians including Birch, Manin, and Cremona.

Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions.

We discuss some of these conjectures and the recent progress and resolution of them.

I will introduce the doubling method for automorphic L-functions of classical groups. In most work concerning special L-values so far, only the partial L-function is considered. Using the doubling method, Yamana defined the `correct’ complete L-function. That is, he defined the ramified local L-factors and proved that the global L-function has meromorphic continuation and satisfies a functional equation. However, these local L-factors are not given explicitly, and his approach cannot be used to study the algebraicity. As my motivation is to study algebraicity/p-adic properties, I construct the ramified L-factors explicitly so that the algebraicity result for partial L-functions can be easily extended to complete L-functions.

We introduce spectral projectors on a torus as a restriction problem with exponential sums, and relate them to other problems including eigenfunction growth estimates and the Gauss circle problem. We discuss a number of approaches for arbitrary or generic tori with roots in number theory.

The spectrum of a commutative unitary ring has a natural partial order induced by inclusions of prime ideals. More generally, we can use this idea to introduce a partial order on any scheme or Kolmogorov space. This allows us to view these spaces as directed graphs. Many interesting arithmetic invariants of varieties such as Berkovich skeleta can be captured purely in terms of this poset structure. In this talk, I will show how to reconstruct this structure using Galois orbits of power series. I will introduce relative decomposition groups, local gluing data, and multivariate symbolic Newton-Puiseux algorithms to compute these posets in many cases of interest. If time permits, I will also show how tropical geometry can be used to find these posets.

I will present recent joint work https://arxiv.org/abs/2203.09434 with Kris Klosin (CUNY) on the modularity of residually reducible ordinary 2-dimensional p-adic Galois representations with determinant a finite order odd character. When this finite order character is quadratic we prove modularity by classical CM weight one forms, otherwise by non-classical weight 1 specialisations of Hida families.

I will report on recent work which shows, under technical hypotheses, that cuspidal automorphic Galois representations over CM fields are rigid in the sense that they have no deformations which are de Rham. This property is predicted by the Bloch-Kato conjecture and the proof uses the Taylor-Wiles method.

Shanks' Conjecture states that the derivative of the Riemann zeta function is, on average, real and positive when evaluated at the non-trivial zeros of the zeta function. (At each individual zero, \zeta'(rho) is complex). A proof of the conjecture was already found in the 1980's. This talk, which is based on joint work with Andrew Pearce-Crump, will give a very simple heuristic using the Landau-Gonek explicit formula that yields the same result, as well as showing how it can be generalised to higher derivatives.

Let \pi be a p-ordinary cohomological cuspidal automorphic representation of GL(n,A_Q). A conjecture of Coates--Perrin-Riou predicts that the (twisted) critical values of its L-function L(\pi x\chi,s), for Dirichlet characters \chi of p-power conductor, satisfy systematic congruence properties modulo powers of p, captured in the existence of a p-adic L-function. For n = 1,2 this conjecture has been known for decades, but for n > 2 it is known only in special cases, e.g. symmetric squares of modular forms; and in all previously known cases, \pi is a functorial transfer via a proper subgroup of GL(n). In this talk, I will explain what a p-adic L-function is, state the conjecture more precisely, and then describe recent joint work with David Loeffler, in which we prove this conjecture for n=3 (without any transfer or self-duality assumptions).

One of the most important results in 20th century number theory is Linnik's theorem, which states that if q is a large modulus, then each invertible residue class mod q contains a prime less than q^L, where L is some absolute constant. In this talk we investigate similar results concerning E_k numbers for small k, where an E_k number is a product of exactly k primes. In particular, we show that each invertible residue class mod q contains a product of three primes, where each prime is less than q^{6/5+epsilon}

An integer is y-smooth (or y-friable) if all its prime factors are at most y in size. Such numbers play an important role in computational number theory. A similar definition, with similar applications, applies for smooth polynomials over a finite field. It has been known for more than 90 years that the densities of smooth numbers and polynomials grow (in certain parameter ranges) like the Dickman function, which will be defined in the talk.

I will survey the tools for studying smooth numbers and polynomials. I'll explain my work (in progress), which studies the relationship between smooth numbers/polynomials and the Dickman function by introducing a new approximation for the number of smooth numbers/polynomials.

In particular, we make progress, conditionally on the Riemann Hypothesis, on the following question of Pomerance: Is the density of smooth numbers always larger than the Dickman function? This turns out to relate, in a subtle way, to the error term arising when counting prime numbers.

I will present recent joint work (https://arxiv.org/abs/2203.09434) with Kris Klosin (CUNY) on the modularity of residually reducible ordinary 2-dimensional p-adic Galois representations with determinant a finite order odd character. When this finite order character is quadratic we prove modularity by classical CM weight one forms, otherwise by non-classical weight 1 specialisations of Hida families.

Let q be a large prime. It is an old and classical problem to understand the distribution of quadratic residues and non-residues modulo q. According to an old and famous conjecture of I.M. Vinogradov, the least quadratic non-residue n modulo q should satisfy n 0, when q is large enough. This statement would be implied by non-trivial upper bounds for averages of the Legendre symbol (n/q) with n 1/4, due to the potential obstruction, difficult to rule out, that (n/q) = +1 for many initial integers n. In this talk I will discuss a generalisation of Vinogradov's conjecture to other primitive Dirichlet characters \chi modulo q, seeking the least n for which \chi(n) is not 1. I will explain some recent work of mine that shows, using techniques from additive combinatorics, that when the order d of \chi grows with q the aforementioned obstruction does not occur, that the analogue of Vinogradov's conjecture for \chi does hold, and that moreover \chi(n) = 1 with n 0. I will also discuss some results related to showing cancellation in short sums of \chi(n) with n 0 arbitrarily small, going beyond Burgess' estimate.

One of the most basic classes of algorithmic problems in combinatorial optimization is the computation of shortest paths. Tropical geometry is a natural language to analyze parameterized versions where some of the arc weights are unknown. I will introduce this point of view, with a link to polyhedral geometry. Moreover, I will present an algorithm that computes these parameterized shortest paths and apply it to real world data from traffic networks. My talk is based on joint work with Michael Joswig.

This talk will take place in MCS2068 (ignore the generic room below).

The Poincaré-type series mentioned in the title refer to "nice" vector valued functions defined on the complex upper half-plane which transform in a suitable way with respect to a multiplier system of real weight k under the action of a Fuchsian group of the first kind. As we will explain, they are very closely related to certain automorphic kernels which admit a spectral expansion with respect to the eigenfunctions of the hyperbolic weighted Laplacian of weight k. Following an approach of Jorgenson, von Pippich and Smajlović (where k=0), we use spectral expansion associated to the Laplacian to first construct wave distribution and then identify conditions on its test functions under which it represents automorphic kernels. As we will see, one of advantages of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. This talk presents joint work with Y. Kara, M. Kumari, K. Maurischat, A. Mocanu and L. Smajlović.

LetLbe a complete discrete valuation field of primecharacteristicpwith finite residue field. Denote by Γ^(v)_L the ramification subgroups of Γ_L= Gal(L^{sep}/L). Consider the category MΓ^{Lie}_L of finite Z_p[Γ_L]-modules H, satisfying some additional (Lie)-condition on theimage of Γ_L in Aut_{Z_p}H. We prove that all information about the images of the ramification subgroups Γ^(v)_L can be explicitly extracted from some differential forms Ω[N] on the Fontaine etale φ-module M(H) associated with H. The forms Ω[N] are completely determined by a connection ∇ on M(H). In the case of fields L of mixed characteristic containing a primitive p-th root of unity the similar problem for F_p[Γ_L]-modules also admits a solution. In this case we use the field-of-norms functor to construct the coresponding φ-module together with the action of a cyclic group of order p coming from a cyclic extension of L. Then the solution involves the above characteristic p part (provided by the field-of-norms functor) and the condition for a “good” lift of a generator of the involved cyclic group of order p. Apart from the above differential forms the statement of this condition also uses a power series coming from the p-adic period of the formal group G_m. This result establishes a link between the Galois theory of local fields and very popular area of D-modules, lifts of Frobenius, Higgs vector bundles etc.

A rigid meromorphic cocycle is a class in the first cohomology of the group SL_2(Z[1/p]) acting on the non-zero rigid meromorphic functions on the Drinfeld p-adic upper half plane by M ̈obius transformation. Rigid meromorphic cocycles can be evaluated at points of “real multiplication”, and their values conjecturally lie in composita of abelian extensions of real quadratic fields, suggesting striking analogies with the classical theory of complex multiplication. In this talk, we discuss the proof of this conjecture for a special class of rigid meromorphic cocycles. Our proof connects the values of rigid meromorphic cocycles to the study of certain p-adic variations of Hilbert modular forms. This is joint work with Henri Darmon and Jan Vonk.

We study sums of one prime and two squares of almost-primes, that is to say, integers whose number of prime factors is less than some absolute fixed bound. We also consider sums of one smooth number and two squares of almost-primes. We employ three main tools: an explicit formula for the number of representations of an integer by a binary quadratic form; results on additive problems with cusp forms which derive ultimately from a trace formula; and a lower-bound sieve which in the case of smooth numbers takes a somewhat nonstandard form. This is joint work with Valentin Blomer and Junxian Li.

Since the seminal works of Wiles and Taylor-Wiles, robust methods were developed to prove the modularity of 'polarised' Galois representations. These include, for example, those coming from elliptic curves defined over totally real number fields. Over the last 10 years, new developments in the Taylor-Wiles method (Calegari, Geraghty) and the geometry of Shimura varieties (Caraiani, Scholze) have broadened the scope of these methods. One application is the recent work of Allen, Khare and Thorne, who prove modularity of a positive proportion of elliptic curves defined over a fixed imaginary quadratic field. I'll review some of these developments and work in progress with Caraiani which has further applications to modularity of elliptic curves over imaginary quadratic fields.

The theory of unlikely intersections makes predictions about how endomorphism algebras vary in families of abelian varieties. I will explain some of these predictions and outline methods used to prove results of this type using reduction theory of arithmetic groups.

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The strong form of Serre's modularity conjecture states that every two-dimensional continuous, odd, irreducible mod p representation of the absolute Galois group of Q arises from a modular form of a specific minimal weight, level and character. We show this minimal weight is equal to two other notions of minimal weight, one inspired by work of Buzzard, Diamond and Jarvis and one coming from p-adic Hodge theory. We discuss the interplay between these three characterisations of the weight for Galois representations over totally real fields and investigate the consequences for generalised Serre conjectures.

In this talk we will discuss Montgomery's pair correlation conjecture for the zeros of the Riemann zeta function. Building on work of Goldston and Gonek (and others) from the late 1990s, we will discuss our new (conditional) partial result concerning the Fourier transform of this pair correlation function. This topic spans both analytic and combinatorial elements of the study of the distribution of primes; the new technical ingredient is a correlation estimate for Selberg sieve weights for which the level of support is beyond the classical square-root barrier.

Kac-Moody groups are generalisations of groups of Lie type, defined via generators (depending on elements of a field) and Steinberg relations, using a generalised Cartan matrix. Such a group is said to be affine when the corank of this matrix is 1, and many of the Dynkin diagrams of these groups can be obtained by adding one extra node to the Dynkin diagram of a finite dimensional simple Lie algebra. Affine Kac-Moody groups over a finite field are finitely generated, but their standard presentation is infinite. This talk will give a brief introduction to these groups, and then show how one can find presentations of bounded length for them by considering overlapping subdiagrams of their Dynkin diagrams.

The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points. The Brauer--Manin pairing cuts out a subset of the adelic points, called the Brauer--Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer--Manin set is empty then we say there's a Brauer--Manin obstruction to the existence of rational points on X. Computing the Brauer-Manin pairing involves evaluating elements of the Brauer group of X at local points. If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p. For p-torsion elements this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision. Classifying p-torsion Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Bloch and Kato. Applications of our work include addressing Swinnerton-Dyer's question about which places can play a role in the Brauer-Manin obstruction. This is joint work with Martin Bright.

(Joint with O. Gorodetsky and B. Rodgers) It is a classical quest in analytic number theory to understand the fine-scale distribution of arithmetic sequences such as the primes. For a given length scale h, the number of elements of a "nice" sequence in a uniformly randomly selected interval (x,x+h], 1 ≤ x ≤ X, is expected to follow the statistics of a normally distributed random variable (in suitable ranges of 1 ≤ h ≤ X). Following the work of Montgomery and Soundararajan, this is known to be true for the primes, but only if we assume several deep and long-standing conjectures such as the Riemann Hypothesis. In fact, previously such distributional results had not been proven for any sequence of number-theoretic interest, unconditionally.

As a model for the primes, in this talk I will address such statistical questions for the sequence of squarefree numbers, i.e., numbers not divisible by the square of any prime, among other related "sifted" sequences called B-free numbers. I hope to further motivate and explain our main result that shows, unconditionally, that short interval counts of squarefree numbers do satisfy Gaussian statistics, answering several old questions of R.R. Hall.

Let G be a finite group, let E/Q be an elliptic curve, and fix a finite-dimensional Q[G]-module V. Let F/Q run over all Galois extensions whose Galois group is isomorphic to G and such that E(F) tensor Q is isomorphic to V as a G-module. Then what does E(F) itself look like "on average" in this family? I will report on joint work with Adam Morgan, in which we consider a particular special case of this general question. We propose a heuristic that predicts a precise answer in that case, and make some progress towards proving it. Our heuristic turns out to be an elliptic curve analogue of Stevenhagen's conjecture on the solubility of negative Pell equations.

Elliptic polylogarithms are analytic functions which play an important role in the study of special values of L-functions of elliptic curves. In the 2000s, Levin-Racinet (and later Brown-Levin) generalized these to multiple elliptic polylogarithms, which are likewise conjectured to be of deep arithmetic-geometric interest.

The original definition of multiple elliptic polylogarithms is analytic and uses the complex uniformization of the underlying elliptic curve in an essential way. The goal of this talk is to give an algebraic-geometric definition of multiple elliptic polylogarithms which gives back the original definition after analytification. This is joint work in progress with Tiago J. Fonseca (Oxford).

In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $p$ the density of entries in the character table of $S_n$ that are divisible by $p$ goes to $1$ as $n$ goes to infinity. I’ll describe a proof of this conjecture, which is joint work with K. Soundararajan. I will also discuss the (still open) problem of determining the asymptotic density of zeros in the character table of $S_n$, where it is not even clear from computational data what one should expect.

Trevor Wooley has shown that every rational cubic hypersurface of dimension at least 35 contains a rational line. In this talk I want to report about recent joint work with Julia Brandes, reducing this 35 to 29 in the generic case of smooth cubic hypersurfaces. One of the key ingredients is a result by Browning, Heath-Brown and myself on intersections of cubic and quadric hypersurfaces. I also want to discuss the related problem of finding lines on cubic hypersurfaces defined over p-adic fields, to give explicit examples of smooth rational cubic hypersurfaces of dimension 9 not containing any rational line, and to mention a few applications of our results.

Schottky uniformization is the description of an analytic curve as the quotient of an open dense subset of the projective line by the action of a Schottky group. All Riemann surfaces can be uniformized in this way, as well as some p-adic curves, called Mumford curves. In this talk, I will present a construction of universal Mumford curves: analytic spaces that parametrize both archimedean and non-archimedean uniformizable curves of a fixed genus. This result relies on the existence of suitable moduli spaces for marked Schottky groups, that can be built using the theory of Berkovich spaces over rings of integers of number fields developed by Poineau.

After introducing Berkovich analytic geometry from the beginning, I will describe universal Mumford curves and explain how these can be used as a framework to study arithmetic-geometric objects such as the Tate curve and Teichmüller modular forms. This is based on joint work with Jérôme Poineau.

In his celebrated proof of Zagier's polylogarithm conjecture for weight 3 Goncharov introduced a "triple ratio", a projective invariant akin to the classical cross-ratio. He has also conjectured the existence of "higher ratios" that should play an important role for Zagier's conjecture in higher weights. Recently, Goncharov and Rudenko proved the weight 4 case of Zagier's conjecture with a somewhat indirect method where they avoided the need to define a corresponding "quadruple ratio". We propose an explicit candidate for such a "quadruple ratio" and as a by-product we get an explicit formula for the Borel regulator of K_7 in terms of the tetralogarithm function (joint work with H. Gangl and D. Radchenko).

Abstract: I will survey some recent vanishing theorems for the mod p cohomology of Shimura varieties. I will mention some p-adic results and some l-adic results, where l is a prime different from p. Both settings rely on the geometry of the Hodge-Tate period morphism, but I will try to highlight the differently flavoured techniques that are needed. This is largely based on joint work with Daniel Gulotta and Christian Johansson, and on separate joint work with Peter Scholze.

A Wieferich prime is a prime number p such that 2^(p-1) is congruent to 1 modulo p^2. These numbers originally arose in the context of Fermat's last theorem. At present very little is known about them, although there are some conjectures. One can analogously define Wieferich primes for 3, or 5, or for a point on an abelian variety. In this talk I will explain what Wieferich primes for abelian varieties have to do with p-adic integrals and rational points on curves, and will also describe some (unconditional) results on the heights of rational points on higher genus curves.

Beginning with the simple question 'when is the sum of the squares of a tuple of integers equal to a multiple of their product?', one arrives at a family of Diophantine equations called Markoff-Hurwitz equations. I will give a `high-level' accessible talk about these Diophantine equations with two broad themes: Firstly, the Markoff-Hurwitz equations are important as a `critical' case in Diophantine geometry, and as such, have strange Diophantine properties. Secondly, the Markoff-Hurwitz equations are intimately connected with a family of fractals that includes the `Rauzy gasket': a fractal that pops up in seemingly disparate areas of mathematics including triply periodic surfaces, dynamics of maps on the circle, higher dimensional generalizations of continued fractions, Teichmuller theory, and now, in Diophantine geometry. This is partly based on joint work with Alex Gamburd and Ryan Ronan.

We will discuss an arithmetic version of Stein's spherical averages. Their associated full maximal function was studied by Magyar--Stein--Wainger and Ionescu who gave the sharp range of estimates. Magyar subsequently proved that the associated ergodic averages converge almost everywhere. In analogy with Stein's spherical averages, one would expect the lacunary averages to have much better behavior. We will show that to some extent they do, but with surprising limitations.

The link between covariants of binary sextics and vector-valued Siegel modular forms of degree 2, obtained in a previous work with G. van der Geer and C. Faber, is an efficient way for producing such modular forms. By using this link, the structure of some graded rings of vector-valued Siegel modular forms of degree 2 with character will be given. This talk is based on a joint work with G. van der Geer and C. Faber.

Markoff surfaces form an interesting class of Diophantine equations which were recently studied by Ghosh and Sarnak. In this talk I shall explain how we have reinterpreted their original work using tools from algebraic geometry, and gained further insights into the existence of integral solutions and failures of the Hasse principle using the so-called Brauer-Manin obstruction. This is joint work with Vlad Mitankin.

The local Jacquet--Langlands correspondence is an instance of Langlands functoriality, relating representations of general linear groups with those of central division algebras over local fields. I will present an effective description of this correspondence in terms of type-theoretic invariants, due to Sécherre--Stevens and myself, and give an application to the geometry of moduli spaces of Galois representations.

It is generally believed that the non-trivial zeros of automorphic L-functions are different for non-isomorphic cuspidal representations. We will verify some consequences of this claim using a characterisation of Maass forms on GL(2). Our proof uses hypergeometric functions and geometry of the hyperbolic plane. This is joint work with Michalis Neururer.

Classically, Saito-Kurokawa lifting is an injective mapping from the space of modular forms of level 1 to the space of Siegel modular forms of degree 2 such that the functions in the image violate generalized Ramanujan-Petersson conjecture. The first construction of such a lifting was given by Maass who exploited correspondences between various modular forms. The image consisted of functions whose Fourier coefficients satisfied what we now call the Maass relations.

One can generalize this mapping to include modular forms of higher levels. However then, classical constructions become fairly complicated and it is not clear whether they still imply (a version of) Maass relations. We show that this is indeed the case by generalizing a representation theoretical approach of Pitale, Saha and Schmidt from level 1 to higher levels.

TBA

Families of Galois representations over p-adic rigid analytic spaces have a number of applications, including to the study of the eigencurve. There are also a number of conjectures about the structure of the eigencurve over the boundary of weight space, which is most naturally studied as an adic space. I will discuss some results about families of Galois representations over certain kinds of adic spaces.

In the 1990s Moy and Prasad revolutionized the representation theory of p-adic groups by showing how to use Bruhat-Tits theory to assign invariants to representations of p-adic groups. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results is an explicit construction of (types for) representations of p-adic groups. In this talk I will indicate why, survey what constructions are known (no knowledge about p-adic groups assumed) and present recent developments based on a refinement of Moy and Prasad's invariants.

I will begin with an introduction about how to attach 2-dimensional representations of Galois groups to modular forms, both in positive and null characteristic, focusing on the important special case of elliptic curves. It turns out that the phenomenon of "complex multiplication", whose definition I will recall, can be read in the shape of the image of the representation. I will then discuss a recent result, obtained with N. Billerey (Clermont-Ferrand), ensuring that if a modular form behaves as having complex multiplication when reduced modulo a certain prime p, then it is congruent (mod p) to another form which truly has complex multiplication.

Orthogonal holomorphic projection of non-holomorphic modular forms is an important tool for arithmetic applications. Sturm's convolution method describes the holomorphic projection reliably for Siegel modular forms of large weights. We discuss Sturm's operator for small weights, the critical weight being one larger than the group rank. In this case it does not only produce the terms of holomorphic projection but also so-called phantom terms. We give a spectral theoretic interpretation of these phantom terms.

The main purpose of this talk is to provide an effective version of a result due to Einsiedler, Mozes, Shah and Shapira, on the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least 3 dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems due to Ratner. Instead, we pursue an alternative strategy based on spectral theory, Fourier analysis and Weil's bound for Kloosterman sums which yields an effective estimate on the rate of convergence in the space of (d+1)-dimensional Euclidean lattices with d>1. This extends my work with Jens Marklof on the 3-dimensional case (2017).

This is a joint work with Daniel El-Baz and Bingrong Huang.

Ihara's lemma is a statement about the structure of the mod l cohomology of modular curves that was a key ingredient in Ribet's results on level raising. I will motivate and explain its statement, and then describe joint work with Jeffrey Manning on its extension to Shimura curves.

Arithmetic Ramsey theory is about large sets necessarily containing given structures, such as arithmetic progressions. These phenomena can be studied using combinatorics, analysis, or ergodic theory. For this reason, Szemeredi's theorem has been described as a Rosetta stone of mathematics. The theory for linear patterns is classical, but over the past two years we've been able to Fourier-analytically transfer the results to higher-degree diophantine equations. This has led to a complete characterisation of density and partition regularity for diagonal equations in sufficiently many variables. This talk is based in part on joint work with Sofia Lindqvist and Sean Prendiville.

In Diophantine approximation, we study questions concerning the approximation of points in the unit interval by rational numbers. Often, one is interested in restricting numerators to lie in some subset of {0, 1, '¦, n} for each denominator n. For example, the Duffin'”Schaeffer Conjecture concerns approximation by reduced fractions, meaning we restrict numerators to the subset consisting of numbers co-prime to n. In this talk I will discuss approximation by random fractions, by which I mean that numerators will be confined to randomly chosen subsets.

Following a celebrated conjecture of Keating and Snaith, expressing elusive moments of the Riemann zeta function on the critical line in terms of moments of characteristic polynomials of Haar-distributed matrices in U(N), much work has been devoted to computing the averages of products and ratios of characteristic polynomials of random matrices from classical groups. We will present an elementary and self-contained approach to this problem, using classical results due to Weyl and Littlewood, and will discuss some (provable) arithmetic applications.

In Diophantine approximation we are often interested in the Lebesgue and Hausdorff measures of certain lim sup sets. In 2006, Beresnevich and Velani proved a remarkable result '” the Mass Transference Principle '”

which allows for the transference of Lebesgue measure theoretic statements to Hausdorff measure theoretic state- ments for lim sup sets arising from sequences of balls in R

k . Subsequently, they extended this Mass Transference

Principle to the more general situation in which the lim sup sets arise from sequences of neighbourhoods of 'ap- proximating' planes. In this talk I will discuss a recent strengthening (joint with V. Beresnevich) of this latter

result in which some potentially restrictive conditions have been removed from the original statement. This improvement gives rise to some very general statements which allow for the immediate transference of Lebesgue measure Khintchine'“Groshev type statements to their Hausdorff measure analogues and, consequently, has some interesting applications in Diophantine approximation.

Given an elliptic curve E over Q and a prime p of good ordinary reduction, there is a natural p-adic analogue of the real canonical height on E. The discriminant of the induced pairing on the free part of the Mordell-Weil group appears in p-adic BSD. However, unlike in the real case, this quantity is only conjectured to be non-zero. I will present a new algorithm to compute p-adic heights in families of elliptic curves, with applications to non-degeneracy. The algorithm uses a modified version of Lauder's deformation method for the computation of the action of Frobenius on an appropriate cohomology group.

The Littlewood and the p-adic Littlewood conjectures are famous open problems on the border between number theory and dynamics. In a joint work with Faustin Adiceam and Fred Lunnon we show that the analogue of the p-adic Littlewood conjecture over function fields is false over \mathbb{F}_3((1/t)). The counterexample is given by the series whose coefficients are the regular paper folding sequence, and the method of proof is by reduction to the non vanishing of certain Hankel determinants. The proof is computer assisted and uses substitution tilings of \mathbb{Z}^2 and a generalisation of Dodson's condensation algorithm for computing the determinant of a matrix.

Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field F_p, but the 'horizontal' distribution as the base field varies over primes remains open. We describe work showing cancellation in the sum over primes if there are exceptional Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a fun blend of ideas from algebraic geometry, the spectral theory of automorphic forms and sieve theory.

The Cohen-Lenstra heuristics, postulated in the early 1980s, say that the sequence of ideal class groups of imaginary quadratic number fields can be modelled as a sequence of random finite abelian groups, where a finite abelian group gets a probability weight that is inversely proportional to the size of its automorphism group. They also propose a model for class groups of real quadratic fields. This was extended in the early 1990s by Cohen-Martinet to much more general families of number fields. I will present very recent joint work with Hendrik Lenstra, in which we are trying to understand the Cohen-Lenstra-Martinet heuristics better. Among other things, this entails disproving them and proposing corrected versions.

We show the existence of abelian varieties over Q_p with good supersingular reduction and non semisimple p-adic Tate module. This result is an application of the characterisation in terms of filtered phi-modules, via p-adic Hodge theory, of the p-adic representations of the absolute Galois group of Q_p coming from abelian schemes. We will show how to obtain varieties having the desired properties for the least possible dimension, namely surfaces. Our constructions easily generalise to higher dimensions.

One way of studying the complex representations of the special linear groups over the integers is to determine the convergence of its representation zeta function. Recently Avni and Aizenbud have given a method that relates special values of the zeta function at even integers $2g - 2$ with the singularities of the representation variety of the fundamental group of a Riemann surface of genus $g$ into the special linear group. This way, Avni and Aizenbud determine that the degree of polynomial representation growth of the special linear group over the integers is smaller than $22$. In this talk I shall report on a recent result obtained in collaboration with N. Budur pushing down this bound to $2$.

The key ingredient in Wiles' proof of Fermat's last theorem was to establish the modularity of elliptic curves. Despite many impressive advances in the Langlands programme the analogous question of modularity for abelian varieties of dimension 2 is far from settled. I will report on work in progress with Kris Klosin on the modularity of Galois representations $G_{\mathbf{Q}} \to {\rm GSp}_4(\mathbf{Q}_p)$ that are residually reducible. I will explain, in particular, how this can be used in certain cases to verify Brumer and Kramer's paramodular conjecture for abelian surfaces over Q with a rational torsion point of order p.

Using a version of the Hardy -- Littlewood circle method over $\mathbb{F}_q(t)$, one can count $\mathbb{F}_q(t)$-points of bounded degree on a smooth cubic hypersurface $X \subset \mathbb{P}^{n-1}_{\mathbb{F}_q}$. Moreover, there is a correspondence between the number of $\mathbb{F}_q(t)$-points of bounded height and the number of $\mathbb{F}_q$-points on the moduli space $\text{Mor}_d(\mathbb{P}^1_{\mathbb{F}_q}, X)$, which parametrises the rational maps of degree $d$ on $X$. In this talk I will give an asymptotic formula for the number of rational curves defined over $\mathbb{F}_q$ on $X$ passing through two fixed points, one of which does not belong to the Hessian, for $n \geq 10$, and $q$ and $d$ large enough. Further, I will explain how to deduce results regarding the geometry of the space of such curves.

in this talk I want to introduce local Borcherds products for idefinite unitary groups U(1,m), m >1. Local here refers to boundary components of the symmetric domain. These functions share some properties with Borcherds products, hence the name. They can be used to study the local Picard group of a boundary component of the local symmetric domain.

I will begin by saying a few words about Borcherds products, and their main properties. Then, I will sketch the construction of the symmetric domain for a unitary group, its boundary components, their stabilizers and a compactification theory for the local symmetric domain. After that, I will introduce Heegner divisors and, for a fixed boundary component, the local Picard group. This will be followed by the definition of local Borcherds and I will show how their transformation behavior can be used to describe the position of Heegner divisors in the local Picard group. As an application, one can further obtain an obstruction statement for a certain space of definite theta-series.

Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F. When E=F this is equivalent to Leopoldt's conjecture for E at p and the '˜p-adic class number formula' of Colmez. In this talk we discuss the p-adic Stark conjecture at s=1 and applications to certain cases of the equivariant Tamagawa number conjecture (ETNC). This is joint work with Andreas Nickel.

Elliptic curves E and E' are said to be n-congruent if their n-torsion subgroups are isomorphic as Galois modules. The elliptic curves n-congruent to a given elliptic curve are parametrised by (the non-cuspidal points of) certain twists of the modular curve X(n). I will discuss methods for computing equations for these curves, and also for the surfaces that parametrise pairs of n-congruent elliptic curves.

It is a standard heuristic that sums of oscillating number theoretic functions, like the Mobius function or Dirichlet characters, should exhibit squareroot cancellation as compared with trivial bounds. It is often very difficult to prove anything as strong as that, and we generally expect that if we could prove squareroot cancellation it would be the best possible bound. I will try to explain some recent results showing that, in fact, certain averages of multiplicative functions exhibit a bit more than squareroot cancellation.

General p-adic L-functions are one of two main ingredients to the Iwasawa main conjectures, but their existence is a non-trivial matter. We'll go through the idea behind a p-adic L-function, what a modular form of half-integral weight even is, and outline how to construct its p-adic L-function. In the end, via the Shimura correspondence, we also end up with an alternative construction of the OG p-adic L-function for modular forms of integral weight.

Let $F$ be a local non-Archimedean field with finite residue field. Let $\mathfrak{O}$ be the ring of integers in $F$. Cuspidal types for $GL_{2}(\mathfrak{O}) plays a special role in inertial Langlands correspondence. We will give an overview of the study of cuspidal types in terms of Clifford theory. Using this theory A. Stasinski and S. Stevens classified special kind of representations of $GL_{n}(\mathfrak{O})$ namely regular representations. I will talk about my work which gives a partial answer on the following question: Which regular representations are cuspidal types?

We will present the topic of 'quantum ergodicity', which has recently gained much attention due to its deep number theoretical connections in ergodic theory (Lindenstrauss's solution to the arithmetic quantum unique ergodicity conjecture), discrete and probabilistic versions, and analogues in the theory of modular forms culminating to the recent works of Holowinsky-Soundararajan (2010) and Nelson-Pitale-Saha (2014). We will review some of the history and current challenges of the problem, and describe a new direction we recently introduced with Etienne Le Masson (Bristol).

Let X be a complex, smooth, projective variety with a non trivial map a: X-->A to an abelian variety (an irregular variety). Let L be a line bundle on X. In the first part of my talk I will introduce some classical problems relating the numerical invariants of L, the so called Clifford-Severi inequalities. Then, I will explain a set of new techniques developed recently to obtain such inequalities in a simpler way and how these techniques allow to understand and classify the extremal cases.

String theory is a physical theory which aims to unify quantum field theory with general relativity. As well as being a promising theory of quantum gravity, string theory has also led to developments in pure mathematics. In this, the first of two consecutive talks, we will recap some historical developments in physics and see how they suggest the need for string theory. A basic description of string theory will be presented, and we will see that a quantity known as the partition function of the theory is invariant under modular transformations. A particular string theory will be presented, whose partition function demonstrates a surprising link between modular forms and group theory. This is the first of numerous interesting connections between string theory and the theory of modular forms, which the second talk will discuss in further detail.

String theory is a physical theory which aims to unify quantum field theory with general relativity. As well as being a promising theory of quantum gravity, string theory has also led to developments in pure mathematics. In this, the first of two consecutive talks, we will recap some historical developments in physics and see how they suggest the need for string theory. A basic description of string theory will be presented, and we will see that a quantity known as the partition function of the theory is invariant under modular transformations. A particular string theory will be presented, whose partition function demonstrates a surprising link between modular forms and group theory. This is the first of numerous interesting connections between string theory and the theory of modular forms, which the second talk will discuss in further detail.

We introduce the higher Chow groups of Bloch (and Landsberg), as a generalization of simplicial homology, and derive the ``calculus'' behind the Bloch map to Hodge cohomology.

Special Values of L-functions can be thought of as generalising such identities as Euler's solution to the Basel problem, that 1+1/2^2+1/3^2 + ... = pi^2/6, which is the special value of the Riemann zeta function at 2. In this talk, the focus will be on a particular L-function one can attach to a Siegel modular form, with the end goal being an explanation of the Rankin-Selberg method employed by Sturm in order to obtain some of its special values.

Most general facts about Galois groups of local fields were obtained in 1960's. They include a functorial description of maximal abelian quotients (local class field theory) and information about the numbers of generators and relations in the case of maximal p-extensions. In the talk it will be explained a further progress in this area based on a generalization of classical Artin-Schreier theory of cyclic p-extensions.

In many ways L-functions have been seen to contain interesting arithmetic information; evaluating at special points can make this connection very explicit. In this talk we shall ask what information is contained in central values of certain automorphic L-functions, in the spirit of the Gan--Gross--Prasad conjectures, and report on recent progress. We also describe some surprising applications in analytic number theory regarding the `size' of a modular form.

We look at the geometry of the space SL_2(Z)\G/K = SL_2(Z)\H and its Eisenstein cohomology, a space in its sheaf cohomology separated from the Eichler-Shimura cuspidal classes. We isolate a rational Eisenstein class analogous to both the classical normalised Eisenstein series E_{2k+2}(z) and the algebraic cohomology, and look at a method developed by Funke-Millson, Alfes-Ehlen and others to find its denominator using Shintani lifts, twisted by quadratic discriminants. This gives a recreation of arithmetic results proven by Harder.

In this talk I will describe a new construction of p-adic heights of p-adic representations and give some applications of this construction to the study of extra zeros of p-adic L-functions (joint work with K. Büyükboduk).

The theory of Shinichi Mochizuki is viewed as the most fundamental development in mathematics for several decades. Its revolutionary conceptual viewpoints drastically extend the range of methods in number theory outside traditional ring-theoretical studies, initiate new important connections between group theory and number theory and open a large territory of potential applications. It provides new group-theoretic links between arithmetic and geometry of elliptic curves over number fields and associated hyperbolic curves. I will discuss some of these aspects.

Are there only finitely many smooth projective hypersurfaces over the ring of integers? If we fix the degree and dimension and assume the Lang-Vojta conjecture, the answer to this question is positive. In this talk I will explain how one proves the latter statement. Furthermore, I will explain how far one can get with current methods in arithmetic geometry without assuming Lang-Vojta's conjecture. This is joint work with Daniel Loughran (Manchester).

We will attach an affine Hecke algebra H to any discrete Langlands parameter of a Levi subgroup of an inner form G of GL(n,F), and show that H admits a specialization that is isomorphic to the corresponding affine Hecke algebra defined by Sécherre and Stevens via the generalization to G of the theory of types due to Bushnell and Kutzko. We will deduce from it a description of the Langlands correspondence for non-supercuspidal irreducible representations of G.

This is joint work with Ahmed Moussaoui and Maarten Solleveld.

We describe a problem of determination of cuspidal Siegel modular forms of degree 2 by their fundamental Fourier coefficients. After a short introduction and motivation for our work, we sketch a couple of methods one could use to tackle this problem: one uses classical tools, the other bases on a strong connection with automorphic representations of GSp(4). In the meantime we present our results related to this topic.

The generalized circle problem asks for the number of lattice points of an n-dimensional lattice inside a large Euclidean ball centered at the origin. In this talk I will discuss the generalized circle problem for a random lattice of large dimension n. In particular, I will present a result that relates the error term in the generalized circle problem to one-dimensional Brownian motion. The key ingredient in the discussion will be a new mean value formula over the space of lattices generalizing a formula due to C. A. Rogers. This is joint work with Andreas Strömbergsson.

Results of Skorobogatov and Zarhin allow one to compute the transcendental Brauer group of a product of elliptic curves. Ieronymou, Skorobogatov and Zarhin used these results to compute the odd order torsion in the transcendental Brauer group of diagonal quartic surfaces. The first step in their approach is to relate a diagonal quartic surface to a product of elliptic curves with complex multiplication by the Gaussian integers. I will show how to extend their methods to compute transcendental Brauer groups of products of other elliptic curves with complex multiplication. Using these results, I will give examples of Kummer surfaces where there is no Brauer-Manin obstruction coming from the algebraic part of the Brauer group but a transcendental Brauer class causes a failure of weak approximation.

Snake graphs are planar graphs first appeared in the context of cluster algebras associated to marked surfaces. In their first incarnation, snake graphs were used to give formulas for generators of cluster algebras. Along with further investigations and several applications of snake graphs, they were also studied from a more abstract point of view as combinatorial objects. This talk will report on their link to continued fractions inspired by planar graphs associated to Markov numbers.

In this talk, I will introduce a notion of the slope for families of curves and review some meaningful bounds of the slope, including the Riemann-Roch theorem for curves and Cornalba-Harris-Xiao inequality for one dimensional families of curves. Then I will state a lower bound of the slope for two dimensional families of curves. If time permits, the Arakelov version of the slope will also be discussed.

The higher Deligne--Lusztig theory is the geometric approach to representations of reductive groups over a finite ring. For a general reductive group, while in the classical level 1 case the geometric method is the unique method to produce almost all irreducible representations, in higher levels there is also an algebraic approach due to Gerardin. As the two constructions share a same set of parameters, a natural question, raised by Lusztig, is whether they are generically equivalent. Recently we made progress on this question (joint with Stasinski); if time permits, I will also discuss an application to Lie algebras.

There is a surprising connection linking period polynomials of modular forms for SL(2,Z) to certain distinguished relations among specific periods, the 2-variable subclass among multiple zeta values. We give an indication--on a rather elementary level--how it comes about. (Joint work with M. Kaneko and D. Zagier.)

The construction of the universal R-matrix for quantum groups produces solutions of the Yang-Baxter equation on tensor products of representations of that quantum group. This gives an action of the braid group of type A, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category. I will explain how the theory of quantum symmetric pairs allows an analogous construction of a universal K-matrix, which produces solutions of the reflection equation on tensor products of representations of that quantum group. This gives a representation of the braid group of type B, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category with a cylinder twist.

Given a group G acting on a space X, one can construct the crossed product algebra G \ltimes C[X] of G with the algebra of functions on X. This generally noncommutative algebra can be seen as a replacement for the algebra of functions on the quotient X/G, which may be badly behaved.

I will speak about deformations of these algebras - both formal deformations (in the sense of Gerstenhaber) and strict deformations (in the sense of Rieffel). I will explain a general construction for producing such deformations and describe some examples. I will then explain how certain geometric and algebraic properties of these deformations are related to certain questions in representation theory, and to the geometry of the original action of G on X.

Weil representations have proven to be a powerful tool in the theory of group representations. They originate from a very general construction of A. Weil, which has as a consequence the existence of a projective representation of the group Sp(2n,K), K a locally compact field. In particular, these representations have allowed to build all irreducible complex linear representations of the general linear group of rank 2 over a finite field, and later over a local field, except in residual characteristic two. I will start by giving a introduction to Weil representations and some examples for classical groups, like SL(2,k), k a finite field. Then we will define SL*(2,A) groups, which are an analogue of SL(2,k) but with entries over an involutive ring A. When these groups have a Bruhat-like presentation, Gutierrez, Soto-Andrade and Pantoja have developed a method for constructing generalized Weil representations for them. I will construct a generalized Weil representation for the finite split orthogonal group O(2n,2n), seen as a SL*-group. Furthermore, we will see that these representation is equal to the restriction of the Weil representation to O(2n,2n) for the dual pair (Sp(2,k), O(2n,2n)). This fact shows an example of compatibility between the mentioned method with 'classical' methods. If time allows we will discuss about an initial decomposition.

Multiple zeta values are a mysterious and intriguing set of real numbers, about which many results are conjectured, but relatively little is proven. One is typically interested in finding all relations between MZVs, and completely understanding them, but transcendentality problems make this difficult to approach directly. One way to make progress with these questions is by lifting MZVs to purely algebraic objects which have additional, more rigid, structure.

I will start by giving an introduction to MZVs and some of the various standard results about them. From here we will lift to Brown's motivic MZVs, and look at the coproduct structure they acquire. Then using this coproduct, I will show how one can sometimes get easy combinatorial proofs of `almost' identities (identities up to a non-explicit rational), even in cases where the explicit identity remains conjectural.

When attacking various difficult problems in the field of Diophantine approximation the application of certain topological games has proven extremely fruitful in recent times due to the amenable properties of the associated 'winning' sets. Other problems in Diophantine approximation have recently been solved via the method of constructing certain tree-like structures inside the Diophantine set of interest. In this talk I will briefly outline how one broad method of tree-like construction, namely the class of 'generalised Cantor sets', can be formalized for use in a wide variety of problems. By introducing a further class of so-called 'Cantor-winning' sets we may then provide a criterion for arbitrary sets in a metric space to satisfy the desirable properties usually attributed to winning sets, and so in some sense unify the two above approaches. Applications of this new framework include new answers to questions within the field of Diophantine apprximation. In particular, I will describe how the method outlined above can be applied to problems surrounding the mixed Littlewood conjecture.

We review part of Shimura's paper "The Special Values of The Zeta Functions Associated with Hilbert Modular Forms". In particular we establish the algebraicity of special values of some Rankin product L-functions, and their analytic continuation.

Given a Riemannian locally symmetric space, bounds for eigenfunctions of the Laplace operator or for joint eigenfunctions of the whole algebra of isometry-invariant differential operators are of great interest in several areas. For example, sup-norm estimates are intimately related to the multiplicity problem and to questions of quantum unique ergodicity. Methods from analysis allow us to provide bounds (nowadays called ``generic'') which are sharp for certain spaces.

If the Riemannian locally symmetric space is arithmetic and one restricts the consideration to the joint eigenfunctions of the algebra of differential operators and of the Hecke algebra then it is reasonable to expect that the generic bounds can be improved.

The archetypical result of such kind is due to Iwaniec and Sarnak in the situation of the modular surface and several other arithmetic Riemannian hyperbolic surfaces, dating back to 1957. In the following years, their way of approach was (and still is) used to deduce many similar results for various spaces of rank at most 1, and was only recently adapted to some higher rank spaces.

The first example of subconvexity bounds for a higher rank setup was provided by our joint work with Valentin Blomer, which we will discuss in this talk.

A basic and important problem in number theory and dynamical systems is to understand the collection of return times to a given region of an irrational rotation of the circle. There is a natural generalization of this problem to higher dimensions, which leads to the study of higher dimensional point patterns called cut and project sets. In this talk we will discuss a connection between frequencies of patterns in cut and project sets, and gaps problems in Diophantine approximation. We will explain how the Diophantine approximation properties of the subspace defining a cut and project set can influence the number of possible frequencies of patterns of a given size. Once this connection is established, we will show how techniques from Diophantine approximation can be used to prove that the number of frequencies of patterns of size r, for a typical cut and project set, is almost always less than a power of log r. Furthermore, for a collection of cut and project sets of full Hausdorff dimension we can show that the number of frequencies of patterns of size r remains bounded as r tends to infinity. For comparison, the number of patterns of size r in any totally irrational cut and project set of dimension d always grows at least as fast as a constant times r^d.

In this talk, I will give a survey on different kinds of lattice point problems in the Euclidean and hyperbolic plane. My viewpoint will be mainly geometric. At times, there will also shine through some deep connections to the spectrum of the Laplacian, but there we can only scratch the surface. I will also present unpublished joint work with C. Drutu (Oxford) relating different geometric counting problems and applications.

The family of holomorphic modular forms defined as sums of -k (k>2) powers of integral quadratic polynomials with positive fixed discriminant was introduced by Zagier in 1975 in connection with the Doi-Naganuma lifting between elliptic modular forms and Hilbert modular forms. Several interesting aspects of these modular forms emerged later, in work of Kohnen--Zagier, and recently Bringmann. I will talk about this. If we consider the same sums with negative discriminants, we obtain meromorphic modular forms, which in several ways are analogues to Zagier's. I will talk about these meromorphic modular forms, particularly their meromorphic part.

Let K be a complete discrete valuation field with a finite residue field of characteristic p>0. If G_K(p) is the Galois group of the maximal p-extension of K then its structure is completely known : it is either free or Demushkin's group. However, this result is not completely satisfactory because the appropriate functor from the fields K to the pro-p-groups G_K(p) is not fully faithful. In other words, in this setting the Galois group does not reflect essential invariants of the original field K. The situation becomes completely different if we take into account an additional structure on G_K(p) given by its decreasing filtration by ramification subgroups G_K(p)^{(v)}, v\ge 0. The importance of explicit description of this filtration was pointed out in 1960-1970's by A.Weil, I.Shafarevich, P.Deligne etc. In particular, if K has characteristic p then we should invent a way to specify a special choice of free generators of G_K(p) which reflects arithmetic properties of K. Suppose C_s, where s\ge 1, is the closure of the subgroup of commutators of order at least s in G_K(p). Then the above problem of "arithmetic description" of G_K(p) can be considered modulo subgroups C_s. If s=1 it is trivial and if s=2 the answer is given via class field theory. In the case s>2 we obtain a long-standing problem of constructing a "nilpotent class field theory". In the talk we discuss the case s=p, in particular, the author new results related to the mixed characteristic case (i.e. when K is a finite extension of Q_p). The quotient G_K(p)/C_p (together with the induced ramification filtration) is complicated enough to reflect the invariants of K. At the same time this quotient comes from a profinite Lie algebra via Campbell-Hausdorff composition law. The description of the appropriate ramification filtration essentially uses this structure of Lie algebra.

Let K be a complete discrete valuation field with a finite residue field of characteristic p>0. If G_K(p) is the Galois group of the maximal p-extension of K then its structure is completely known : it is either free or Demushkin's group. However, this result is not completely satisfactory because the appropriate functor from the fields K to the pro-p-groups G_K(p) is not fully faithful. In other words, in this setting the Galois group does not reflect essential invariants of the original field K. The situation becomes completely different if we take into account an additional structure on G_K(p) given by its decreasing filtration by ramification subgroups G_K(p)^{(v)}, v\ge 0. The importance of explicit description of this filtration was pointed out in 1960-1970's by A.Weil, I.Shafarevich, P.Deligne etc. In particular, if K has characteristic p then we should invent a way to specify a special choice of free generators of G_K(p) which reflects arithmetic properties of K. Suppose C_s, where s\ge 1, is the closure of the subgroup of commutators of order at least s in G_K(p). Then the above problem of "arithmetic description" of G_K(p) can be considered modulo subgroups C_s. If s=1 it is trivial and if s=2 the answer is given via class field theory. In the case s>2 we obtain a long-standing problem of constructing a "nilpotent class field theory". In the talk we discuss the case s=p, in particular, the author new results related to the mixed characteristic case (i.e. when K is a finite extension of Q_p). The quotient G_K(p)/C_p (together with the induced ramification filtration) is complicated enough to reflect the invariants of K. At the same time this quotient comes from a profinite Lie algebra via Campbell-Hausdorff composition law. The description of the appropriate ramification filtration essentially uses this structure of Lie algebra.

A representation zeta function \zeta_G(s) is a (meromorphic continuation of a) Dirichlet series whose nth coefficient r_n(G) counts the number of irreducible representations of dimension n of the group G, provided this number is finite. Currently little is known, or even conjectured, about the zeros of representation zeta functions. Very recently Kurokawa and Ochiai conjectured that if G is an infinite compact group such that r_n(G) is finite for all n, then the representation zeta function of G has a zero at -2. We will present a proof of this for compact p-adic analytic groups, due to Gonzalez-Sanchez, Jaikin-Zapirain and Klopsch. For finite groups it is a classical result that \zeta_G(-2) equals the order of the group, so this result can be interpreted as saying that the order of certain infinite groups is zero.

If classical Diophantine approximation aims to give quantitative versions of the statement "the rationals are dense in the reals", then diagonal Diophantine approximation can be thought of as trying to give quantitative version of the weak approximation theorem. In its most basic form (with the rational numbers as the ground field), the weak approximation theorem states that if we take elements from finitely many distinct completions of the rationals, we can always find a sequence of rationals which approaches each of them simultaneously.

We explain the general setup of diagonal Diophantine approximation, and sketch proofs of some of the main results in the setting of diagonal approximation over the rationals.

Classical Eichler cohomology has many applications in the study of modular forms and their L-functions. After a brief introduction to the classical case of integral weight I will talk about an analouge of this theory for modular forms of general real weight first studied by Knopp in 1974. I will discuss possible applications to the study of L-functions of half-integral weight modular forms and present a new proof of Knopp and Mawi's theorem on an isomorphism between the space of modular forms and certain cohomology groups

The celebrated connection between elliptic curves and weight 2 newforms over the rationals has a conjectural extension to general number fields. For example, over odd degree totally real fields, one knows how to associate an elliptic curve to a weight 2 newform with integer Hecke eigenvalues. Conversely, very recent work of Freitas, Hun and Siksek show that over totally real fields, most elliptic curves are modular (in fact, over real quadratic fields, "all" are modular).

Beyond totally real fields, we are at a loss at associating elliptic curves to weight 2 newforms. The best one can do is to "search" for the elliptic curve. In joint work with X.Guitart (Essen) and M.Masdeu (Warwick), we generalize Darmon's conjectural construction of algebraic points on elliptic curves to general number fields and then use this conjectural construction to analytically construct the elliptic curve starting from a weight 2 newform over a general number field, under some hypothesis. In the talk, I will start with a discussion of the first paragraph and then will sketch our method.

For a long time we have known about the existence of congruences between the Hecke eigenvalues of elliptic modular forms. Of course the most famous of these is the Ramanujan congruence for the tau function mod 691. Such congruences are important in describing, in some sense, the structure of Galois representations.

Around ten years ago, a well known paper by Harder exploited the cohomology of Siegel modular varieties in order to predict a far reaching generalization of Ramanujan's congruence. His conjecture describes a specific congruence between the Hecke eigenvalues of Siegel modular forms and elliptic modular forms (both of level 1).

In this talk I will briefly discuss Harder's conjecture along with a paramodular version (for prime levels). Then using conjectural work of Ibukiyama I show how we may translate into the realms of algebraic modular forms via something akin to the Eichler/Jacquet-Langlands correspondence. In this setting I provide a strategy for collecting evidence for the new conjecture, giving examples at previously unseen levels.

I will present results that give upper bounds for the torsion in homology of nonuniform arithmetic lattices, and explain how together with recent results of Calegari-Venkatesh this can be used to obtain upper bounds on K2 of the ring of integers of totally imaginary fields.

Modular forms of integral weight and half integral weight have many interesting applications in number theory. Shimura in 1973 defined a very important correspondence between the two which can be defined in the framework of theta lifts. More recently harmonic weak Maass forms (generalisations of classical modular forms) and their uses have been studied. In this talk I will discuss these objects and their properties and describe my work on a theta lift which links all of them together.

Polylogarithms are a class of special functions which have applications throughout the mathematics and physics worlds. I will begin by introducing the basic properties of polylogarithms and some reasons for interest in them, such as their functional equations and the role they play in Zagier's polylogarithm conjecture. From here I will turn to Aomoto polylogarithms, a more general class of functions and explain how they motivate a geometric view of polylogarithms as configurations of hyperplanes in P^n. This approach has been used by Goncharov to establish Zagier's conjecture for n = 3.

The Deligne-Lusztig generalized induction is a generalization of the parabolic induction for finite subgroups of reductive groups, e.g. GL_n(F_q), SL_n(F_q), U_n(F_q). This induction functor is based on using the l-adic cohomology (with compact support) of certain varieties attached to those groups as (virtual) bi-modules. In the talk I would like to give an introduction to this construction, together with some of its significant features, and some explicit computations.

Associated to an algebraic variety over a number field, one has a family of Hasse-Weil L-functions. Such L-functions are "motivic", and, according to the Langlands program, should be identified with their "automorphic" counterparts. One of the consequences of such an identification would be the conjectural analytic continuation and functional equation of the L-functions. On the other hand, by so-called "converse theorems", such analytic properties are a stepping stone to general automorphic properties. In practice it is very difficult to prove that a general Hasse-Weil L-function comes from an automorphic representation. The most up-to-date results have a heavy dependence on the base number field, the Euler characteristic and the dimension - A recent example of such a statement being "an elliptic curve over a totally real field is potentially modular". By viewing an algebraic curve as the generic fibre of an arithmetic surface, I will show how to understand certain analytic properties of Hasse-Weil L-functions in terms of "mean-periodicity", regardless of base field or genus, and provide comparisons to the conjectural automorphicity of the generic fibre. Time permitting, I will show how to interpret mean-periodicity as a statement of analytic two-dimensional adelic duality.

Brumer and Kramer have formulated a conjecture on the modularity of abelian surfaces involving paramodular Siegel modular forms. I will report on joint on-going work with Lassina Dembele, Ariel Pacetti and Haluk Sengun to provide further evidence for this conjecture, using theta lifts of Bianchi modular forms.

In this talk I will discuss the so-called "Torsion Congruences" for the Tate motive. I will briefly discuss their relation to the existence of the non-abelian $p$-adic $L$-function for the Tate motive and then, following the work of Ritter and Weiss, I will explain how one can use the theory of Eisenstein series to prove them. After that I will also consider the Torsion Congruences for other motives, as for example the ones attached to Hecke Characters of CM fields.

The p-adic Littlewood conjecture was firstly posed by de Mathan and Teulie in 2004 and it is often considered as a "simplified version" of a famous Littlewood conjecture. In the series of two talks we'll consider the set "mad" of the counterexamples to this conjecture (which is believed to be empty). Thanks to the results of Einsiedler and Kleinbock we already know that the Haudorff dimension of "mad" is zero, so this set is very tiny. In the talk we'll see that the continued fraction expansion of every element in mad should satisfy some quite restrictive conditions. As one of them we'll see that for these expansions, considered as infinite words, the complexity function can neither grow too fast nor too slow.

The p-adic Littlewood conjecture was firstly posed by de Mathan and Teulie in 2004 and it is often considered as a "simplified version" of a famous Littlewood conjecture. In the series of two talks we'll consider the set "mad" of the counterexamples to this conjecture (which is believed to be empty). Thanks to the results of Einsiedler and Kleinbock we already know that the Haudorff dimension of "mad" is zero, so this set is very tiny. In the talk we'll see that the continued fraction expansion of every element in mad should satisfy some quite restrictive conditions. As one of them we'll see that for these expansions, considered as infinite words, the complexity function can neither grow too fast nor too slow.

The p-adic Littlewood conjecture was firstly posed by de Mathan and Teulie in 2004 and it is often considered as a "simplified version" of a famous Littlewood conjecture. In the series of two talks we'll consider the set "mad" of the counterexamples to this conjecture (which is believed to be empty). Thanks to the results of Einsiedler and Kleinbock we already know that the Haudorff dimension of "mad" is zero, so this set is very tiny. In the talk we'll see that the continued fraction expansion of every element in mad should satisfy some quite restrictive conditions. As one of them we'll see that for these expansions, considered as infinite words, the complexity function can neither grow too fast nor too slow.

p-adic L-functions play a central role in classical (abelian) Iwasawa Theory since they constitute the analytic input in the so-called Main Conjectures. Conjecturally one can attach such a p-adic L-function to any critical motive. Known examples are the p-adic L-function of an elliptic curve, of a Hecke character of a CM or totally real field, or of an elliptic modular form. The p-adic L-function, say of an elliptic curve E defined over the rationals, encodes information about the critical values of the L-function of E as we base change E over the p-cyclotomic tower. In the context of non-abelian Main Conjectures, as they were formalized in the work of Coates, Fukaya, Kato, Sujatha and Venjakob, one is interested in replacing the p-cyclotomic tower with a Galois extension whose Galois group is a p-adic Lie group. This leads to the notion of non-abelian p-adic L-functions, which are higly conjectural and only a few examples are known. In this talk I will start by explaining the notion of p-adic L-functions in the abelian setting and their role in the Main Conjectures. Then I will discuss the non-abelian setting and the recent progress made by various people in this direction with respect to the existence of non-abelian p-adic L-functions.

The theory of theta functions tells us that the representation numbers of positive definite quadratic forms are modular forms. This idea can be extended to quadratic forms of indefinite signature in a coherent way, and this gives rise to the idea of a theta lift. We will define a theta lift in the setting of a complex vector space of signature (1,1), which takes weakly holomorphic modular functions to meromorphic modular forms of weight 2, and briefly explain the construction of the lift. We will also present some results of lifting different modular functions e.g. 1, real analytic Eisenstein series, Klein j-invariant. This follows on from the work of Kudla, Millson, Funke, Bruinier and others.

(postponed from last week)

Abstract: I will briefly explain the ideas behind the relative trace formula and outline a specific case of application. Namely: the Saito-Kurokawa lifting of automorphic representations from PGL(2) to the projective symplectic group of similitudes PGSp(4) of genus 2 established by the use of the relative trace formula, thus characterising the image as the representations with a nonzero period for the special orthogonal group SO(4,E/F) associated to a quadratic extension E of the base field F, and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup. The image is nongeneric and almost everywhere nontempered, violating a naive generalization of the Ramanujan conjecture.

This talk is being postponed (new proposed time 18 Nov) due to a time clash.

Resolution of singularities is a topic in Algebraic Geometry that has classical appeal and is also a current area of research. In 1964 Heisuke Hironaka proved that all varieties over a field of characteristic zero can be resolved, and went on to receive the Fields medal for this work. We will look at the basic techniques of resolution, and use them to outline proofs of resolution for curves and surfaces which demonstrate the basic idea behind Hironaka's proof. We will also give an overview of current progress in extending the proof to positive characteristic.

We prove the upper bound of p-adic multiple zeta (resp. L-) value spaces. This is a p-adic analogue of Goncharov, Terasoma, and Deligne-Goncharov's result (resp. Deligne-Goncharov's result). In the proof, we use the motivic Galois group of the Tannakian category of mixed Tate motives over Z (resp. over a ring of S-integers of a cyclotomic field). We also formulate a p-adic analogue of Grothendieck's conjecture on a special element in the motivic Galois group.

Since their introduction in the mid-20th Century, Hodge structures have been a fundamental tool in transcendental algebraic geometry, for example in the study of algebraic cycles and moduli of complex algebraic varieties. Mumford-Tate groups are the symmetry groups of Hodge theory, and their orbits (Mumford-Tate domains) are the moduli spaces for Hodge structures with given symmetries.

The 'classical' case of Hodge structures of weight 1 (and those they generate by linear-algebraic constructions) has been thoroughly studied. In this case, the MT-domains are Hermitian symmetric spaces whose arithmetic quotients yield algebraic (Shimura) varieties. The many beautiful results facilitated by MT groups in this setting include Deligne's theorem on absolute Hodge cycles and the resolution (by many authors) of the full Hodge conjecture for various classes of abelian varieties.

Following on a review of this history, I will describe recent joint work with P. Griffiths and M. Green on the "nonclassical" higher weight case. The corresponding theory is in its early stages and is of an entirely different character: Shimura varieties are replaced by global integral manifolds of an exterior differential system, and nonclassical (exceptional) Lie groups turn out to occur as MT groups. In addition to the general context mentioned above, part of the motivation for our project was to better understand the very interesting special features of period domains associated to Calabi-Yau 3-folds, and I will explain a classification result for the MT subdomains in an important special case.

In this talk we extend the Shintani lift from cusp forms to arbitrary modular forms. In particular, we use a (co)homological approach (joint work with John Millson).

I will explain the Ihara question/Oda-Matsumoto conjecture, and its pro-l abelian-by-central variant. Then I will give some hints about the proof.

Modular symbols are geodesics (both, closed or infinite) in a non-compact quotient X of the Poincare upper half plane by a subgroup of SL_2(Z). The systematic study of modular symbols was initiated by Manin who in particular showed that they span the first (relative) homology of X. In this talk we extend the Shintani lift from cusp forms to arbitrary modular forms. In particular, we use a (co)homological approach. This is joint work with John Millson.

I will try to cover part of what could called geometric (as contrasted with `motivic') Grothendieck-Teichmueller theory, which started around twenty years ago, partly following (with delay) Grothendieck's `Sketch of a program'. Among the topics that will be touched upon are: the algebraic fundamental group, arithmetic Galois action on the fundamental group, the case of the projective line with three points and `dessins d'enfants', moduli stacks of curves and the Grothendieck Teichmueller group. I will explain in particular how one can topologically understand (and prove a version of) the `two level principle', which lies at the root of the very existence of the Grothendieck-Teichmueller group and its ubiquity.

The Shafarevich Conjecture pertains to the non-existence of abelian varieties over $\Bbb Q$ with everywhere good reduction. We discuss its generalization to proper smooth schemes over $\Bbb Z$ (i.e. projective varieties over $\Bbb Q$ with everywhere good reduction) and recent progress towards varieties which have bad semi-stable reduction in p=3 and good reduction in all $p\ne 3$. Main attention will be paid to the roles of finite flat group schemes, crystalline and semi-stable representations on the one side and of Odlyzko estimates for minimal discriminants of algebraic number fields on the other side.

The Shafarevich Conjecture pertains to the non-existence of abelian varieties over $\Bbb Q$ with everywhere good reduction. We discuss its generalization to proper smooth schemes over $\Bbb Z$ (i.e. projective varieties over $\Bbb Q$ with everywhere good reduction) and recent progress towards varieties which have bad semi-stable reduction in p=3 and good reduction in all $p\ne 3$. Main attention will be paid to the roles of finite flat group schemes, crystalline and semi-stable representations on the one side and of Odlyzko estimates for minimal discriminants of algebraic number fields on the other side.

The Shafarevich Conjecture pertains to the non-existence of abelian varieties over $\Bbb Q$ with everywhere good reduction. We discuss its generalization to proper smooth schemes over $\Bbb Z$ (i.e. projective varieties over $\Bbb Q$ with everywhere good reduction) and recent progress towards varieties which have bad semi-stable reduction in p=3 and good reduction in all $p\ne 3$. Main attention will be paid to the roles of finite flat group schemes, crystalline and semi-stable representations on the one side and of Odlyzko estimates for minimal discriminants of algebraic number fields on the other side.

In this talk, after a review of ramification theory of torsion Galois representations over a local field, we give a ramification estimate of the torsion semi-stable representations with Hodge-Tate weights in {0,...p-2}.

Euler, the father of multiple zeta values, mainltreated non-strict multiple zeta values (MZSVs) in his article. The Q-algebras spanned by strict (ordinary) multiple zeta values (MZVs) and MZSVs are the same to each other. I am planning to review and compare various relations among MZVs and MZSVs and explain an advantage of MZSVs in studying the explicit structure of the algebra. I will also introduce new identities and prospects which are proper to MZSVs.

We discuss an explicit formula for a map from the motivic cohomology of an algebraic variety to its rational Deligne homology. This generalizes the Abel-Jacobi map of Griffiths to (essentially) certain "relative algebraic cycles" living over X. We will work out some simple (but interesting) examples related to polylogarithms and hypergeometric integrals. While arithmetic issues are necessarily involved, the flavor of the talk will be primarily algebro-geometric (in characteristic 0).

In this talk, we present a bird's-eye view of the theory of motives. We begin with an overview of the theory of pure motives based on correspondences on algebraic cycles (as envisioned by Grothendieck in the 1960s in-order to prove the so-called "standard conjectures".). We then introduce mixed Hodge structures and using the definition of pure (Tate) motives and mixed Hodge structures over the rationals, we explain what mixed Tate motives are, and list the desirable properties of the conjectural abelian category of mixed motives of which mixed Tate motives are a subcategory. (All through this we treat Voevodsky's construction of a derived triangulated category of mixed motives as a "black-box"- in a later talk, we will return to Voevodsky's theory.) We finish by mentioning a few applications of mixed Tate motives to questions about special values of zeta and multizeta functions, especially with a teaser on the recent work of Bloch-Esnault-Kreimer.

In this talk, we present a bird's-eye view of the theory of motives. We begin with an overview of the theory of pure motives based on correspondences on algebraic cycles (as envisioned by Grothendieck in the 1960s in-order to prove the so-called "standard conjectures".). We then introduce mixed Hodge structures and using the definition of pure (Tate) motives and mixed Hodge structures over the rationals, we explain what mixed Tate motives are, and list the desirable properties of the conjectural abelian category of mixed motives of which mixed Tate motives are a subcategory. (All through this we treat Voevodsky's construction of a derived triangulated category of mixed motives as a "black-box"- in a later talk, we will return to Voevodsky's theory.) We finish by mentioning a few applications of mixed Tate motives to questions about special values of zeta and multizeta functions, especially with a teaser on the recent work of Bloch-Esnault-Kreimer.

A.B. Goncharov and Manin have shown that the moduli spaces of curves of genus 0 with n marked points are natural spaces in which one can find multiple zeta values as periods and in which one can build a motivic avatar (over $\Z$) of the multiple zeta values. In this talk, we will describe the geometry of those spaces needed in order to sketch a proof of the main result of Goncharov and Manin on the algebraic aspect of their work. The motivic part of the article will be discussed in a later talk.

A.B. Goncharov and Manin have shown that the moduli spaces of curves of genus 0 with n marked points are natural spaces in which one can find multiple zeta values as periods and in which one can build a motivic avatar (over $\Z$) of the multiple zeta values. In this talk, we will describe the geometry of those spaces needed in order to sketch a proof of the main result of Goncharov and Manin on the algebraic aspect of their work. The motivic part of the article will be discussed in a later talk.

Physicists and mathematicians alike have encountered "periods" (arising from integrating an algebraic integrand against an algebraically defined domain), on the one hand from considering Feynman graphs, and on the other hand from associating interesting invariants to "motives". This term in the seminar series, we would like to understand work that has been done relating the two--rather different--points of view. As a main example, Bloch-Esnault-Kreimer have linked the period zeta(3) arising from the "wheel of spokes graph" in physics to algebraic-geometric constructions.

"This talk will be concerned with detecting cycles modulo rational equivalence on a (complex) projective algebraic variety -- that is, elements in the Chow group. We will stick to zero-cycles (Q-linear combinations of points) for simplicity. The work described builds on that of Green, Griffiths, Lewis, Voisin and others. The problem is completely solved by Abel's theorem when the variety is a curve. But already for a surface of positive geometric genus, Mumford's theorem says that the kernel of the Albanese map (i.e. the Abel-Jacobi map on 0-cycles) is "huge". The problem of detecting this kernel leads us to consider "spreads" of cycles which take into account their field of definition. The present talk will be devoted to explaining how a filtration on the Chow group and "higher" Abel-Jacobi maps emerge from this construction. (Our approach is to avoid actually "using" the arithmetic Bloch-Beilinson conjecture or the Hodge conjecture.) We will conclude by describing some applications to 0-cycles on products of curves, where there are links to regulators on algebraic K-theory, iterated integrals, and transcendental number theory (and of course some beautiful applications of Hodge theory). "

"There are generalizations of the Riemann zeta function to functions of several variables called the multiple zeta functions. Like the usual zeta function, their values at positive integer points, called multiple zeta values, are interesting. While originally defined by Euler, recently these numbers have been studied from a different point of view. It turns out that there are several relations between them and the algebra of multiple zeta values is very interesting. Further they are periods in the sense of Kontsevich and Zagier and in fact they appear as periods of a mixed Hodge structure on the fundamental group of the complex projective plane with the points 0,1 and the point at infinity removed. In this talk we define a generalization of such numbers called multiple L-values of modular forms. We show that they have similar properties to the multiple zeta values and further, some of the values are periods. In some cases we can show that these numbers are periods of a mixed Hodge structure on the fundamental group of a modular curve."

Abstract: There are many good reasons to believe that black holes are physical systems that can distribute (``scramble'') information among their internal degrees of freedom at maximal possible rate. This rate can be quantified in terms of the Maldacena-Stanford-Shenker bound $\lambda_L < 2 \pi T$ on Lyapunov exponents $\lambda_L$ in quantum systems. However, most of the proofs of this bound rely on classical gravitational description within the AdS/CFT holographic duality. This work is an attempt to understand ``fast scrambling'' in terms of microscopic degrees of freedom that make up black holes. Motivated by its holographic description in terms of black D-branes, we consider Banks-Fischler-Susskind-Stanford (BFSS) matrix model as a microscopic model of black holes. First, we use numerical real-time simulations to demonstrate that the Hamiltonian of the BFSS model leads to fast scrambling and entanglement generation at all temperatures. We then consider a dramatic simplification of the BFSS model down to two bosonic and one fermionic degrees of freedom, which allows to completely diagonalize the Hamiltonian and demonstrate fast scrambling at the level of random-matrix-type statistical fluctuations of energy levels. Amazingly, even in such extremely simple supersymmetric system we are able to identify the regimes of graviton gas, Schwarzschild black hole and black D-branes as predicted by holographic duality. We demonstrate how supersymmetry ensures that quantum chaos persists all the way down to zero temperature, in contrast to non-supersymmetric gauge theories which become non-chaotic in the confinement regime.

LHC collisions can act as a source of photons in the initial state, in addition to the more common quark and gluon-initiated processes. Indeed, photon-intitiated production is a promising search channel for BSM states as well as probe of the EW couplings of the SM particles. Due to the colour singlet nature of the photon, a key feature of this process in proton-proton collisions is the possibility for leaving the protons intact and/or producing rapidity gaps in the final state. Indeed, the outgoing intact protons can be measured by dedicated `tagging' proton detectors in association with ATLAS and CMS. Moreover, the possibilities are not limited to proton collisions: in heavy ion collisions, the ions can act as a strong source of photon radiation, and the photon-initiated channel can play a significant role. In this talk I will overview the current status and prospects for photon-initiated production at the LHC. I will discuss the theoretical foundations underlying the modelling of such processes and their implementation in a Monte-Carlo event generator. I will in particular demonstrate that the underlying theory is well understood, with limited sensitivity to unconstrained region of QCD due to the strong interaction of the colliding hadrons. We are therefore well justified in viewing such processes as photon-photon collisions, even if the devil is as always in the detail, as I will discuss.

In two dimensional topologically ordered matter, processes depend on gross topology rather than detailed geometry. Thinking in 2+1 dimensions, particle world lines can be interpreted as knots or links, and the amplitude for certain processes becomes a topological invariant of that link. While sounding rather exotic, we believe that such phases of matter not only exist, but have actually been observed in quantum Hall experiments and quantum simulations, and could provide a route to building a quantum computer. Possibilities have also been proposed for creating similar physics in systems ranging from superfluid helium to topological superconductors to semiconductor-superconductor junctions to quantum wires to spin systems to graphene to cold atoms.

Diverse astrophysical and cosmological observations indicate that most of the matter in the Universe is cold, dark and non-baryonic. Traditionally the most popular dark matter candidates have been new elementary particles, such as WIMPs and axions. However Primordial Black Holes (PBHs), black holes formed from over densities in the early Universe, are another possibility. The discovery of gravitational waves from mergers of ~10 Solar mass black hole binaries by LIGO-Virgo has generated a surge in interest in PBH dark matter. I will overview the formation of PBHs, the observational limits on their abundance and the key open questions in the field.

In the search for a complete description of quantum mechanical and gravitational phenomena we are inevitably led to consider observables on boundaries at infinity. This is the holographic principle: A purely boundary--or holographic--description of physics in the interior. The AdS/CFT correspondence provides an important working example of the holographic principle, where the boundary description of quantum gravity in anti-de Sitter (AdS) space is an ordinary quantum mechanical system that is, in particular, given by a Conformal Field Theory (CFT). This is particularly striking as CFTs are important and well-studied landmarks in the landscape of QFTs, where any given CFT can describe a variety of physical systems of criticality from boiling water to ferromagnets - all of which are much less daunting than the question mark that is quantum gravity. It is natural to ask if AdS/CFT correspondence could be used to improve our understanding of the universe we actually live in. I will explain how spatial correlations at the end of the inflationary epoch can be (formally) recast as correlation functions on the boundary of anti-de Sitter space, opening up the possibility to import techniques, results and understanding from AdS/CFT to inflationary cosmology.

Since man first became aware of his existence, time has been one of his primary obsessions. While our watches and clocks remind us every day of this permanent and ineluctable flow, with a precision that is now atomic, it has not always been so. In this conference we retrace the epic of humanity's quest to master time and its measurement. From Mesopotamian clepsydras to atomic clocks, via Egyptian sundials and the discovery of quartz, mankind has drawn on its greatest geniuses, mathematicians, physicists, craftsmen and astronomers to enslave this time which will always elude us.

I’ll review a new, simpler explanation for the large scale geometry of spacetime, presented recently by Latham Boyle and me in arXiv:2201.07279. The basic ingredients are elementary and well-known, namely Einstein’s theory of gravity and Hawking’s method of computing gravitational entropy. The new twist is provided by the boundary conditions we proposed for big bang-type singularities, respecting CPT and conformal symmetry (traceless matter stress energy) as well as analyticity at the bang. These boundary conditions allow gravitational instantons for universes with positive Lambda, massless (exactly conformal) radiation and positive or negative space curvature. Using these new instantons, we are able to infer the gravitational entropy for a complete set of quasi-realistic, four-dimensional cosmologies. If the total entropy in radiation exceeds that of Einstein’s static universe, the gravitational entropy exceeds the famous de Sitter entropy. As it increases further, the most probable large-scale geometry becomes increasingly flat, homogeneous and isotropic. I’ll summarize recent progress towards elaborating this picture into a fully predictive cosmological theory.

Strong magnetic fields can catalyse non-perturbative quantum field theory processes which would otherwise be exponentially suppressed. In this talk, I will discuss two examples of this: production of magnetic monopoles (if they exist), and baryon number violation (within the Standard Model itself). I will present results of numerical calculations in which we found the explicit instanton and sphaleron solutions describing these processes, which demonstrate that they become unsuppressed at sufficiently strong magnetic fields. This was the basis of a recent monopole search by the MoEDAL collaboration in heavy ion collisions at the LHC, which have the strongest known magnetic fields in the Universe. The results placed new model-independent lower bounds on the mass of magnetic monopoles. I will discuss the prospects of improving these bounds and also of achieving baryon number violation in future experiments.

After a brief introduction to some of the impact which integrable methods and the Bethe ansatz have had on the study of the AdS/CFT correspondence in string theory, we will focus on the axiomatic approach to S-matrix theory in 1+1 dimensions. We will highlight the issues that arise when the particles are massless, and how this is in fact connected to Zamolodchikov's way of describing two-dimensional conformal field theories by means of integrability techniques. We will then mention how the axiomatic approach extends to form-factors, which are the gate to access the n-point functions of the theory. If time permits, we will briefly depict how this finds a contemporary application in the area of the AdS_3/CFT_2 correspondence.

Symmetries have frequently aided our study of physical systems. For conformally invariant quantum field theories there has been a lot of recent progress in what can be broadly described as "bootstrapping" these theories from their symmetries. I will review this progress and how it can be used to learn about strongly coupled theories, for which we often cannot rely on traditional perturbative methods, with a special focus on supersymmetric conformal field theories.

Zoom: https://durhamuniversity.zoom.us/j/96642285471?pwd=OU5XVVVKSzhEczVTSHBzb25PWlk2Zz09

Quantum mechanical potentials with multiple classically degenerate minima lead to spectra that are determined by the pertaining tunneling amplitudes. For the strong interactions, these classical minima correspond to configurations of a given Chern-Simons number. The tunneling amplitudes are then given by instanton transitions, and the associated gauge invariant eigenstates are the theta-vacua. Under charge-parity (CP) reversal theta changes its sign, and so it is believed that CP-violating observables such as the electric dipole moment of the neutron or the decay of the eta-prime meson into two pions are proportional to theta. Here we argue that this is not the case. This conclusion is based on the assumption that the path integral is dominated by saddle points of finite action and fluctuations around these. In spacetimes of infinite volume, this leads to the requirement of vanishing physical fields at the boundaries. For the gauge fields, this implies topological quantization corresponding to homotopy classes or all integers. We consequently calculate quark correlations by first taking the spacetime volume to infinity and then summing over the sectors. This leads to an absence of CP violation in the quark correlations, in contrast to the conventional way of taking the limits the other way around. While there is an infinite number of homotopy classes in the strong interactions, there is only a finite number of classical vacua for quantum mechanical systems. For the latter the order of taking time to infinity and summing over the transitions is therefore immaterial.

Zoom: https://durhamuniversity.zoom.us/j/95941846325?pwd=a1FnNEJzNENwcWNnbmhvQUFxV0FOUT09 Password: see email announcement or ask an organiser.

This lecture focuses on the scientific legacy of the Arab polymath Alhazen (Ibn al-Haytham; b. ca. 965 CE in Basra, d. ca. 1041 CE in Cairo). A special emphasis will be placed on his mathematical approaches to natural philosophy in the context of his studies in optics, and by way of his geometrization of the inquiries in classical physics and establishing the methodological rudiments of experimentation and controlled testing. To illustrate some of the principal aspects of his geometrical redefinition of the key concepts of natural philosophy qua physics, I shall consider the analytical case of his positing of place (al-makān) as a postulated geometric void in the context of his critical refutation of the definition of topos in Book IV of Aristotle’s Physics.

After the discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012, we have now entered a new era of precision high-energy physics. This precision is the key to making new discoveries, as even slight deviations from the Standard Model will provide important hints towards as yet unknown particles and interactions. However, with experimental precision at the high luminosity upgrade to the LHC (HL-LHC) expected to outstrip theoretical uncertainties, the success of this program will rely on our ability to overcome the immense challenges involved in improving the accuracy of our theoretical predictions. In particular, we will need to calculate higher-order quantum corrections which are beyond the scope of current methods.

The application of integrability methods in string theory has significantly advanced our understanding of strings moving on symmetric flux backgrounds. Such backgrounds play a central role in the AdS/CFT correspondence and can be used to explore strongly coupled gauge theory. We will introduce some of the key ideas behind these developments in string theory and discuss the generalisation to the integrable deformations and duals of these models.

For many years, physicists thought that all particles fall into two kingdoms: bosons and fermions. They were wrong. Anyons – emergent particles that have a kind of memory – form a new kingdom. Recently the predicted appearance of anyon behavior in the fractional quantum Hall effect was confirmed, in beautiful experiments. Vigorous efforts are afoot to observe anyons in other states of matter, and to mass-produce for use in quantum computing. The age of anyons is upon us.

Understanding the structure and the dynamics of the Earth’s atmosphere is a task of great importance. On the one hand, it allows us to predict the weather a few days in advance, which often translates in saving human lives. On the other hand, it gives us the opportunity to understand how the climate will evolve over the next century, as the impacts due to global warming become progressively stronger. In spite of the progress that has been made, however, our understanding of the atmosphere remains incomplete, and many fundamental questions unanswered. In this talk, I will give an overview of some aspects of modern atmospheric physics. I will begin by introducing some concepts and terminology commonly used in the field. I will then discuss how theoretical developments in the last century made weather forecasts possible (and how von Neumann played a crucial role there too!). Finally, I will present a number of problems and open questions about atmospheric phenomena that we still do not understand, with a particular focus on the tropics.

I will give an overview of how upcoming neutrino and gravitational wave experiments can be used to improve our knowledge of Standard Model particle physics and the evolution of Universe. I will begin by discussing new methods to improve the detection of the least understood Standard Model particle: the tau neutrino. I will then discuss how data from upcoming neutrino oscillation experiments and gravitational wave detectors can be used to understand the unification of matter and forces at the highest energy scales and how our Universe came to have more matter than anti-matter.

In recent years, remarkable algebraic structures have been discovered in the quest to perform exact non-perturbative computations in supersymmetric quantum field theory, with deep connections to geometry and representation theory. I will try to explain how some these algebraic structures arise in simple examples, including Landau levels on a sphere, monopoles creating and destroying vortices, and interfaces colliding with boundaries.

The existence of axions is well motivated from particle physics, string theory and cosmology. I will describe ongoing research in astrophysical and experimental searches for axions.

In modern high energy physics abelian gauge theories are usually considered as boring cousins of non-abelian gauge theories. I will argue that they are under-appreciated, and much more versatile than naively thought

The pioneering work of Bekenstein and Hawking in the 70s showed that black holes have a thermodynamic behavior. They produced a universal area law for black hole entropy valid in the limit that the black hole is infinitely large. Quantum effects induce finite-size corrections to this formula, thus providing a window into the fundamental microscopic theory of gravity and its deviations from classical general relativity. In this talk I will discuss recent advances in high-precision computations of quantum black hole entropy in supersymmetric theories of gravity, using new localization techniques. These calculations allow us to test the suggestion that black holes are really ensembles of microscopic states in a very detailed manner, much beyond the semi-classical limit.

I will then discuss how one can independently verify these calculations using explicit models of microscopic ensembles for black holes in string theory constructed in the 90s. These investigations throw up a surprising link to number theory and the so-called Mock modular forms of Ramanujan. I will end by sketching some research directions that these ideas lead to.

In this talk I'll describe how to use atomic clocks and co-magnetometers for direct detection of light dark matter candidates. These candidates are well motivated theoretically but are hard to detect in more traditional searchers due to their small momentum.

The LHC has completed its second run at the unprecedented energy of 13 TeV and prepares for the upcoming Run-3, with the High-Luminosity upgrade on the horizon. While the search for new physics continues in the soon-to-be-accessible high-energy regime, precision measurements of inclusive observables are likewise on the experiments physics programme. In this presentation I will review the role and properties of the electroweak half of the Standard Model and detail how its precise understanding is crucial to the success of both objectives in these seemingly so very dissimilar regimes.

String theory is traditionally presented as a candidate for a theory of unification of all forces, but it can also serve as a framework providing a deeper understanding of quantum field theories. In this colloquium, I will exemplify how non-trivial aspects of quantum field theories can be given a (often surprisingly simple) geometric origin within string theory by focussing on the classic story of electric-magnetic duality. I will then explain the beautiful geometric relation to superconformal field theories in six dimensions and highlight recent achievements concerning the construction of these otherwise elusive theories.

During the last few years there has been a transformation in our understanding of symmetries and anomalies. The fundamental ideas ultimately originate from condensed matter physics, and more specifically from the classification of topological phases of matter. I will review the basics of the modern viewpoint, and explain a number of results that follow from applying this philosophy to particle physics.

Briefly: we will find that the Standard Model is anomaly-free on arbitrary spacetime topologies (oriented and unoriented), a natural Z_4 refinement of (-1)^F (with F the fermion number) is anomaly free only if the number of fermions in the SM is a multiple of 16, and proton triality being anomaly-free implies that the number of generations is a multiple of 3.

Software for quantum computation has a layered structure. Closest to the hardware, and strongly device-dependent, is the `quantum control' of physical qubits. One layer up is the compilation of `native' quantum gates into a universal gate library and quantum circuits of increasing complexity. The top layer is a quantum algorithm for a concrete computational task. Recent experimental progress has allowed the execution of all layers of this software stack, and the comparison of the performance of different qubit platforms.

In most cases, quantum control is played out through one and two-qubit operations. We present a framework for quantum control directly at the level of N qubits, relying on ideas from quantum many-body theory. An example is a protocol for a gate called iSWAPn, using a linear qubit array with so-called Krawtchouk couplings.

Determining the whole spectrum of stable excitations of a quantum field theory (QFT) is a well-known open problem. To tackle this question a good theoretical laboratory is provided by supersymmetric field theories (SQFTs) with enough conserved supercharges to constrain the QFT dynamics towards exact results. In this context, string theory techniques can be exploited to compute the spectrum of excitations of infinitely many classes of SQFTs in various dimensions. After a brief overview of these methods, we will discuss applications to four-dimensional SQFTs. In this context string theory can be viewed as a tool to predict non-perturbative properties of the spectrum of QFTs, that provides several surprising insights about the physics of this problem.

Massive neutrinos are our first open door into physics beyond the standard model. They also have intriguing consequences in Astroparticle Physics and Cosmology. In this talk I will first review our present understanding of neutrino masses, the leptonic mixing structure and the possibility of leptonic CP violation from a global interpretation of neutrino oscillation results. I will then discuss some avenues open by these results such as the sensitivity to non-standard neutrino interactions, their use in improving the modeling of the Sun, the study of the Earth interior..., as well as the possibility of directly testing beyond the standard models built to explain these results at colliders.

In quantum field theory, conserved particle numbers are associated with ordinary global symmetries. However, many theories can also possess a conserved density of extended objects such as strings, branes, etc. The generalized symmetry principle associated with such conservation laws is just as powerful as that for ordinary symmetries but has only recently been systematically explored. I will explain some of the resulting insights, discussing the classification of low-energy phases and discussing the emergence of gapless Goldstone modes when they are spontaneously broken. Such a symmetry also plays an important role in characterizing the long-distance physics of familiar Maxwell electrodynamics in four dimensions; as an application, I will discuss the realization of this symmetry at finite temperature and provide a reformulation of magnetohydrodynamics from the point of view of symmetry and effective field theory.

After the completion of the Standard Model (SM) through the Higgs discovery particle physicists are waiting for the discovery of new particles either directly with the help of the Large Hadron Collider (LHC) or indirectly through quantum fluctuations causing certain rare processes to occur at different rates than predicted by the SM. While the later route is very challenging, requiring very precise theory and experiment, it allows a much higher resolution of short distance scales than it is possible with the help of the LHC. In fact in the coming flavour precision era, in which the accuracy of the measurements of rare processes and of the relevant lattice QCD calculations will be significantly increased, there is a good chance that we may get an insight into the scales as short as 10^-21 m (Zeptouniverse) corresponding to energy scale of 200 TeV or even shorter distance scales. In particular we emphasize the correlations between flavour observables as a powerful tool for the distinction between various New Physics models. We will summarize the present status of deviations from SM predictions for a number of flavour observables and list prime candidates for new particles responsible for these anomalies. A short outlook for coming years will be given.

The Standard Model of elementary particle physics has proven to be remarkably durable. Starting with a theoretical structure established in the early 1960's, the SM has expanded to accommodate the discovery of new particle generations and symmetry violations. The missing particle content, the Higgs boson, was finally discovered by the ATLAS and CMS experiments in 2012. Since then experiments at the CERN Large Hadron Collider have collected data that have been used to test SM prediction with exquisite precision. Searches for non-SM particles now extend into the multi-TeV mass range and measurements have probed distant scales down to 10−4 fermi.

A review will be made of the SM's development and recent experimental tests using data collected with LHC operation at a proton-proton center of mass energy of 13 TeV.

Particle physics and cosmology have succeeded in creating standard models for the structure of matter and the universe. Such models integrate known data and provide a consistent theoretical description of the field. Standard models are then used as benchmarks to design future experiments or observations that expose them to stringent tests which may in turn lead to extensions or possibly even failures of the models.

Understanding the brain is a problem of equal importance but research has so far not succeeded in producing a comparable condensation of knowledge into a consistent theoretical framework. Will this be possible at all ? In the lecture I will discuss state of the art of brain theories, propose novel computer based methods for their test and express my personal opinion on the possibility of a future standard model of the brain.

Physics at the Large Hadron Collider is dominated by enormous amounts of strongly integrating radiation which must be modelled precisely using Quantum Chromodynamics (QCD). Experiments continue to gather data and reduce errors on their measurements, challenging the theoretical predictions and probing our understanding of the Standard Model.

The fundamental building blocks for these predictions are perturbative scattering amplitudes. The complexity of these objects - especially when considering the necessary quantum corrections - can grow quickly beyond the reach of traditional methods. Understanding the mathematical structure of scattering amplitudes has often led to new developments which can some of which are now at work in experimental analyses.

I will take a look at some modern methods for scattering amplitude computations and their role in making precision predictions at the LHC.

Scattering amplitudes are the basic observables measured by particle colliders and have remarkable mathematical structure which is fascinating in its own right. I will review recent progress in the calculation of scattering amplitudes in gauge theory and gravity using a branch of mathematics called twistor theory.

One-hundred years after Einstein formulated General Relativity, the pivotal role of its most fundamental and fascinating objects --- the black holes --- is nowadays recognized in many areas of physics, even beyond astrophysics and cosmology. Still, solving the theory that governs their dynamics remains a formidable challenge that continues to demand new ideas. I will argue that, from many points of view, it is natural to consider the number of spacetime dimensions, D, as an adjustable parameter in the theory. Then we can use it for a perturbative expansion of the theory around the limit of very many dimensions, that is, considering 1/D as a small number. We will see that in this limit the gravitational field of a black hole simplifies greatly and its equations often turn out to be analytically tractable. A simple picture emerges in which, among other things, the shape of the black hole is determined by the same equations that describe soap bubbles.

Many natural systems are far from thermodynamic equilibrium and keep on exchanging matter, energy or information with their surroundings. These exchanges produce currents, or fluxes, that break time-reversal invariance. Such systems lie beyond the realm of traditional thermodynamics and the principles of equilibrium statistical mechanics do not apply to them. In fact, there exists no general conceptual framework à la Gibbs-Boltzmann to describe these systems from first principles.

The last two decades, however, have witnessed remarkable progress. The aim of this lecture is to explain some recent developments, such as the Work Identities (Jarzynski, Crooks), the Fluctuation Theorem (Cohen, Evans, Gallavotti and Morriss) and the Macroscopic Fluctuation Theory (Jona-Lasinio et al.) which represent the first steps towards a unified approach to non-equilibrium behaviour.

Although there is substantial evidence for the existence of vast amounts of Dark Matter in the Universe, we still ignore its nature. The detection and identification of this new type of matter constitutes one of the greatest challenges in modern Physics, as it can only be explained with Physics beyond the Standard Model.

Dark matter can be searched for directly, through its scattering off nuclei inside direct detection experiments. I will summarise the current experimental situation, with special emphasis on the SuperCDMS detector, and adopt an optimistic point of view, assuming that future detection is possible.

Does this mean that dark matter can be identified? I will address the reconstruction of dark matter properties, the uncertainties involved, and the necessity of data from different detectors.

The structure of light nuclei provides a microcosm for our understanding of the strong interaction, but importantly one in which it is possible to apply state-of-the-art nuclear models. The correlations that are responsible for binding of nuclei also yield complex structures linked to the formation of clusters. The clusters, typically alpha-particles, can arrange themselves in geometric structures, with dynamical symmetries. This talk will explore how these clusters precipitate and how the cluster structures might be imaged and their impact of stellar processes and even the origins of organic-life.

Over the past decade, spatial entanglement entropies have been revealed as a powerful tool for understanding the structure of quantum field theories, giving new perspectives on old questions and leading to interesting new ones. A particularly interesting set of theories is the so-called holographic ones, which admit a dual description in terms of classical gravity. It turns out that entanglement entropies in such theories are relatively easy to study and exhibit intriguing information-theoretic properties, which may offer clues to nature of holographic dualities. I will review these developments and comment on important open problems.

The conductivity of strongly coupled materials, such as the high Tc superconductors, exhibit fascinating properties which are not compatible with those of a weakly coupled Fermi liquid. Signature properties include the linear dependence of the resistivity in temperature and also the anomalous scaling of the Hall angle with temperature. It has been argued that these properties are due to strong coupling and the lack of quasiparticles. I will discuss the way strongly coupled field theories conduct heat and electric current at finite density using holographic techniques and show the promise of this approach to capture some of these features.

I give an introduction to studies of non-equilibirum dynamics in isolated many-particle quantum systems. These have recently attracted a lot of theoretical attention, which is motivated by experiments on systems of ultra-cold trapped atoms (an example being the now famous "Quantum Newton's Cradle"). I focus on how, and in which sense, such isolated systems relax and eventually can be described by statistical mechanics. Time permitting I will discuss the time evolution of observables, which displays interesting phenomena related to the spreading of information out of equilibrium.

I will tell the story of three best friends in 19th century Scotland and their attempt to develop an atomic theory based on knots and links. Tait, Kelvin and Maxwell were inspired by a fantastic experiment involving smoke rings, and their theories, whilst being completely wrong, inspired a new field of mathematical study which is once again becoming important in physics, chemistry and biology.

The entanglement entropy has been very important in various subjects such as the quantum information theory, condensed matter physics and quantum gravity. Especially, for more than twenty years, this quantity has been studied by many people in order to obtain a quantum mechanical interpretation of the gravitational entropy such as the black hole entropy. We will introduce recent progresses toward this long-standing problem in quantum gravity by applying the idea of holography, especially the AdS/CFT correspondence found in string theory. In this talk, we will give an overview of recent progresses in this subject.

Gauge/string duality arguably provides our best framework for computing quantum properties of gravity and black holes from first principles. In the simplest instances it reformulates the problem of understanding a quantum gravity black hole in terms of understanding certain (rather special) gauge theories at finite temperature. I will review this surprising and powerful duality, and then discuss progress over the last 5 years in both analytic and numerical attempts to directly solve such gauge theories with the aim of performing direct quantum gravity calculations.

String theory, famously, has a great many ground states. So many, in fact, that some argue that we should seek information in the statistical properties of these vacua, or worse, argue that we should abandon string theory as a theory with predictive power. On the other hand, very few vacua are known that look like the observed world of particle physics. In this talk I will review this situation and show that there are intriguing, seemingly realistic, models at the tip of the distribution of vacua, where topological complexity is minimised.

The last 15 years in observational Cosmology have been extremely fruitful with key measurements such as the determination of the abundance of invisible matter (the so-called 'dark' matter) in our Universe today. Yet the main issues of Cosmology remain unsolved. In particular the nature of dark matter and dark energy is still mysterious despite strong evidence for their existence. In this talk, I will review some of the progress made in the field of dark matter to identify its nature and summarise the main challenges that need to be tackled to determine what the dark matter is made of.

I will describe the status of current Cosmic Microwave Background observations. I will then focus on recent results from the Atacama Cosmology Telescope, which has mapped the microwave sky to arcminute scales. I will present results from ACT, as well as the South Pole Telescope, on the angular power spectrum of the Cosmic Microwave Background fluctuations, measuring primordial acoustic oscillations well into the Silk damping tail. I will also describe the extraction of a gravitational lensing signal from these observations, and the detection of galaxy clusters via the Sunyaev-Zel'dovich effect. I will describe the implications of these various measurements for cosmology, and discuss prospects for the Planck satellite, and upcoming ground-based experiments.

Einstein's general theory of relativity (GR) is one of the most successful and well-tested physical theories ever developed. Nevertheless, modern cosmology poses a range of questions, from the smallest scales to the largest, that remain currently unresolved by GR coupled to the known energy and matter contents of the universe. This raises the logical possibility that GR may require modification on the relevant scales.

I will discuss the status of some modern approaches to alter GR to address cosmological problems. We shall see that these efforts are extremely theoretically constrained, leaving very few currently viable approaches. Meanwhile, observationally, upcoming missions promise to constrain allowed departures from GR in exciting new ways, complementary to traditional tests within the solar system.

In the recent years theoretical studies and astrophysical observations have confirmed that unknown constituents of our universe like dark matter may find its explanation not only at large-scale experiments at highest energies, but could also show up at the opposite energy scale. In many laboratories world-wide searches for axions, axion-like particles, hidden photons, chameleons or other so-called WISPs with masses below the eV scale are ongoing. Examples at DESY are the experiments ALPS ("Any Light Particle Search") and SHIPS ("Solar HIdden Photon Search"). In all these experiments new particles could manifest themselves in a very spectacular manner. Light would apparently shine through thickest walls. The results of a first generation of laboratory and astrophysics experiments will be summarized and plans for future enterprises be discussed.

N=4 SYM has been dubbed the "Hydrogen Atom" of gauge theories. It provides a playground to find and test new techniques for eventual use in QCD as well as holding the tantalizing possibility of being a solvable, four dimensional gauge theory, a close cousin to QCD. If we want to understand quantum field theories in four-dimensions in a deeper way than given by Feynman diagrams, N=4 SYM is the place to start. In this talk I will try to review recent progress made, especially over the last couple of years in computing scattering amplitudes, their relation to both Wilson loops and Correlation functions and discuss progress made in computing high loop amplitudes and correlation functions in N=4 SYM.

Phenomena in the highly non-linear viscoelastic flow of entangled macromolecular fluids motivate a fundamental programme of theoretical and experimental work on the Brownian dynamics of entangled Gaussian classical strings. An effective (topological field) approach proves effective at addressing first the anomalous linear response of highly branched polymers, then surprisingly provides a way to capture the essential non-linearities as well. Very recently this has paid-off with a major joint university-industry project tackling the molecular engineering, in silico, of fluids with industrial complexity of branching.

For a while now LHC has been answering physics questions in and beyond the Standard Model. I will go through different aspects of our theoretical understanding of high energy physics, including QCD, Higgs searches, and new physics searches. In the absence of a revolutionary discovery I will illustrate how we nevertheless learned much more from the early running phase than we would have expected. This promises a bright future once the LHC runs closer to design energy and luminosity.

The long-standing proposed gravitational wave observatory LISA has been redesigned following the withdrawal of NASA from the project. I shall describe the implications that the new eLISA mission proposal would have for fundamental physics. These include strong-field tests of general relativity, constraints on scalar gravitational fields, searches for exotic objects like cosmic strings, and the determination of the epoch at which seeds for supermassive black holes began to form. This last topic involves observing individual black-hole binaries at redshifts beyond 10, more distant than any astronomical objects seen up to now. If black hole seeds are found to form too early, it will challenge the standard Lambda-CDM cosmology. The talk will review ground-based gravitational wave detection progress, as well as efforts using pulsar timing.

The AdS/CFT correspondence is one of the most important discoveries of string theory. In its simplest form it states that string theory propagating on an anti-de-Sitter spacetime is equivalent to a conformally invariant quantum field theory living on the boundary. It provides a beautiful and powerful tool to study strongly coupled quantum field theories using classical gravity. I will review some of my work that aims to utilise the framework to study strongly coupled systems arising in condensed matter.

String theory provides an ultraviolet complete extension of Yang-Mills theory and general relativity. This talk will describe the structure of scattering amplitudes in string theory and contrast them with corresponding field theory amplitudes. The talk will focus in particular on the structure of graviton scattering which has a rich perturbative and non-perturbative structure that is strongly constrained by "duality" symmetries. This provide an intriguing insight into the ultraviolet divergences of the corresponding (super)gravity field theory.

With recent theoretical and computational advances we have been able to calculate the properties of condensed matter systems from first principles. The first-principles approach is vastly ambitious because its goal is to model real systems using no approximations whatsoever. That one can even hope to do this is down to the accuracy of quantum mechanics in describing the chemical bond. Dirac's apocryphal quip that after the discovery of quantum mechanics the rest is chemistry sums it up: if one can solve the Schrodinger equation for something an atom, a molecule, assemblies of atoms in solids or liquids one can predict every physical property. Dirac's statement doesn't quite show how difficult doing the rest is, and it has taken great effort and ingenuity to take us to the point of calculating some of the properties of materials with reasonable accuracy. The impact of simulations on our thinking about condensed matter problems is immense. Here I shall concentrate on just a few elements of what is a very large subject. First I shall discuss the first-principles rationale and what makes the task so difficult. I shall focus on one of the most successful approaches, the application of density-functional theory and consider why this method turned out to be so important. I shall also spend some time discussing the simulation approach in general, and the types of information that come out of a calculation. To illustrate the usefulness of some of the methods I shall present highlights of a number of simulations to indicate the wide applicability of the method.

The talk reviews physical and mathematical aspects of Generalised Compactifications in String Theory

Professor Martin Utley completed a PhD in High Energy Physics at Glasgow in 1996. Since then, he has worked exclusively on problems in health and health care, applying, adapting and developing simple analytical techniques in collaboration with clinicians. He now leads the Clinical Operational Research Unit at UCL. He will discuss a range of projects and will discuss important differences he sees between modelling in the physical sciences and modelling in health care.

I will discuss some recent results concerning the statistical properties of quantum wavefunctions on networks/graphs. Most of the talk will be introductory, but I will give a birds-eye overview of how field-theoretic techniques have led to some significant steps forward.

Cardiac arrhythmias and sudden cardiac death is the leading cause of death accounting for about 1 death in 10 in industrialized countries. Although cardiac arrhythmias has been studied for well over a century, their underlying mechanisms remain largely unknown. One of the main problems in studies of cardiac arrhythmias is that they occur at the level of the whole organ only, while in most of the cases only single cell experiments can be performed. Due to these limitations alternative approaches such as mathematical modeling are of great interest. From mathematical point of view excitation of the heart is described by a system of non-linear parabolic PDEs of the reaction diffusion type with anisotropic diffusion operator. Cardiac arrhythmias correspond to the solutions of these equations in form of 2D or 3D vortices characterized by their filaments. In my talk I will present a basic introduction to cardiac modeling and mechanisms of cardiac arrhythmias and briefly report on main directions of our research, such as development of virtual human heart model, modeling mechano-electric feedback in the heart using reaction-diffusion mechanics systems and filament dynamics in anisotropic cardiac tissue.

It is a remarkable fact that the standard model (SM) of particle physics is classically conformally invariant - except for a single term: the scalar mass term, commonly introduced for electroweak symmetry breaking. This work is based on the hypothesis that classically unbroken conformal invariance, in conjunction with the Coleman-Weinberg mechanism and the conformal anomaly, can explain the observed hierarchy of scales. I will present evidence that such a scenario might be viable, provided (1) there are no intermediate scales of any kind between the weak scale and the Planck scale, and (2) the RG evolved couplings exhibit neither Landau poles nor instabilities over this whole range of energies. I will also comment on the issue of embedding such a scenario into a UV finte theory of quantum gravity.

The cosmic microwave background (CMB) provides much of the impressive evidence for an enormously hot and dense early stage of the universe - referred to as the Big Bang - but was this singular event actually the absolute beginning? Observations of the CMB are now very detailed, but this very detail presents new puzzles, one of the most blatant being an apparent paradox in relation to the Second Law of thermodynamics. The hypothesis of inflationary cosmology has long been argued to explain away some of these puzzles, but it does not resolve some key issues, including that raised by the Second Law. In this talk, I describe a quite different proposal, which posits a succession of universe aeons prior to our own. The expansion of the universe never reverses in this scheme, but the space-time geometry is nevertheless made consistent through a fundamental role for conformal geometry. Black-hole evaporation turns out to be central to the Second Law. Some analysis of CMB data, obtained from the WMAP satellite provides a tantalizing input to these issues.

The black-hole information paradox has fueled a fascinating effort to reconcile the predictions of general relativity and those of quantum mechanics. Gravitational considerations teach us that black holes must trap everything that falls into them. Quantum mechanically the mass of a black hole leaks away as featureless (Hawking) radiation. However, if Hawking's analysis turned out to be accurate then the information would be irretrievably lost and a fundamental axiom of quantum mechanics, that of unitary evolution, would likewise fail. Here we show that the information about the matter that collapses to form a black hole becomes encoded into pure correlations within a tripartite quantum system, the quantum analog of a one-time pad until very late in the evaporation, provided we accept the view that the thermodynamic entropy of a black hole is due to entropy of entanglement. In this view the black hole entropy is primarily due to trans-event horizon entanglement between external modes neighboring the black hole and internal degrees of freedom of the black hole.

It is well-known that through stereographic projection, one can put a complex coordinate z on a spherical surface. Felix Klein studied the complex coordinates of the vertices, edge centres and face centres of each platonic solid this way, and found that they are the roots of rather simple polynomials in z. Related to these Klein polynomials there are some further, rational functions of z (ratios of polynomials), which have the same symmetries as the platonic solids.

Recently, it has been discovered that various model physical systems, in chemistry, condensed matter, nuclear and particle physics, have smooth structures with the same symmetries as platonic solids. The Klein polynomials and related rational functions are very useful for describing them mathematically.

The talk will end with a discussion of a model for atomic nuclei in which the protons and neutrons are regarded as close enough together to partially merge into one or other of these symmetric structures. Various small nuclei, up to carbon-12 and a bit larger, have been modelled this way.

Hamilton's first application of the concept of phase space - later so fruitful in physics - was a prediction in optics: conical refraction in biaxial crystals. This was one of the first successful predictions of a qualitatively new phenomenon using mathematics, and created a sensation. At the heart of conical refraction is a singularity, anticipating the fermionic sign change underlying the Pauli exclusion principle and the conical intersections now studied in quantum chemistry. The light emerging from the crystal contains many subtle diffraction details, whose definitive understanding and observation have been achieved only recently.

Generalizations of the phenomenon involve radically different mathematical structures.

Owing to fossil-fuel use, land-use change and agriculture, global atmospheric concentrations of carbon dioxide, methane and nitrous oxide have increased markedly since 1750 and now far exceed pre-industrial values determined from ice cores spanning many thousands of years. Warming of the climate system is unequivocally evident from observations of increases in global average air and ocean temperatures, widespread melting of snow and ice, and rising global average sea level. Paleoclimate information supports the interpretation that the warmth of the last half century is unusual compared with at least the previous 1300 years.

Most of the observed increase in globally averaged temperatures since the mid-20th century is very likely due to the observed increase in anthropogenic greenhouse gas concentrations. There are discernible human influences on other aspects of climate, including ocean warming, continental-average temperatures, temperature extremes and wind patterns. For the next two decades a warming of about 0.2°C per decade is projected for a range of emission scenarios. Continued greenhouse gas emissions at or above current rates would cause further warming and induce many changes in the global climate system during the 21st century that would very likely be larger than those observed during the 20th century. Anthropogenic warming and sea level rise would continue for centuries due to the timescales associated with climate processes and feedbacks, even if greenhouse gas concentrations were to be stabilized.

John Cockcroft's splitting of the atom and Ernest Lawrence's invention of the cyclotron in the first half of the twentieth century ushered in the grand era of ever higher energy particle accelerators to probe deeper into matter. It also forged a link, bonding scientific discovery with technological innovation that continues today in the twenty first century. In the second half of the twentieth century, we witnessed the emergence of the photon and neutron sciences driven by accelerators built-by-design producing tailored and ultra-bright pulses of bright photons and neutrons to probe structure and function of matter from aggregate to individual molecular and atomic scales in unexplored territories in material and life sciences. As we enter the twenty first century, the race for ever higher energies, brightness and luminosity to probe atto-metric and atto-second domains of the ultra-small structures and ultra-fast processes continues. We give a glimpse of the recent developments and innovations in the conception, production and control of charged particle beams in the service of diverse scientific disciplines.

Superstrings is a lecture that links Einstein's favourite instrument, the violin, with many of the concepts of modern physics that he did so much to found. The performance begins with an introduction to Einstein's life and involvement with music and how his ideas have shaped our concepts of space, time and the evolution of the Universe. These slides are accompanied by selections from J.S. Bach's Sonatas and Partitas for Solo Violin, some of Einstein's favourite music.

The lecture then proceeds with a discussion of some of our modern ideas that build on the structures of Einstein and define the so-called "Standard Model" of particle physics, in which the evolution of the Universe after the Big Bang can be understood by the interplay of a small number of fundamental forces on a few point-like "elementary" particles, the quarks and leptons, and their antimatter equivalents.

At several points in the performance Jack uses his J.B. Guadagnini violin, the "ex-Wilhelmj", to illustrate some of the ideas discussed by Brian in the lecture by analogy.

Although in many ways a fantastic success, the "Standard Model" leaves many questions unanswered and leads to several paradoxes. Modern ideas of Superstrings may well lead to a much more satisfactory theory, although at the cost of prediciting a whole host of new particles as yet undiscovered. Superstring theory also predicts that the universe has extra "hidden" dimensions of space whose size is so small that they are invisible to our everyday experience. Nevertheless, they may give rise to measureable effects in the next generations of "atom smashers" due to start operation at CERN in Geneva in a couple of years time. The lecture ends by looking at these possible effects and with a duet for two violins by Mozart in which lecturer and soloist join forces and pay tribute to Einstein's lifelong love of chamber music.

Over the past ten years Paul Sutcliffe and I have studied an elementary question of Euclidean geometry. The problem remains unsolved but we have produced a string of further conjectures. I will survey these conjectures and show how they are related to various aspects of physics.

The detection of Gravitational Radiation remains one of the major challenges for experimental astrophysics. This will provide a unique tool for looking into the heart of some of the most violent events in the Universe by detecting changes in the fabric of space-time. Detectors are needed which can measure the relative lengths of perpendicular arms of kilometre scale to about 10-19 m on multi-millisecond timescales. A global network of such detectors - GEO, LIGO, Virgo - are now in operation around the globe, with enhanced versions being developed.

In this talk a review of the status of this emerging new field will be given.

We can describe the growth of a simply connected set in the plane by thinking about how the conformal transformation, which maps it to some standard set like the unit circle, evolves. For the scaling limit of sets which arise 2d statistical mechanics (for example spin clusters in the Ising model), this is conjectured to be particularly simple, and is called Schramm-Loewner Evolution (SLE). However the scaling limit of such models is also supposed to be described by conformal field theory (CFT). We show that a link between these two can be made through so-called parafermionic holomorphic observables, which can already be identified on the lattice.

X-rays, having a wavelength comparable to the spacing of atoms in solids and liquids, are a natural probe of condensed matter. Laboratory x-ray sources have been available for over a century but have been largely superseded by synchrotron radiation sources in the last twenty years. Now a new revolution is upon us with the advent of x-ray free electron lasers. These new linear accelerators promise billion-fold gains in source brilliance and the opportunity to study femtosecond dynamics. How do x-ray lasers work, what will be the first experiments, and what new science is likely to emerge?

Entanglement is a fundamental charasteristic of quantum mechanics: a measurement at a point in space may affect instantaneously measurements performed elsewhere in a way that cannot be described by local variables. The entanglement entropy is a proposed measure of entanglement in a pure quantum state, and it also occurs in the study of black hole entropy. I will explain what entanglement entropy is and some of its basic properties, and I will describe what happens with it for the ground state of quantum chains near to a critical point. It is related to interesting geometries and it turns out that it encodes neatly important universal information about the region around the critical point.

To describe black hole thermodynamics in any quantum theory of gravity, one must introduce constraints that ensure that a black hole is actually present. I show that for a very large class of dilaton black holes, the inclusion of such ``horizon constraints'' allows us to use conformal field theory techniques to compute the density of states, reproducing the correct Bekenstein-Hawking entropy in a nearly model-independent manner. This picture suggests an elegant description of the relevant degrees of freedom, as Goldstone-boson-like excitations arising from symmetry breaking by a conformal anomaly induced by the horizon constraints.

Abstract: In recent years there's been much progress on understanding the dynamics of solitons in gauge theories. I will review some of this work, describing instantons, and monopoles, and vortex strings, and domain walls, and monopoles threaded on vortex strings, and vortex strings ending on boojums on domain walls, and instantons trapped inside domain walls, and many more. I'll also explain how the quantizing vortex strings can be used to understand the quantum dynamics of four-dimensional gauge theories.

The holographic principle states that any $d+1$ dimensional quantum theory of gravity should have a description in terms of a $d$-dimensional quantum field theory without gravity. In this talk we discuss how holography is realized in string theory and how one extracts quantum field theory data from a gravitational solution. We illustrate our discussion with examples.

Ideas about a duality between gauge fields and strings have been around for many decades. During the last ten years, these ideas have taken a much more concrete mathematical form. String descriptions of the strongly coupled dynamics of semi-realistic gauge theories, exhibiting confinement and chiral symmetry breaking, are now available. These provide remarkably simple ways to compute properties of the observed strongly coupled quark-gluon plasma phase, and also shed new light on various phenomenological models of hadron fragmentation. I will present a review and highlight some exciting recent developments.

I will discuss some ideas and problems in the theory of wetting and capillary phenomena. I will comment on potential analogies with problems encountered in present-day string theory.

Cosmological observations show that most matter in the Universe is non-baryonic and "dark" (really: transparent). This requires the existence of a new kind of matter, beyond that described by the Standard Model of particle physics. The existence of such matter is predicted by supersymmetric extensions of the Standard Model. In this talk I will discuss how the relic density of supersymmetric Dark Matter is calculated, and how this calculation may constrain the parameter space of supersymmetric models. I will also discuss possible ways to detect these particles, and what can be learned from such a detection.

Would you like to see some toys?

I will do many demos with a wide range of objects from sand to coins to turtles, and discuss theoretical issues, some still open, such as finite-time singularity, integrability that does not seem to come from any symmetry, and chirality.

"We will review the present status of high energy photon, hadron, and neutrino astronomy and discuss its implications for astrophysics, particle physics, and cosmology."

"This talk will explain why a new type of inflation has completely changed traditional ideas about inflationary models. I will explain why reheating often takes place at the same time as inflation. Constraints on supersymmetric models from cosmology are quite different from what we thought previously. CMB fluctuations now have their origin in thermal fluctuations in the hot big bang, rather than quantum fluctuations. "

"In the recent years strong evidence has been obtained of the existence of neutrino oscillations, implying that neutrinos are massive and mix. This provides the first evidence of Physics Beyond the Standard Model of Particle Interactions. I will briefly present the status of neutrino physics and in particular the results from atmospheric, solar and reactor neutrino experiments and their implications for our understanding of neutrino physics. I will discuss the questions which need to be addresses in the future, namely the nature of neutrinos, the number of neutrinos, the values of their masses and the issue of CP-violation. A wide experimental program has been proposed for answering these questions and many experiments are already taking data or are under construction. New exciting results are expected soon. "

" "

"The radical new description of gauge theory as twistor string theory has provided a new framework for the study of gauge theories and gravity. There has been much progress in the past year, such as the use of twistor-inspired ideas to obtain many new results in perturbative Yang-Mills. Applications to gravity are also starting to emerge. This talk will review developments in this field."

"There are a wide variety of naturally occuring excitable media which possess spiral wave vortices. Examples include oxidation waves in chemical reactions, aggregation patterns in amoebae, and electrical depolarization waves in cardiac tissue (believed to play a role in sudden cardiac death). These examples will be discussed and vortices studied as solutions of reaction-diffusion equations. Three dimensional solutions will also be considered in which vortex strings form knots. "

"I will give an overview of work done in the last few years on a novel class of black holes in five dimensions with ring-shaped horizons. In particular, I will discuss their implications for black hole uniqueness, as well as their role within string theory. "

"In recent years the new field of quantum gravity phenomenology has shown that experiments at energies much below the Planck scale may be able to probe effects arising from quantum gravity. In this talk we explore the sensitivity of high energy neutrinos to two postulated consequences of quantum gravity, namely quantum decoherence and Lorentz invariance violation. "

"Until recently, we knew of only two types diagram allowing all-orders summation: ladders and chains. We now know that the Hopf algebra of rooted trees organizes the iterated subtraction of subdivergences generated by all nestings and chainings of primitive divergences. It thus offers the prospect of more powerful summations of renormalized perturbative quantum field theory. I shall describe the analytical, combinatoric and Hopf-algebraic structure of a summation of diagrams whose divergence structure is described by undecorated rooted trees, generated by a single skeleton term. The exact results will be compared with Pade-Borel approximations. The Hopf algebra reveals a remarkable structure that enables the momentum dependence of the sum of diagrams to be reconstructed from a non-perturbative result for an anomalous dimension."

"Recently it has been shown that there are some interesting connections between one-dimensional spin chains, supersymmetric field theories and strings propagating in a certain curved space. In this talk I will give an introductory discussion about these connections and how they arise. "

"This year marks the 30th anniversary of the formulation of QCD for numerical simulation on a space-time lattice, but only recently has it become possible to do the calculations with few percent accuracy required to contribute to high precision tests of the Standard Model. I will outline how lattice calculations are done and the breakthrough that has meant agreement with experiment for simple hadron masses at last. I will review recent results and what can be expected in the near future for the hadron spectrum and the form factors needed for CKM tests."

"Until recently, we knew of only two types diagram allowing all-orders summation: ladders and chains. We now know that the Hopf algebra of rooted trees organizes the iterated subtraction of subdivergences generated by all nestings and chainings of primitive divergences. It thus offers the prospect of more powerful summations of renormalized perturbative quantum field theory. I shall describe the analytical, combinatoric and Hopf-algebraic structure of a summation of diagrams whose divergence structure is described by undecorated rooted trees, generated by a single skeleton term. The exact results will be compared with Pade-Borel approximations. The Hopf algebra reveals a remarkable structure that enables the momentum dependence of the sum of diagrams to be reconstructed from a non-perturbative result for an anomalous dimension. "

I shall try to present an insight on the relevance of Supersymmetry in nature.

"In theories that postulate the existence of extra, spacelike dimensions in nature, the production of black holes may be greatly enhanced. Like their four-dimensional analogues, these black holes emit Hawking radiation in the form of particle modes. The detection of these modes in the laboratory can give us valuable information concerning the dimensionality of spacetime since both the amount and type of radiation emitted strongly depends on the number of extra dimensions that exist in nature."

"Precision calculations in quantum field theories have played an important role in the last fifty in the development of QFT itself as well as in the testing of both quantum electrodynamics and the electroweak theory. I will survey some of these contributions, and then describe the developments that are ushering in the era of precision calculations in quantum chromodynamics, the remaining component of the Standard Model of particle physics. I will discuss the prospects for precision physics in QCD and its uses. "

"We implement mathematical techniques developed for the study of quasicrystals and fullerenes to address open problems in virology. In particular, we use tiling theory (a theory that considers tessellations of surfaces by a set of basic shapes) to explain the location of the protein subunits in the viral capsids, that is in the protein shells protecting the viral genome. We furthermore show how affine extensions of noncrystallographic Coxeter groups can be used to obtain shell models for the packing of the viral genome. Finally, we point out an intriguing connection between the geometry of fullerenes, viral capsids and Skyrmions. The presentation will be elementary, and will not require any previous knowledge in any of the above areas. "

"One approach to solving the problem of quantum gravity (reconciling and extending General Relativity and Quantum Theory) is based on the causal set hypothesis, which states that the deep, quantum structure of spacetime is discrete and is what is known in mathematics as a ``partial order'' or ``poset'', a kind of extended family tree. Causal set theory has now reached a stage at which questions of phenomenology are beginning to be addressed. This talk will introduce the basic concepts and motivations behind the hypothesis and address some of the latest developments which include: (i) an apparently confirmed order of magnitude prediction for the cosmological constant, the only prediction made in any proposed theory of quantum gravity that has been subsequently verified by observation; (ii) a classical stochastic causal set dynamics which is the most general consistent with the discrete analogs of general covariance and classical causality; (iii) the formulation of a ``cosmic renormalization group'' which indicates how one might in principle solve some of the ``large number puzzles'' of cosmology without recourse to post-quantum-era inflation; and (iv) a rigorous characterisation of the ``observables'' (or ``physical questions'') of causal set cosmology, at least in the classical case. "

"Beyond-the-Standard-Model-physics is required to accomodate neutrino masses and the excess of matter over antimatter observed in the Universe (baryon asymmetry). These data can be fit by the supersymmetric seesaw, a theoretically attractive extension of the Standard Model which induces neutrino masses, a baryon asymmetry and lepton flavour violation (eg mu --> e gamma). I will consider the question: ``can the baryon asymmetry produced in the SUSY seesaw be predicted from laboratory observations?'' "

"The status of dark matter, both baryonic and non-baryonic, will be discussed. I will review various aspects of galaxy formation, including the successes and current challenges, and discuss how the evolution of the baryonic component of galaxies could impact these issues."

"Recent developments have shown that string theory is really a theory of particles, strings, membranes and other higher dimensional objects, generically called branes. I will review some properties of these solitonic extended objects. Interestingly, understanding some features of these branes has led to new approaches to more conventional lower dimensional physics. In particular, I will describe how branes have recently proved very useful in understanding properties of gauge theories and black holes."

"During the past decade, scientists and engineers have assembled a toolkit of experimental techniques which allows direct access to some of the smallest things in nature. We can now see individual atoms, pick them up, and build new structures atom by atom. This ability to work at sub-microscopic lengthscales is called nanotechnology, and promises a revolution in computing, medicine, manufacturing and environmental science to rival that of the Industrial Revolution and the Internet. In this lecture I answer the two questions: how does one make and study tiny things and why might it be useful?"

"The 8th of August this year was the centenary of the birth of Paul Dirac, one of the founders of quantum theory and the author of many of its most important subsequent developments. This talk will give some account of his early development, the influences on him and how he came to make his early great discoveries. "

"We will review the scientific case for neutrino astronomy. It has been made since the 1950's by pioneers who realized that, of all high-energy particles, only neutrinos can directly convey astrophysical information from the edge of the Universe and from deep inside its most cataclysmic high-energy regions near black holes. With the Antarctic Muon And Neutrino Detector Array (AMANDA), we have performed the first scans of the sky using neutrinos of TeV-energy and above as cosmic messengers. We have searched with improved sensitivity for magnetic monopoles, cold dark matter and TeV-scale gravity. Most importantly, by observing neutrinos produced by cosmic rays in the Earth's atmosphere, we present a proof of concept for an expandable technology with which to build the ultimate kilometer-scale neutrino observatory, IceCube. "

"E-science is a new approach to science, in which geographically distributed researchers exploit collaboratively computers, data and instruments, wherever they may be in the world. A new infrastructure called the Grid, a much-enhanced world-wide web, will be created to access these computational resources and extract knowledge from them. E-Science and the Grid will facilitate the formation of virtual organisations, transient groups co-operating on challenging problems. In due course, the Grid will revolutionise the business world and transform our daily lives by making information as commonplace as electricity. The Universities of Edinburgh and Glasgow have established the National e-Science Centre to lead the UK effort, to align it with international developments and to propagate e-science techniques rapidly to industry and commerce. Richard Kenway will describe the concept of e-science and our first steps towards this IT revolution."

"The theory of symmetry breaking in presence of gauge fields is presented, following the historical track. Particular emphasis is placed upon the underlying concepts."

"Beside the well known crystallographic solid, nature has in the last two decades revealed to us another solid phase: the quasicrystal. It behaves in almost all aspects as an ordinary crystal: it can be cleaved only in certain discrete directions, it grows in nicely symmetric structures, and its diffraction patterns consist of Bragg peaks. The defining difference with ordinary crystals is that its rotational symmetries are forbidden by the theory of crystallography. Quasicrystals seem to show both periodicity and rotational symmetry, but mathematically these symmetries are not compatible. In this colloquium I present some tiling models that show these same properties of physical quasicrystals. A tiling is a complete and non-overlapping covering of space by copies of a few geometrical objects. Here they are studied as statistical models, so that a whole ensemble of tilings is considered. We will focus on cases where the thermodynamic quantities can be calculated exactly. "

"QCD sum rules are a very versatile tool for the calculation of nonperturbative quantities in QCD. Their application ranges from hadron masses and decay constants to wave functions and form factors. I will give an overview over both the basics and more recent developments of the field."

"One of the most important observations in nature is that CP is violated. String theory does not predict this a priori. In this talk I discuss why, and show how CP may be spontaneously broken in the effective action. Nevertheless there remains a serious conflict with experiment that makes it difficult to reconcile string theory with reality."

"The expansion of the universe can convert virtual particles in the quantum foam of vacuum quantum fluctuations into real particles. Although Schroedinger studied this phenomenon in 1939, only recently have we been able to observe the effects of particle creation in the expanding universe. Perhaps the universe displays the pattern of early-universe vacuum quantum fluctuations. "

"Noncommutative geometry of some form is expected to be relevant for the description of spacetime beyond the Planck scale. Recently it was realized that noncommutative geometry also arises naturally in non-gravitational settings and plays an important role in the physics of D-branes. Field theories on noncommutative spacetime have been constructed and studied extensively. Due to their "stringy" and nonlocal nature, they exhibit intriguing perturbative and nonperturbative properties. More formal developments as well as phenomenological aspects of the physics of noncommutative geometry will be discussed."

"One of the more remarkable results of the last years is the emergence of correspondences between gravity and gauge theories. These dualities allow one to translate many deep problems in quantum gravity, such as the quantum mechanical behaviour of black holes and the sum over different space-time geometries, into often equally deep issues in local quantum field theory. Vice versa, typical quantum effects in gauge theory dynamics such as confinement and chiral symmetry breaking can be reformulated in a geometric language. Many of these dualities make use of string theory as an overarching structure."

In this talk I will discuss recent work with Aleix Gimenez-Grau and Pedro Liendo (ArXiV:2012.00018). In this work, we have studied boundaries in three-dimensional N=2 superconformal field theories, which preserve one half of the original bulk supersymmetry. There are two possibilities which are characterized by the chirality of the leftover supercharges. Depending on the choice, the remaining 2d boundary algebra exhibits N=(0,2) or N=(1,1) supersymmetry. For N=(1,1) supersymmetry, some of our results can be analytically continued in the spacetime dimension while keeping the codimension fixed. This opens the door for a bootstrap analysis of the ϵ-expansion in supersymmetric BCFTs. Armed with our analytically-continued superblocks, we prove that in the free theory limit two-point functions of chiral (and antichiral) fields are unique. The first order correction, which already describes interactions, is universal up to two free parameters. As a check of our analysis, we study the Wess-Zumino model with a supersymmetric boundary using Feynman diagrams, and find perfect agreement between the perturbative and bootstrap

[based on work in 2010.11813]

Developments in N=4 super Yang-Mills have yielded a huge variety of ways to calculate and express amplitudes, showcasing a number of different properties and connecting to many areas of pure mathematics. Of particular interest are the amplituhedron and related geometry-based approaches. We would like to develop a similar understanding of supergravity. In this talk, I will present some steps mirroring one approach taken in N=4. Starting with the N=7 theory and on-shell diagrams, I will talk through a new recursion scheme which automatically incorporates the bonus relations. This gives us a convenient way to recover known formulae. I will then show how the use of momentum twistors defined over different coordinate patches highlight some interesting features, including a geometrical interpretation of the 6pt spurious poles and their cancellations.

We will review the recent progress in the understanding of irrelevant deformations of two-dimensional CFTs and its holographic semiclassical gravity description in AdS. In particular, we will consider a class of solvable deformation in the CFT given by coupling the product of the left- and right- moving stress tensor. We show that, holographically, the deformation removes the asymptotic region of AdS leaving the boundary at finite radial distance. We will present some precise computations from both sides that support the proposal.

I will discuss recent progress towards understanding the thermodynamics of accelerating black holes. Starting with an asymptotically AdS bulk, for which one has good computational control, I will explain how the conical deficit responsible for acceleration may be interpreted as a thermodynamic parameter and elucidate the origin of its conjugate potential, the "thermodynamic length". The impact of acceleration on the black hole phase space will be discussed. I will then move onto the asymptotically flat case, presenting some of the challenges and a proposal for a first law of black hole mechanics.

I will introduce the AdS/CFT correspondence with particular focus on the connections between entanglement and classical geometry. I will then explain the Gao, Jafferis, Wall construction for making a traversable AdS wormhole by turning on a double trace deformation in the boundary CFT. If there is time I will discuss my recent work on applying this construction to near-extremal charged black holes.

My aim is to make this talk understandable for people with no prior knowledge of AdS/CFT and to highlight the things that I think are really cool while avoiding too much technical detail.

Machine learning is soooooo 2019, Quantum Machine learning is what all the hip cool kids do now.

In this talk we will briefly introduce "classical" neural networks and a quantum extension known as a Variational Quantum Classifier. By combining quantum computing methods with classical neural network techniques we aim to foster an increase of performance in solving classification problems.

The talk will mostly be pedagogical, acting as an introduction to the subject, and hopefully will be approachable to those who haven't used "regular" ML before. I will also discuss the results of some of my recent research. We have applied our QML model to a resonance search in di-top final states. We find that this method has a better learning outcome than a classical neural network or a quantum machine learning method trained with a non-quantum optimisation method.

Quantum machine learning may sound like a meme but I promise its mostly not.

[talk based on https://arxiv.org/pdf/2010.07335.pdf]

In flat space, four-point amplitudes of closed strings in type IIB string theory are described by the Virasoro-Shapiro amplitude. It is of great interest to generalise this to curved backgrounds and in this talk we focus on string theory in AdS5xS5, which is dual to N=4 SYM. We introduce a simple 10d scalar effective field theory describing the stringy corrections to supergravity in AdS5xS5 from which we can systematically derive all four-point 1/2-BPS correlators described by tree-level string theory. To do this we introduce a new 10d formulation of Witten diagrams and the Mellin transform which treats AdS and S on equal footing.

Symmetry is one of the most useful and fruitful tools in the analysis of quantum field theory. In the recent months, there has been a rapidly growing interest in generalised global symmetries, also known as higher form symmetries. In this talk I will introduce the concept of higher form symmetries by first recalling the definition of ordinary global symmetries and then generalising it. I will then discuss the example of 4d Yang-Mills theory to further elaborate. In the second part of the talk, I will explain how higher form symmetries can be found systematically from geometric engineering and how flux non-commutativity in type IIB results in mixed 't Hooft anomalies for the defect group which constrain the global structures of the corresponding field theories.

As the search for physics beyond the Standard Model (BSM) continues, the Standard Model Effective Field Theory (SMEFT) has become a useful tool to constrain deviations from the SM in a model-independent way. In this talk, we consider the associated production of a Higgs boson and a photon in weak boson fusion (WBF), with the Higgs boson decaying to a pair of bottom quarks. I will present a cut-based analysis and multivariate techniques to determine the sensitivity of this process to the bottom-Yukawa coupling in the SM and to possible CP-violation mediated by dimension-6 operators in the SMEFT.

Models of gauged U(1)Lµ−Lτ can provide a solution to the long-standing discrepancy between the theoretical prediction for the muon anomalous magnetic moment and its measured value. In this talk, we explore ways to probe this solution via the scattering of solar neutrinos with electrons and nuclei.

In this discussion, we shall first review various kinds of entropies and a pictorial representation of path integrals. Then we shall review the classical information paradox (in terms of an entropy inequality). We shall then see how the 'Island formula' gives the page curve for the entropy of black hole radiation. Finally, we shall present a sketchy proof (using replica wormholes) of the Island Formula.

Inspired by AdS/CFT correspondence, we review the holographic dual of the two dimensional Schwinger model. Using a Chern-Simons theory in one dimension higher we will compute the vector and axial currents through the holographic Ward identities. We find that the result reproduces the correct expression for the chiral anomaly of the boundary theory.

Outreach and public engagement are becoming more and more popular amongst researchers, and are now even included on the REF. I will discuss what doing outreach and public engagement involves and whether or not they achieve their goals, looking at data from both scientific studies and from the IPPP's Modelling the Invisible exhibitions.

We shall discuss a method of classifying '2-character' 2D RCFTs based on the Modular Linear Differential Equations (MLDEs) that the characters of their respective partition functions satisfy. This method is popularly known as the MMS Classification. This method has helped not only in the classification of 2D RCFTs but has also given rise to new '2-character' 2D RCFTs which were not known before.

This talk discusses on how to identify events with fatjets from charming Higgs decays, H→cc, at the LHC. To reduce the overwhelmingly large backgrounds and to reduce false positives, we consider applying a combination of jet shape observables and imaging techniques, using a selection of neural network architectures.

In recent years it has become increasingly apparent that the study of BPS states is highly applicable not only to physics but several areas of mathematics as well. For example, BPS states are important objects in black hole physics, homological mirror symmetry and enumerative geometry. It is therefore important that we develop an efficient method of calculating the BPS states of a theory. This is made more complicated by the fact that the BPS spectrum discontinuously varies upon crossing certain surfaces in parameter space- giving rise to the so called 'wall-crossing phenomenon'. In this talk we develop the method of BPS quivers which gives us a way to understand all the BPS chambers of a theory in a purely combinatorial way.

On-shell diagrams are a useful tool for calculating and manipulating amplitudes. For N=4 SYM, they can be used to recurse amplitudes to all loop orders but their application to supergravity is less clear.

I will review how to calculate tree level amplitudes in these theories using recursion relations and on-shell diagrams. I'll then look at what they can tell us about 1-loop amplitudes and their leading singularities, hinting at possible new expressions for n-point MHV supergravity amplitudes at 1-loop.

In this talk I will give an introduction to B-meson mixing, focusing on the determination of non-perturbative input through HQET sum rules. I will demonstrate how the sum rule works and then highlight the advantages of using it over alternative methods, i.e Lattice QCD, in the context of the bag parameter. Finally, I will illustrate the importance of mixing constraints for the determination of the CKM matrix and in testing the Standard Model.

In this talk we will derive an expression for conserved charges in Lovelock anti'“de Sitter gravity for solutions having k-fold degenerate vacua, making manifest a link between the degeneracy of a given vacuum and the nonlinearity of the energy formula. The level of degeneracy fixes the relevant order in the curvature where the mass of a black hole solution is contained. As a matter of fact, the full charge can be consistently truncated and expressed in terms of powers of the Weyl tensor. This may be interpreted as a natural generalization of the Ashtekar-Magnon-Das formula for any Lovelock-AdS gravity theories.

I will review a technique to obtain analytical spinor-helicity rational coefficients for loop amplitudes from floating-point numerical evaluations, with explicit examples for QCD processes. Afterwards, I will discuss the Cachazo-He-Yuan formalism for massless scattering. I will show how arbitrary-precision numerical solutions of the scattering equations lead to compact analytical tree-level amplitudes in a variety of theories, including the first complete set of five-point (DF)^2 amplitudes.

Compact stellar objects such as white dwarfs (WDs) have been proposed as potential probes to set constraints on dark matter (DM) particles. When DM scatters off nuclei, kinetic energy is transferred to the star that can give rise to an observational signal. Previous works did not consider relativistic effects on the calculation of the DM capture rate in WDs. However, since WDs are very dense objects, these effects can lead to sizeable corrections to the DM scattering cross-section. We present preliminary results of such computation and also study the impact of the inner structure and finite temperature of these stars on the DM capture rate.

The 2D Yang-Mills theory is an example of a quantum field theory which can be solved exactly without resorting to perturbation theory. '˜Solving' the theory means finding an exact expression for the partition function of the theory on a Riemann surface Σ of genus g and area A. The free 2D Yang-Mills theory partition function only depends on A and g, and not the special geometry of Σ, so in the zero area limit it is a topological quantum field theory. I will calculate the partition function by first considering the theory on a cylinder in the Hamiltonian formulation. Then, once we have the partition function on a cylinder, I will use the '˜gluing rule' to find the partition function on a Riemann surface of genus g.

Lifetimes are among the most fundamental properties of elementary particles. Our project aims to carry out a precise determination of the lifetime ratio Ï„(Bs) / Ï„(Bd), which happens to be very sensitive to higher power corrections because of multiple cancellations arising. A comprehensive study of this observable could then provide a unique way to test the theoretical framework and indirectly constrain the size of possible new physics contributions. In this talk I will present the status of our project with some preliminary results.

Meson decay rates are often difficult to calculate using QCD, especially in the case of high spin mesons. However, the problem may instead be studied by modelling the meson as a string in an holographic background. Recent work suggests that the problem may be further simplified by Wick rotating the time coordinate of the spacetime and using an instanton method. This talk will demonstrate a simple example of how to build this model, starting with a simple toy 2D flat spacetime example, before moving on to a more realistic example in Sakai Sugimoto spacetime for zero temperature. A possible extension to finite temperature will also be discussed.

We highlight that the anomalous orbits of Trans-Neptunian Objects (TNOs) and an excess in microlensing events in the 5-year OGLE dataset can be simultaneously explained by a new population of astrophysical bodies with mass several times that of Earth. We take these objects to be primordial black holes (PBHs) and point out the orbits of TNOs would be altered if one of these PBHs was captured by the Solar System, inline with the Planet 9 hypothesis. Capture of a free floating planet is a leading explanation for the origin of Planet 9 and we show that the probability of capturing a PBH instead is comparable. The observational constraints on a PBH in the outer Solar System significantly differ from the case of a new ninth planet. This scenario could be confirmed through annihilation signals from the dark matter microhalo around the PBH.

Tree level amplitudes in quantum field theory are represented by rational functions that depend on the momenta of the external states participating in the scattering process. In the last two years, unexpected combinatorial structures have been found for tree-level amplitude's poles of the planar phi^3 [arXiv:1711.09102v2] and phi^4 theory [arXiv:1811.05904v2] from which the full tree amplitudes can be extracted. This combinatorial structure fully characterizes the amplitude at tree level, allowing to make no reference to space-time, but its boundary structure, i.e. external particle states. Moreover, new recursion relations are made possible by this picture and various properties are made manifest.

This type of structure also arise in correlators function (Cosmological Polytope) [arxiv:1709.02813] and famously for amplitudes in N=4 SYM through the Amplituhedron.

In this talk, I will show the main features of this approach to scattering amplitudes analyzing the bi-adjoint phi^3 case. We will finish by looking to how this kind of analysis extends to loop-level for the amplitude integrand in the planar limit.

High Energy Jets (HEJ) provides all-order summation of the perturbative terms dominating the production of well-separated multiple jets at hadron colliders to leading log accuracy. We will present the first calculation of all the real next-to-leading high energy logarithms to the processes of pure jet and W-boson production in association with at least two jets.

The amplituhedron provides a beautiful description of perturbative superamplitude integrands in N=4 SYM in terms of purely geometric objects, generalisations of polytopes. On the other hand the Wilson loop in supertwistor space also gives an explicit description of these superamplitudes as a sum of planar Feynman diagrams. Each Feynman diagram can be naturally associated with a geometrical object in the same space as the amplituhedron (although not uniquely). This suggests that these geometric images of the Feynman diagrams give a tessellation of the amplituhedron. I will begin by introducing the amplituhedron and the twistor Wilson loop, and explain how to associate a geometry to each Wilson loop diagram (WLD). I will go through the NMHV case, then go on to show that beyond NMHV the WLDs do not give a tessellation of the amplituhedron.

Machine learning (ML) is used everywhere in our everyday life nowadays, from image recognition to language translation. What is it though and how does it work? In my talk I'll give a short introduction on machine learning and more specifically on artificial neural networks (aNN), the most common structure of ML. In the second part of the seminar, I'd like to go through an example of writing the code of an aNN and how you can implement it on actual data. For this you will only need to bring your laptop.

Geometry has long been an important tool in physics, finding it's place in everything from Einstein's theory of General Relativity to the structure of Lie groups in Quantum Field Theory. In this talk I will extend the reach of geometry even further by presenting the Eisenhart lift. This formalism allows the effect of any conservative force to be re-expressed as a consequence of the geometry of a curved manifold. I will present our recent work on extending the applicability of the Eisenhart lift to scalar field theories. I will show how the Eisenhart lift allows us to write any scalar field theory in a kinetic-only form where the effect of the potential is incorporated instead into the kinetic terms. Finally, I will show how applying this formalism to the theory of inflation can offer a novel solution to the measure problem. By incorporating the inflationary potential into the geometry of phase space we find the total volume of this space becomes finite. We can thus unambiguously distinguish finely-tuned and generic sets of initial conditions for inflation. This talk is based on arXiv:1806.02431 and arXiv:1812.07095.

Neutrino physics is a field very familiar with experimental anomalies. In this talk, I will discuss the latest and most discussed anomaly, the 4.7 sigma excess of electron-like events in MiniBooNE. After showing that minimal scenarios with extra sterile neutrinos are not viable, I will present a new class of BSM models where "dark" neutrinos are introduced to explain the anomaly. We then derive novel constraints on these models and end with a discussion on why solving the MiniBooNE puzzle is so challenging.

I will give an insight into how we can use the technology of holographic duality to study U(N) or SU(N) gauge theory when N is "large". One aspect we can hope to study is its so-called "generalised global symmetry" (GGS) structure. GGS is also a useful tool in its own right for constructing effective field theories for various physical problems. If time allows I will briefly describe some of these applications.

Generalised unitarity and on-shell recursion relations have led to the automation of numerical computations for high multiplicity NLO matrix elements, whereas analytical expression are often still too complicated to be determined. I will introduce a new technique aimed at obtaining such analytical expressions through the analysis of numerical spinor helicity amplitudes. I will first discuss how the structure of poles and zeros can be determined from single and double collinear limits in complexified momentum space. Secondly, I will show how sufficiently high precision floating point arithmetic can be used to reconstruct the amplitude.

Twistor theory emerged in the 1960s as a possible method of understanding quantum gravity, and more generally physics, in terms of holomorphic geometry. While little progress has been made in understanding quantum gravity, twistor geometry led to several interesting results in mathematical physics and geometry. In this talk I will discuss one such result, the remarkable Penrose-Ward correspondence, together with the foundations of twistor theory and its applications to the study of Yang-Mills instantons.

In this piece of performance art, I will talk about two of my recently completed projects on direct detection of dark matter and how the dark matter problem can be connected with flavour anomalies. Both pieces utilize the non-relativistic effective field theory formalism that has consumed much of my time in the last few years.

I will review the scattering equations, which give a unified description for calculation of tree-level amplitudes of massless particles, going over basic mathematical properties of the equations on the blackboard. I will then pose the problem of numerical solutions to the equations and describe my current ideas for efficient algorithms, by gradient descent and by soft limits. I plan to publish a paper with a corresponding computer package to implement these algorithms in the next few months. I know some C++, but I'm hoping to find someone who has more experience than I do to collaborate with, if anyone is interested.

The KKLT (Kachru, Kallosh, Linde, Trivedi) construction of de-Sitter vacua (hep-th/0301240 ~ 3000 citations) is very important for string theory. One key ingredient of the KKLT paper is the result from a previous paper (hep-th/0112197) by KPV (Kachru, Pearson, Verlinde). However, the KPV paper is highly controversial so the KKLT construction is thrown into question as well. In this talk, I'll (schematically) consider the debate around KPV and talk about our recent paper (1812.01067) on the topic. I'll present our results, which I believe greatly affect the KPV debate, and also the conceptual ideas of our analysis.

Dark matter and neutrinos provide the two most compelling pieces of evidence for new physics beyond the Standard Model of Particle Physics but they are often treated as two different sectors. In this talk, I will review the observables associated to these interactions and discuss different UV-complete models where neutrino-DM interactions lead to the strongest experimental signatures.

String theory has an enormously rich structure, which incorporates all known physics in some form or other. However, a major difficulty that phenomenologists have faced is to produce stable de Sitter vacua. After some very general comments that set the stage for string phenomenology, with an emphasis on some contemporary developments, I will proceed to review some of the basic ingredients for model building in type I string theory. In particular, I will discuss orientifold theories with toroidal compactifications, the inclusion of branes and half-branes, as well as a SUSY-breaking mechanism known as coordinate dependent compactification. This discussion will provide us with a basis to review an upcoming paper written in collaboration with Steve Abel, Emilian Dudas and Herve Partouche. In the paper we classify a certain class of `near'-stable non-supersymmetric open string theories with near vanishing cosmological constant.

In this talk I will introduce alpha space in one dimension by solving the Sturm-Liouville problem for the SL(2,R) Casimir. I will then explain how conformal blocks map to simple poles in alpha space before deriving bounds on OPE coefficients and analysing the crossing kernel in the integral version of the bootstrap equation. If I have time, I will also talk about how we can learn about anomalous dimensions from higher-order poles, about alpha space in d > 1 and about how we can naturally access the Regge limit.

As direct detection experiments improve, the sensitivity of our searches is rapidly approaching a region of the dark matter parameter space known as the "neutrino floor". Known to many as "that brown line at the bottom of all the dark matter plots", I believe that the neutrino floor represents a dynamic and interesting area of physics. In this talk I argue that rather than being afraid of the neutrino floor, dark matter physicists should be excited by the prospect of putting competitive constraints on the physics of a phenomenon which until last year was entirely unobserved, and should at least be cautious of dismissing the neutrino floor as merely a problem for the far future. Much of the content of the talk will be based on my recent paper [1809.06385].

In this talk I want to present a conjectural relation between conformal correlation functions and S-matrix in flat space. I will first introduce some basics of Witten diagram and a little about holographic duality. Then I will discuss the flat space limit of Witten diagram (with two heuristic calculations) and its relation to the more familiar Feynman diagram. Hopefully at the end of the talk I can make the conjecture somewhat convincing.

Come for an easy going introduction to the coherent state formalism in QFT. Bring a pen and paper if you like as there will be the odd short optional problem along the way. We start with a quick review of coherent states in QM before moving to QFT. The coherent state formalism we will build is essential in the semiclassical calculation of multi-particle amplitudes in my recent paper [1810.017222].

In this talk, we will introduce and review some of the known instances of Moonshine, describing their main features and similarities. Mathieu Moonshine concerns a surprising observation relating superstring theory to the representation theory of a particular sporadic group, Mathieu 24. This is reminiscent of Monstrous Moonshine, in which it was discovered that the coefficients of the modular j-function are related to the representation theory of the Monster group, and whose physical interpretation is linked to the bosonic string. After reviewing these various instances of Moonshine, we will then discuss recent and ongoing work aimed at extending the observations of Mathieu Moonshine to a new class of theories. In practice, this means we compute a suitable supersymmetric index for a class of non-linear sigma models whose current algebras are described by a large N=4 superconformal algebra.

The aim of this talk is to discuss the reconstruction of analytical one loop amplitudes from numerical results obtained by BlackHat. This allows to avoid large intermediate expressions that traditionally appear in this type of analytical calculations.

To keep things accessible, in the first part of the talk I am going to review Colour Ordering, the Spinor Helicity Formalism, Little Group Scalings, BCFW recursion and Generalised Unitarity. Afterwards I will explain: 1) how exploit the structure of poles and zeros, arising from single and double collinear limits in complexified momentum space; 2) the importance of introducing spurious singularities and of partial fractioning large expressions in order to respect the pole structure; 3) how to fit numerators by repeatedly evaluating generic ansätze in particular collinear limits. This will be done by running parts of the code I wrote on an iPython session.

This work aims to tackle a problem which many of us face, in generality. In particular, Direct Dark Matter Detection is usually framed in an over simplified way, which makes unclear the actual power of this technology. The reason for keeping things simple is to make our lives easier and our calculations finish quicker. However, a small group in the IPPP have managed to show that you can circumvent the Direct Detection calculation altogether using PROFESSOR, giving you huge speed up factors and allowing you to do more general analyses than ever before. Also you'll get the results before pub time.

I will spend some time recapping the physics and motivating the need for more complexity. Time permitting, I will also have a discussion on the non-relativistic effective field theory basis, since we have many who work with EFTs in some way. Most importantly, I will give you the opportunity to play with my code and I'll set you some fun challenges!

The first half of this talk is on principles of functional programming in Mathematica, in which we will discuss the use of options and attributes, the development of custom notations, and how functional programming fits into the practice of physics. The second half will discuss techniques for organising computations, including namespaces and scoping, debugging techniques, package-generation by notebooks, and the implementation of multiple systems of backups.

The first half of the talk will be an introduction to resurgence. Resurgence is a way of dealing with asymptotic series, which are ubiquitous in physics, appearing in fluid mechanics, condensed matter, the Standard Model, String Theory and everything in between. I hope to explain most of the basic concepts in a pedagogical manner. I will then discus the contents of arXiv:1711.04802, the first example of resurgence in a QFT. This is done by way of Chesire Cat Resurgence, which I hope to explain.

In this talk I will discuss the Standard Model Effective Field Theory (SMEFT), a very general model to describe the interaction of the Standard Model with fields at masses far higher than those of the SM. I will discuss the necessity for, and techniques of using dimension-6 operators to compute observables at next-to-leading order (NLO) and the phenomenological applications of such calculations.

In recent years a lot of progress was made in the calculation of scattering amplitudes without the use of Feynman diagrams. In this talk I will discuss non-perturbative calculations in strongly coupled N=4 super Yang-Mills theory in the high-energy regime or more specifically, the multi-Regge limit. I will give a brief introduction to amplitudes in N=4 SYM and to the multi-Regge limit and why we can find all-loop results in this regime. Since we investigate scattering amplitudes at strong coupling, we can make use of the AdS/CFT-correspondence where the calculation reduces to the solution of a system of non-linear, coupled integral equations which simplify in the multi-Regge limit. I will review the calculation of the six-point amplitude which is fully known at all loop orders before we will investigate higher point amplitudes about which much less is known.

Fluid dynamics is a very old subject, with thousands of papers being written on it every year. Landau and his contemporaries compiled the underlying principles of fluid dynamics into a coherent framework of hydrodynamics, and until very recently, most of the following work was on the application of their ideas in the real world. But with the introduction of the fluid/gravity correspondence in 2008, fluid dynamics regained the attention of fundamental physicists. This has lead to many new insights and developments in our understanding of fluids over the past decade. In this talk, I will try to forget everything we already know about fluids from our daily lives, and develop them from a fundamental perspective of quantum field theories. Hopefully, this will allow the audience to better appreciate some of the recent advancements in hydrodynamics. The talk is going to be extremely basic and hand-wavy, but if time permits, I will comment towards the end on how my work fits into this bigger picture.

Neutrinos have a non-zero mass, this is a very well established concept. However, we are still far from understanding why. Also numerous experiments have reported anomalous results in the last decade, hinting at physics beyond the standard model. So, long story short, there are strong motivations to modify (read extend) the standard model. In this talk, I will show a method to estimate the sensitivity of future neutrino experiment (like DUNE) to searches of new physics in a pseudo model-independent way.

Astrophysics has already told us a great deal about dark matter, including how much there is, and where it is located. Astrophysics also has the potential to constrain the nature of the dark matter particle. In this talk I will discuss historical and current efforts to do exactly this, with a focus of numerical simulations of galaxy formation and gravitational lensing.

In this talk I will consider transport of conserved charges in strongly coupled quantum systems with broken translations, using holographic techniques. Such systems are relevant in condensed matter physics in the context of spontaneous symmetry breaking as well as in the context of momentum relaxation through a lattice. After introducing the relevant concepts, I will give the precise identification of the hydrodynamic modes that diffuse heat and electric charge. As an aside, in the case of explicit breaking I will connect with previous results of DC conductivities from black holes horizons via an Einstein relation for the diffusion constants.

The short answer: A fair amount.

The longer answer: Using a simplified model approach, I will show how the complementarity of indirect DM searches and Large Scale Structure formation can rule out a large region of the parameter space for DM and mediator masses in models with neutrino-DM interactions. It will be my first attempt at giving a whiteboard talk so I hope I can get into the interesting details of some calculations and keep a chilled tone, being very open to discussions.

What is Higgsplosion? Why should you care? By adding literally nothing to the SM it can be argued that there is some physical minimum resolvable scale. Beyond this scale, particles 'Higgsplode' into a large number of soft quanta. This has interesting implications for the UV behaviour of the theory. I will focus on the 1->n calculation in phi4 theory and the effect of Higgsplosion on RG running.

Will Higgsplosion change your life? Probably not, but it's an interesting idea. Come for the memes, stay for the physics.

Crossing symmetry can provide highly non-trivial constraints on many physical systems, in the form of bootstrap equations. By (slowly) introducing a type of Jacobi transform, I will explain how we can rephrase crossing symmetry as an eigenvalue problem for some kernel K and will discuss the merits of this approach.

Recent work has show that the AdS/CFT correspondence can be used to successfully model mesons. In particular, the combination of this method and the instanton method shows particular promise in calculating meson decay rates. I will present the background of this technique, beginning with introducing a toy 2 dimensional model. I will then present my more recent work on a more realistic model using the Sakai-Sugimoto spacetime in for the string picture

In this talk we are going to study hydrodynamics on curved manifolds. We place the underlying quantum field theory on curved space with a spatially periodic metric and chemical potential and we derive the Navier-Stokes equations after the application of an electric field and a thermal gradient. We show how the diffusive dispersion relations are related to the DC conductivity and certain thermodynamic susceptibilities, thus obtaining generalised Einstein relations. Finally, we comment on the derivation of these relations in the context of holographic CFTs.

I will be reviewing methods for NLO QCD calculations, emphasizing the differences between subtraction and slicing methods, introducing Catani Seymour subtraction and N-Jettiness slicing as examples of these methods. I will the move on to discuss the current status of an implementation of N-Jettiness slicing within the Sherpa framework

With the Higgs boson discovered, one of the aims of current collider experiments is to pin down its properties. Its mass has been measured, and its couplings to the gauge bosons and heavy fermions have been determined to be within 10% and 20% of their SM values, respectively. However, the self-couplings of the Higgs boson are in a much worse situation. Due to the small cross section for di-Higgs production (the SM expectation is O(10 fb)), this process has not been seen directly and so the limits that can be extracted from it on the trilinear self-coupling are weak. In this talk, I will outline an alternative approach of using electroweak precision measurements to set constraints on the trilinear Higgs self-coupling. This involves a calculation of the indirect effects that arise in the oblique parameters S and T via the two-loop gauge boson self-energies. The limits that we are able to find are competitive with the constraints from di-Higgs production, and provide complementary information due to the orthogonal approach.

There are a number of intriguing anomalies in rare B meson decays, which could indicate the presence of beyond the Standard Model physics. I will give some background on these anomalies, and then describe some recent work where we attempted to explain one of the anomalies using a model-independent approach. I will show how by looking at 4-quark operators of the form (bs)(cc), we can explain one of the anomalies while still agreeing with very strong constraints from other rare B decays, due to large renormalisation group effects.

This review talk is on the relationships between Diffeomorphism symmetry, Weyl symmetry and conformal symmetry. After getting their precise relationships clear, we will investigate their realizations in classical field theories, their anomalous breaking in the corresponding quantum theories, and their implications for RG flow. We will end with several illustrative examples including QCD, Einstein gravity in various dimensions, and a curious theory with broken 1-dimensional conformal invariance.

Five years ago, a Higgs boson was found at the LHC. Better precision is now required to disentangle the properties of the Higgs and find flaws in the Standard Model. In this talk I will introduce some of the challenges that arise with Next-to-Next-to-Leading Order calculations for Higgs phenomenology.

When you regularise a quantum field theory by truncating your Hilbert space, you generate non-local counterterms. I will discuss whether or not this is as disastrous as it first seems in the context of the Truncated Conformal Space Approach

The Cachazo-He-Yuan (CHY) formalism is a 2d CFT (string theory) which allows the computation of scattering amplitudes. It is equivalent to, and at the same time fundamentally different from, the perturbative treatment of quantum field theory using Feynman diagrams (up to tree-level). It deals in particular with the scattering of n massless particles in an arbitrary D-dimensional flat space-time. This is achieved by a map from momentum space to the Riemann sphere with punctures. Starting from this map, I will discuss the so-called Scattering Equations, the proof for their polynomial form by Dolan and Goddard, and their general solution in terms of the determinant of a (n−3)!×(n−3)! matrix. Finally, I will discuss of how the scattering amplitudes can be obtained.

I will introduce the language of on-shell diagrams for calculating scattering amplitudes via BCFW recursion in N = 4 super Yang Mills theory, and then explain how they can be extended to N = 8 supergravity. I will describe how this approach relates scattering amplitudes to the Grassmannian Gr(k,n), a purely geometric object describing the space of k planes in n dimensions. This link to the Grassmannian introduces a new planar object into the theory which generates physical amplitudes, and exposes a duality between on-shell diagrams and ambi-twistor string theory. I will present my work in progress in this area.

We show how one-loop amplitudes with massive fermions can be computed using generalised unitarity. With this approach, the divergent on-shell cuts can be avoided and the additional information is extracted from the universal IR poles in 4-2ε dimensions and UV poles in 6-2ε dimensions. The aim is to address the formal problem of whether a purely on-shell formulation of amplitudes with masses is possible or not.

In this talk I will discuss how n-body simulations can be used to constrain the nature of dark matter. I will describe the current progress in the field, and where its limitations lie. I will then discuss gravitational lensing as a powerful new probe of dark matter physics, as well as how n-body simulations are being adapted as a tool in this research frontier

The AdS/CFT revolution has triggered a lot of interest in the area of black hole thermodynamics (BHTds). The thermodynamic quantities generally associated to black holes have been worked out for most generic black holes, static, rotating or charged, in either de Sitter or anti-de Sitter universes, as well as in varying numbers of dimensions. The accelerated black hole, represented by the so-called C-metric, is a solution to Einstein's equations which has been known since 1917, with a deeper understanding of it only provided in 1970. I will present the C-metric, along with an introduction to black hole thermodynamics to explain how we recently extended the realm of BHTds to include the accelerated black hole.

The QCD axion is among the best motivated candidates for Dark Matter. In a scenario, where the Peccei Quinn symmetry is restored after inflation the axion field acquires random initial values in causally disconnected patches of our universe. When the axion potential develops around the QCD phase transition fluctuations in the axion field are transferred into order 1 differences in the density contrast on comoving scales of roughly 0.02 pc. Besides, the decay of cosmic strings and domain walls, which are present as remnant of the phase transition, might add further inhomogeneities to the axion density. The regions of high overdensity collapse already around matter radiation equality, forming so called axion miniclusters. The existence of axion miniclusters is crucial to the outcome of axion dark matter direct detection experiments but also of possible indirect signatures. In order to accurately predict the properties of miniclusters detailed knowledge of the density contrast previous to gravitational collapse is crucial. In this talk I explain the production of axions from misalignment, string and wall decay and the difficulties in modeling these processes numerically. I continue by showing recent results of our numerical simulations, which follow the evolution of the axion field around the time of the QCD phase transition and determine the resulting density contrast, for the first time including all three relevant production processes. Our simulations indicate that the inclusion of strings and domain walls puts fluctuation power in scales, which are smaller than the horizon at the time of the QCD phase transition and we expect a large hierarchy of masses extending down to those smaller scales.

Given a metric, what is its energy? The answer to this question is not straightforward to pin down. I shall contrast various constructions of conserved quantities in GR and the way in which they are couched in Hamiltonian frameworks.

In the absence of the direct discovery of a new particle at the LHC, it is possible to parametrise the possible impact of new physics on various Standard Model processes, while being somewhat agnostic regarding the UV origin of such effects. This talk will discuss the extension of the SM with all dimension-6 operators and then focus on the impact this has on the decay of the Higgs to b quarks. In particular, this will be done at NLO including both QCD and a subset of electroweak corrections and the dimension-6 extensions thereof. The talk will cover aspects of the renormalisation within this framework before presenting the final answer. Finally, a short discussion on possible phenomenological impacts will be presented.

n this talk I'll talk about the topic of my MSc: What is the fundamental structure of space time? After reviewing some aspects of "Black hole mechanics", I will comment on how this leads us to suspect that spacetime is fundamentally a discrete entity. I'll further comment briefly (and for more advance students) some other hints that suggest that the metric is not the most fundamental feature of spacetime. With all these in mind I will then explain the foundations of Causal Set Theory, which takes all these ideas seriously and has had a fair amount of success (although "success" here is a very ambitious term).

Rather than discuss my recent research, I will return to a topic from the first term's CPT lectures that caused me much confusion in my MSc year at Durham: the energy-momentum tensor. Starting with what this tensor actually is and why we need such a tensor, I will (hopefully) clarify its canonical definition as a Noether current and give additional detail regarding the use of and differences between 'active' and 'passive' transformations. I will then consider alternative definitions of the energy-momentum encountered later in the lectures and explore the relationships between these energy-momentum tensors.

The first part of the talk will be aimed at MSc and first year PhD students who should have recently encountered the energy-momentum tensor in lectures, with later parts suitable for anyone who, like me, still has questions regarding this bothersome object.

We propose a new mechanism to generate a lepton asymmetry based on the vacuum CP-violating phase transition (CPPT). This approach differs from classical thermal leptogenesis as a specific seesaw model, and its UV completion, need not be specified. The lepton asymmetry is generated via the dynamically realised coupling of the Weinberg operator during the phase transition. This mechanism provides strong connections with low-energy neutrino experiments.

In this talk we will briefly review our understanding of neutrino oscillations and discuss some of the anomalies at short baseline experiments. These anomalies point towards the existence of a sterile neutrino with a mass at the eV scale and have been the motivation behind many of the efforts in the neutrino community. Some of the motivations and implications of such a sterile are discussed. We will then present our most recent work on NuSTORM, an experimental proposal that looks to search for such steriles in a novel way.

We introduce two new graphical-level relations among possible contributions to the four-point correlation function and scattering amplitude in planar, maximally supersymmetric Yang-Mills theory. When combined with the rung rule, these prove powerful enough to fully determine both functions through ten loops. This then also yields the full five-point amplitude to eight loops and the parity-even part to nine loops. We'll outline a derivation for some of the rules, illustrate their applications, compare their relative strengths for fixing coefficients, and survey some of the features of the previously unknown nine and ten loop expressions.

In this talk I will explain what M-theory is and review some of the novel gauge theories that have been invented in order to describe the interactions of multiple branes within the theory. I will begin by setting out what M-theory is posited to be, and show how it links to both 11d Supergravity and string theory. After reviewing some mathematical concepts- Chern- Simons gauge theories and 3- algebras- I will go on to discuss two candidates for theories to describe multiple M2 branes. These are the BLG theory with N=8 supersymmetry, and the ABJM theory with N=6 supersymmetry. I will conclude by discussing future directions for research in the subject.

The Mathematica package HypExp is currently limited by the small number of basis functions on which it operates in order to produce its series expansions. We examine what these basis functions are, how their series expansions are calculated, and present a new generalised method for calculating them. This new method significantly increases the number of hypergeometric functions that can be expanded using HypExp.

The AdS/CFT revolution has triggered a lot of interest in the area of black hole thermodynamics (BHTds). The thermodynamic quantities generally associated to black holes have been worked out for most generic black holes, static, rotating or charged, in either de Sitter or anti-de Sitter universes, as well as in varying numbers of dimensions. The accelerated black hole, represented by the so-called C-metric, is a solution to Einstein's equations which has been known since 1917, with a deeper understanding of it only provided in 1970. I will present the C-metric, along with an introduction to black hole thermodynamics to explain how we recently extended the realm of BHTds to include the accelerated black hole. If time permits, I shall also talk about the interesting limit of rotating and/or accelerating black holes which prompted this research.

In this work we study a classically scale invariant extension of the Standard Model that can explain simultaneously dark matter (DM) and the baryon asymmetry in the universe (BAU). In our set-up we introduce a dark sector, namely a non-Abelian SU(2)_DM hidden sector that is coupled to the SM via the Higgs portal, and a singlet sector with a real singlet sigma and three right-handed Majorana neutrinos N_i. Due to a custodial symmetry all three gauge bosons Z'^a have the same mass and are absolutely stable, making them suitable dark matter candidates. The lepton flavour asymmetry is produced during CP-violating oscillations of the right-handed neutrinos, which have masses of a few GeVs. All the scales in the theory are dynamically generated and related to each other via scalar portal couplings.

In this short talk we review the concept of hadron-quark duality in B and D meson physics. The discussion will be centered on possible violations on this duality using mixing observables leading to bounds for new physics in experimental searches. One of the most interesting results is the possibility of explaining the experimental measurement for the life-time splitting of neutral D mesons using only a 20% effect of duality violation, this result deserves some consideration taking into account that the theoretical methods available nowadays give answers that are in disagreement with the experimental measurements by several orders of magnitude.

Mathieu Moonshine concerns a surprising observation relating string theory to the representation theory of a particular sporadic group, Mathieu 24. This is reminiscent of Monstrous Moonshine in which it was discovered that the coefficients of the modular j-function are related to the representation theory of the Monster group. In this talk we will introduce a topological invariant of string theories compactified on K3 surfaces, called the elliptic genus of K3, and see how Mathieu 24 appears in this context. To this date, the role of the large discrete symmetry M24 in String Theory is not properly understood. We will then discuss Umbral moonshine, which comprises of 23 examples of moonshine in which the Niemeier lattices are used to connect certain mock modular forms to finite groups.

b-tagged jets, i. e. jets containing a b-hadron, are an important final state at high energy particle colliders, providing insight into some of the more interesting Standard Model processes as well as opening up a channel to test many BSM physics scenarios. I will spend some time reviewing the techniques and current status of b-tagging, focussing on the general purpose detectors ATLAS and CMS, a topic that is often glossed over in theory classes. I will then move on to motivate why simple b-tagging might not be the end of the story, as it does not discriminate against the large QCD backgrounds from gluon splitting. I will show how jet substructure observables can be used to distinguish these cases and will introduce a new observable that might improve the current methods considerably.

NNLO predictions are the bleeding edge of theoretical predictions for the run 2 of the LHC. These calculations give rise to many theoretical and numerical challenges which need to be addressed.

I present an overview on a Monte Carlo integrator for NNLO production. What methods can (or should) be used? What do we expect to achieve? Do we have a way of testing a brand new calculation is actually correct?

The observed leptonic mixing pattern could be explained by the presence of a discrete flavour symmetry broken into residual subgroups at low energies. In this scenario, a residual generalised CP symmetry allows the parameters of the PMNS matrix, including Majorana phases, to be predicted in terms of a small set of input parameters. We study the mixing parameter correlations arising from the symmetry group A5 including generalised CP subsequently broken into all of its possible residual symmetries. Focusing on those patterns which satisfy present experimental bounds, we then provide a detailed analysis of the measurable signatures accessible to the planned reactor, superbeam and neutrinoless double beta decay experiments.

Meson decay rates are often difficult to calculate using QCD, especially in the case of high spin mesons. However, the problem may instead be studied by modelling the meson as a string in an AdS background. Recent work suggests that the problem may be further simplified by Wick rotating the time coordinate of the spacetime and using an instanton method. This talk will demonstrate how a simple toy model of the meson as a string in a flat Euclidean background demonstrates the promise of this method. A way to extend this work to a more realistic model will also be introduced.

Standard Model Effective Field Theory (SMEFT) is a method to parametrise the impact of new physics which may become accessible at higher energies without specifying its UV origin. The new physics is said to be integrated out, leaving behind effective non-renormalisable operators. In this talk, we supplement the Standard Model with all (baryon number conserving) operators which appear at dimension-6 and calculate, to one-loop, the amplitudes for Higgs decays to bottom quarks and tau particles in the limit of vanishing gauge couplings. Special attention will be given to the set-up and renormalisation of the amplitudes in the context of SMEFT.

Despite being a subdominant production mode at the LHC, the ZH-channel is of vital importance for measurements in certain Higgs decay channels. Searches for invisible Higgs decays as well as analyses of H->bbar decays heavily rely on this channel. The presence of non-negligible loop-induced terms in the "Higgsstrahlung" process calls for advanced calculational techniques but also allows for the determination of the sign of the top Yukawa coupling. In this talk I will address both aspects in the context of invisible Higgs decays and decays to bottom-quark pairs.

To understand how the holographic principle encodes bulk geometry holographically in a boundary field theory, one can consider the entanglement properties of states dual to interesting bulk geometries. It has been recently proposed by Susskind that entanglement between multiple field theories is holographically intimately linked to connectivity in the bulk. We considered the entanglement properties of states holographically dual to multiboundary wormholes in the setting of 3D gravity. I will introduce multiboundary wormholes and how to utilise the beautiful structure of 3D gravity to construct them there. I'll additionally talk about constructing and interpreting their holographically dual states. What we find is that the entanglement structure in the limit where all of the horizons become very large is extremely simple. I'll mainly use the three-boundary wormhole as an example. Coming down from this limit, we expect multipartite entanglement have some manifestation in the dual state. In this case we decided to utilise tensor networks to approximate the dual state, so I'll also give an introduction to tensor network representations of quantum states.

Dark matter models are often studied in a simplified form which prevent new physics appearing in flavour measurements (e.g. meson mixing, rare decays). Even in more complex models, minimal flavour violation is generally invoked to achieve the same result. I will talk about an extension of minimal flavour violation that allows for sizeable contributions to flavour observables, and explain how neutral meson mixing can constrain certain dark matter models in this extended framework. I will also present an initial look at the constraints on my model from both flavour and dark matter observables.

It is possible for some classical 1+1-dimensional integrable field theories to accommodate discontinuities in the fields and yet remain integrable, with the fields on either side of the defect related by some set of defect conditions. In this talk, momentum conserving defects in the ATFTs based on the A series of Lie algebras and in the Tzitzeica model are reviewed, and the fact that the defect conditions give a Backlund transformation for the bulk theory is noted. A more general form of the defect is then considered, which is momentum conserving for ATFTs based on the B, C and D series of Lie algebras.

Through an exhaustive exploration of simplified models, we show that the WIMP assumption of thermal production severely restricts the allowed parameter space, once combined with direct and indirect limits.

Neutrinos are one of the least understood particles within the Standard Model. The recent discovery of their massive nature raises the question of whether they are Dirac or Majorana particles. If massive neutrinos are Majorana particles, processes where total lepton number is violated will occur in nature. A particular relevant example of such processes is the neutrinoless double beta decay (0vbb) since it would confirm the Majorana nature of neutrinos.

Beyond the standard mechanisim that drives 0vbb some other exotic contributions arise if one considers alternatives to the SM. An attractive possibility is the Left-Right Symmetric Model (LRSM) which extends the SM electroweak gauge group and has rich phenomenology. In this talk, I will present the new contributions to 0vbb within the LRSM framework and update the limits on the new physics parameters introduced by this model using the latest released data.

The Ryu-Takayanagi hypothesis and its covariant generalization state that the entanglement entropy of a region in a CFT is given by the surface area of a minimal extremal surface in the holographic dual gravitational theory. But what is entanglement entropy? The first half of this presentation will be a gentle introduction to the notion of entanglement entropy as a measure of the amount of entanglement. The second half will be a not so gentle exploration of the entanglement structure of perturbed thermofield double states in 1+1 dimensions, via analysis of their holographic duals - perturbed BTZ black holes.

Unitarity is a fundamental property of any theory required to ensure we work in a theoretically consistent framework. In comparison with the quark sector, experimental tests of unitarity for the 3x3 neutrino mixing matrix are considerably weaker. It must be remembered that the vast majority of our information on the neutrino mixing angles originates from electron and tau neutrino disappearance experiments, with the assumption of unitarity being invoked to constrain the remaining elements. New physics can invalidate this assumption for the 3x3 subset and thus modify our precision measurements. I will discuss where such non-unitarity can originate from and give results on a reanalysis to see how global knowledge is altered when one refits oscillation results without assuming unitarity. There is significant room for new low energy physics, especially in the tau neutrino sector which very few current experiments constrain directly.

The importance of non-relativistic systems in physics cannot be overstated. Although the universe we live in is relativistic, at sufficiently low 'day to day' energy scales, it is governed by non-relativistic laws. Non-relativistic (or more precisely Galilean) physics has been a topic of interest for centuries, and its implications have been well studied and tested. But in last century there has been an exponential increase in our understanding of relativistic phenomenon. Hence it is important to ask if these new exotic relativistic phenomenon (like anomalies) leave any signature on the Galilean physics.

In this talk we will explore a systematic mechanism to translate relativistic theories to Galilean, hence allowing us to study the effect of various relativistic phenomenon on Galilean systems. Rather than the usual 'low velocity limit' this approach is based on 'null reduction' which maps a relativistic theory to a Galilean theory in one lower dimension. We will pay special attention to Galilean hydrodynamics, and construct a relativistic system which will give rise to the most generic Galilean fluid upon null reduction. We will also discuss the shortcomings of the most obvious candidate for such a relativistic system - a relativistic fluid, and explain why its null reduction fails to give the most generic Galilean fluid.

The discussion will be based on an extremely simple case of chargeless non-anomalous Galilean fluid. If time permits, we might comment on charged anomalous Galilean fluids as well. The talk is based on recent papers: arXiv:1505.05677, arXiv:1509.04718, arXiv:1509.05777.

The Inert Doublet Model (IDM) is a minimal extension of the Standard Model that can account for the dark matter in the universe. In this work we study a classically scale invariant version of the IDM with a minimal hidden sector, which has a $U(1)_{\text{CW}}$ gauge symmetry and a complex scalar $\Phi$. The mass scale is generated in the hidden sector via the Coleman-Weinberg mechanism and communicated to the two Higgs doublets via portal couplings. Since the CW scalar acquires a vev and mixes with the Standard Model Higgs boson, we analyse the impact of adding this CW scalar and the $Z'$ gauge boson on the calculation of the relic density and on the spin independent nucleon cross section for direct detection experiments. Finally, by studying the RG equations we find that some points in parameter space remain valid all the way up to the Planck scale.

Skyrmions are candidates for a solitonic description of nuclei, with the topological charge or number of solitons being identified with the baryon number. I will present work on the 2-dimensional analogue, the baby Skyrme model, in an AdS_3 background. I will demonstrate that stable solutions can be formed with no mass term, due to the addition of the metric and that the numerical solutions now take the form of slightly perturbed concentric rings. I will then propose a simple numerical model, that will allow transitions between the different forms of ring solutions to be predicited. Finally, I will touch upon my current work, on how this extends to the full 3-dimensional Skyrme model in AdS_4. I propose that the results in 2-dimensions suggest that Skyrmions in this space, should take the form of multi-layered concentric rational maps.

In this talk I will introduce Moonshine from a physical point of view by introducing the partition function of different (super-)conformal field theories. The non-trivial topological invariant partition function, i.e. elliptic genus will be introduced when we consider N=2 and N=4 superconformal algebras. I will then decompose the elliptic genus of K3 surface in terms of small N=4 superconformal characters and show how to find the Mathieu moonshine explictly from the decomposition.

The Monte Carlo event generator 'High Energy Jets' (HEJ) is unique in its attempt to calculate cross sections (for processes involving two or more jets) to leading logarithmic accuracy in the limit of large invariant mass between the jets. In this talk, I shall review how such large logarithms can arise and how they can be summed to all orders in perturbation theory. Typically this procedure involves stringent kinematic requirements. I shall explain how such assumptions may be relaxed, and discuss how this can lead to improvements in predictions for observables at the LHC.

In this talk I aim to provide a general introduction to the process of computing scattering amplitudes in string theory, with an emphasis on one loop order. I will then briefly discuss how this can be used to aid in the determination of the corresponding low-energy effective field theory.

In the gauge/gravity formalism probe particles for the field theory are realized as strings or D-branes in the gravity side. This talk will be focused on the applicability, recent developments and open questions concerning D3 branes representing probe operators.

In this talk I will discuss why is it important to consistently include b-mass effects in pp->h especially in view of discarding/founding any discrepancies with the Santard Model.

In particular I will present a method called FONLL, which has been applied to other processes, and how this can be extended to this process.

This talk will discuss how threshold logarithms, which become large as the invariant mass of the produced top quarks approaches the partonic centre of mass, can be resummed using techniques from Soft Collienar Effective Theory (SCET). The resummation is possible because a factorisation takes place in the threshold limit which allows one to derive and solve RG equations. It will be shown how the solutions to these equations lead to the inclusion of large logs to all orders in the perturbative expansion and a finite answer obtained. Such results can then be matched with an exact NLO to produce NLO+NNLL results, providing a more accurate determination of differential cross-sections. Finally it is also possible to use the RG equations to obtain approximate fixed order results at higher orders in perturbation theory. The results and discussion will also feature boosted top quark production where approximate N^3LL and N^3LO results are obtained.

We consider the partition function for a conformal field theory with c = 6, N = 4 which describes the internal worldsheet theory of the superstring compactified on K3 and write the partition function as a quadratic function of the N = 4 characters. We then consider the elliptic genus of this model and discover a connection to the sporadic group M24.

In this talk I aim to provide a general introduction to the process of computing scattering amplitudes in string theory, with an emphasis on one loop order. I will then briefly discuss how this can be used to aid in the determination of the corresponding low-energy effective field theory.

In compressed spectra SUSY scenarios, standard LHC searches based on missing transverse energy are not effective. In this seminar, I will introduce two alternative search strategies that use monojet and monotop probes and, for the former case, show that the simulation of high transverse momentum radiation can have a significant impact on exclusion boundaries.

Inspired by the AdS/CFT correspondence and by Skyrme theory (a low-energy effective field theory for baryons), there have been many attempts to use holography as a way of studying strongly-coupled QCD. The pre-eminent example of this is the Sakai-Sugimoto model, in which bulk Yang-Mills instantons in five spacetime dimensions are dual to boundary Skyrmions (which in turn represent baryons). In this talk I will discuss some lower-dimensional analogues of this model, in which modifications to an O(2) sigma model in three spacetime dimensions take the place of the 5-d Yang-Mills instanton of the Sakai-Sugimoto model.

An affine Toda field theory (ATFT) is simply a field theory based on the affine root vectors of a Lie algebra. A defect in a system is a discontinuity with some defect conditions relating the fields on either side of the defect. Defects in ATFTs have been found to have a momentum-like conserved quantity, which is surprising as the system is no longer translationally invariant. The equations of motion at the defect also give a Backlund transformation for the bulk theories. Solitons can be delayed or advanced by the defect. Integrable defects have already been found for ATFTs based on the An root vectors. In `a new class of integrable defects' (Corrigan and Zambon, 2009), an extra field which exists only at the defect was introduced and this allowed a description of defects in the Tzitzeica model. Using this method I find a momentum conserving defect for the An, Bn, Cn and Dn ATFTs.

This talk will be a gentle introduction to the idea of mass factorisation, initial state collinear singularities and why they can be a massive pain. We will be considering them in the context of a non-abelian SU(N) gauge theory in the limit of N -> 3 with 5 light quark flavours.

Instantons (static solutions to 5d Yang-Mills theory) may have great utility in unravelling the mysteries of M-theory, but are also interesting objects in their own right. In this talk, I'll motivate and describe the construction, dynamics and scattering of two instantons in a noncommutative space, where very different behaviour emerges compared to other known soliton solutions.

The observed neutrino mixing pattern could be explained by the presence of a discrete flavour symmetry broken into residual supgroups at low energies. In this scenario, the presence of a residual generalised CP symmetry allows the phases of the PMNS matrix to be predicted as well as the mixing angles. In this article, we study all such predictions associated with the symmetry group A$_5$. We present a derivation of the most general CP symmetry allowed in this context, and compute the predictions of all possible preserved subgroups. We identify those patterns which satsify the present experimental bounds on the mixing parameters and discuss the predicted correlations between angles and phases, including the Majorana phases $\alpha_{21}$ and $\alpha_{31}$. We find that there are $8$ patterns of mixing angles and phases described in terms of a single unknown parameter which can be brought into agreement with current global data. These patterns describe certain correlations between mixing parameters which can be tested by high-precision measurements. To assess this potential, we then focus on upcoming superbeam, reactor and neutrinoless double beta decay experiments, and highlight a number of experimental observations, both from experiments in isolation and in combination, which will allow these predictions to be thoroughly investigated.

Hydrodynamics is the low-energy effective field theory which describes the long-wavelength fluctuations of any interacting QFT. It is characterized by the gradient expansion of an energy-momentum tensor and charge current which satisfy certain dynamical equations. On top of this, an additional constraint has to be imposed which ensures the second law of thermodynamics is respected by any fluid flow. In this talk I will describe a complete solution to hydrodynamic transport at all orders in the gradient expansion compatible with the second law constraint. A key ingredient is the notion of adiabaticity, which allows to take hydrodynamics off-shell. I will furthermore argue for a new symmetry principle, an Abelian gauge invariance that is the underlying reason for adiabaticity in hydrodynamics and elucidates the origin of the second law constraint. This new symmetry should be viewed as the macroscopic manifestation of the microscopic KMS condition. In non-equilibrium situations the macroscopic "KMS gauge invariance" enables us to keep Feynman-Vernon influence functionals under control and to formulate an off-shell effective action that encompasses the entirety of adiabatic fluids in a consistent way.

Constructing full analytic solutions to the Einstein equation is difficult - it requires a lot of symmetries and luck. The classification of near horizon geometries (NHG) allows one to classify possible extreme black hole horizon geometries and topologies in a much simpler set up. Nevertheless given a NHG, there may or may not exist a corresponding black hole solution, let alone uniqueness. I will talk about this inverse problem and discuss the special case in 3d where one can "integrate out" completely to find the most general black hole solution.

Subject of this talk will be the structure of the loop-induced coupling between QCD gluons and the Higgs boson, on which the dominant Higgs production channel at the LHC relies. An infinite Top mass approximation is customarily applied in order to render higher order calculations in this loop-induced channel feasible. We will present our most recent improvements on these approximations in Monte Carlo simulations. Furthermore, the phenomenological potential of studying the gluon fusion Higgs production channel in search for BSM signals in the Higgs-gluon coupling will be discussed.

Recent work interpreting the cosmological constant as a thermodynamical pressure term has yielded interesting results regarding the universality class of AdS black holes. I will focus on the path of expanding this field to black holes with non-AdS asymptotics. Specifically we will go through a thermodynamic definition of mass that is consistent with the first law and a modified Smarr relation.

In this talk the assumption that no new physics is acting in tree-level B-meson decays will be reviewed. The consequences of beyond standard model tree level effects allowed by current experimental data for the precision in the direct determination of the CKM angle $\gamma$ will be also discussed. Interestingly tree level deviations consistent with the experimental measurements available nowadays lead to a non negligible intrinsic uncertainty $\delta \gamma \approx \pm 4^{\circ}$, which can affect the sensitivity expected by LHCb and Belle II for the determination of this observable in the near future.

In this talk, we will introduce the concept of entanglement entropy and the basic techniques involved in its computation under a global quantum quench, focusing mainly on (1+1)d CFTs. Then, we will see how we can use the AdS/CFT correspondence to descibe similar processes from a gravitational perspective.

I will present a way to construct non-supersymmetric models and I will discuss their phenomenological properties.

Precise QCD calculation is a major research area for LHC analysis. It aims on testing our understanding of the Standard Model and develop new tools for analysing experimental results. My talk will focus on the cutting edge progress for Higgs phenomenology, higher order calculations and most importantly the impact of those calculations. For those who have seen LO, NLO, NNLO and NNNLO papers but not sure what it is talking about. This talk would be a introduction of modern QCD to you. For those who have been wondering how does string theory, Yang-Mills and gravity theories apply to LHC study, this talk would illustrate amazingly how much string theorists have contributed to LHC analysis.

I will give a brief heuristic overview of gauge gravity duality and its applications to light quark jet quenching in relativistic heavy ion collisions.

In this talk I will discuss the suitability of the collective coordinate approximation when modelling two interacting solitons in interesting modifications of the non-linear Schrodinger and sine-Gordon equations. These equations have been chosen as systems where the concept of quasi-integrability can be explored and I will give some background to this idea.

Can Dark Matter be made up of magnetic monopoles? In this talk I will explore the cosmological implications of extending the Standard Model to include a dark sector with magnetic monopoles. These models will also introduce vector dark matter and dark radiation. We find that magnetic monopoles, with masses of the order of 100 TeV, can make up a significant fraction of Dark Matter. In addition the long range interactions between dark matter in these models can help solve some of the problems at small scales in theories with cold collisionless dark matter.

A deeply intriguing aspect of the holographic principle is the question as to its generality. It is well-known that there is a correspondence between quantum gravity in asymptotically AdS spacetime and a conformal field theory living at the boundary of AdS. Does such a correspondence exist for theories of quantum gravity whose geometry asymptotes to some background other than AdS? One hopes that by attempting to answer this question one may learn something about why the holographic principle works at all! The familiar AdS/CFT correspondence is fleshed out in terms of a relation between bulk and boundary objects which is called a holographic dictionary. Our enquiry then concerns the existence of a holographic dictionary for asymptotically non-AdS spacetimes. Of particular interest to us are certain backgrounds with non-relativistic isometries for which the corresponding dual quantum theories are therefore non-relativistic. I will briefly talk about Non-relativistic Holography for both asymptotically Lifshitz and asymptotically Schrodinger spacetimes, the latter of which is our current research topic.

I will introduce two formalisms that are often used in QCD and give some simple examples. I will then show how they can be extended to higher dimensions and why this is useful.

Topological solitons are stable, particle-like solutions of field theories, where stability is a consequence of the topological characteristics of a solution. In this talk I will introduce a number of different theories which contain topological solitons including the Skyrme-Faddeev model. In this model string-like solutions are know to exist and to form knotted structures. I will then discuss my recent work in finding new knotted structures within this model.

Non-geometric spaces arise as consistent string theory backgrounds in p-form flux compactifications. In this talk I will explain how these spaces can be geometrised using membrane models. I will then show how to perform quantization using various deformation quantization techniques and discuss topics such as nonassociative field theory.

The problem of "dark energy" is rather simple: we don't know what substances are in the Universe which could make it accelerate (or, look like its accelerating): but, we know about 70% of the Universe must be made of it. There are many scalar field and modified gravity models on the market trying to describe these observations.

In this talk, I will look at a radically different type of theory: material models of dark energy. The theory of relativistic solids is used in a cosmological context, and is built up so that the theory of a relativistic viscoelastic solid can be used as a candidate dark energy model.

The rough idea is simple: generalise Hooke's law (built for a non-relativistic elastic solid), and make the theory relativistic. The work is based on my recent publication, arXiv 1403.1213.

Higher-spin gauge theories have become a lively area of research in recent years. I will try and give a very brief and pedagogical introduction to the subject, in particular talking about exactly what "higher-spin" means, how to write down free theories of higher-spin gauge fields and a brief review of some of its seductively unusual features. If the feeling takes us I might also mention something about Vasiliev theory.

Lifshitz and hyperscaling violating geometries, which provide a holographic description of non-relativistic field theories, generically have a singularity in the infrared region of the geometry, where tidal forces for freely falling observers diverge, but there is a special class of hyperscaling violating geometries where this tidal force divergence does not occur. I will give a short introduction about these properties in Lifshitz spacetime and review some material in Schwarzchild black hole spacetime. Then I will show how to construct the nonsingular coordinate for hyperscaling violation spacetime.

Scattering amplitudes have recently made enormous conceptual progress, mainly by being reformulated in an intrinsically combinatorial way. Some similar formulation is likely to work with much less supersymmetry than N=4. I will quickly outline some of the excitement, reveal some of the mathematical tools that need developing, in particular on how to go beyond the current limitations, and show lots of pretty pictures.

In the last few years there has been a departure from Feynman Integrals as the most efficient way to calculate scattering amplitudes in supersymmetric gauge theories. I will do a whiteboard talk to introduce some of the new diagrammatic methods for calculating these quantities and ask several questions such as "How easy is it to dimensionally regularise these new representations?" Then, finally, can we build these objects primarily from symmetry considerations in an algorithmic way which avoids too much effort? Pictures will be involved in more than one colour!

First I will tell you about some elegant ideas that do work: affine Toda field theory, solitons and defects. Towards the end I will stray into the dangerous territory of having my own ideas. These do not work but they're still elegant.

In the talk I will list and briefly describe a list of useful applications available on the Linux computers in maths. The lists extends from tools to manipulate pdf files to mathematical packages like mathematica or maxima.

I will provide an introduction to Linux, including information on developing advanced skills with the Linux system that can be utilized on any distribution. I will then detail some real-life senarios that happened to Durham students, and how Linux was utilized to quickly and easily provide a solution. I will then detail some security related issues and solutions that all academics should know given the security threats facing academia today. Using the information in the talk, which will provide guided outside work, any attendee will quickly have competency in Linux, including more advanced command-line functions.

I'll give a quick tutorial on 'regular expressions', a fundamental concept in string processing that amounts to an advanced find-and-replace procedure. This is often indispensable for manipulating strings longer than a few characters, or processing long raw text documents. I'll show how to use it to make speedy changes to LaTeX documents.

The human voice has a higher channel capacity than the human fingers. I developed an app which takes strings of natural-language spoken mathematics and formats the result in LaTeX, used standalone or in conjunction with a keyboard. I will give a short demo and any suggested improvements will be appreciated.

A short talk about Mathematica, how it can be used, and some tricks to help solve problems.

Parallel programming for distributed memory machines (e.g. the hamilton cluster) is an indispensable skill for the modern applied mathematician. In this talk, I've tried to distill the 'theoretical minimum' that you need to know to get started, using the most common interface: MPI. This is a library that works with either Fortran or C. The talk will be structured around a simple example of the one-dimensional heat equation. I've uploaded the example code (both Fortran 90 and C++ versions) to https://github.com/antyeates1983/mpi_seminar, in case you want to follow along.

I plan to explicitly show how to install Linux from a Live CD. How to use the repository in Linux to install programs (both via terminal and GUI interface), how to use the command line to install software not included in Linux repositories. I also want to show how to edit configuration files in Linux, and plan to use ClamAV as the example. I also will talk about different layouts with linux (aka Gnome vs. KDE, etc.).

The talk will provide a short description of the Linux operating system, explaining some of the basic features of Linux. I will then talk about software and accomplishing some basic tasks on Linux, such as installing Linux, installing software from repository vs. compiled package vs. source, converting from image, ps files to pdf, basic terminal commands, taking ownership of locked files on linux, and encryption to safeguard files and research.

In the talk I will list and briefly describe a list of useful applications available on the Linux computers in maths. The lists extends from tools to manipulate pdf files to mathematical packages like mathematica or maxima.

Since 2001 postgrad students have passed down an old tattered thesis template from year to year. Last year Steven Charlton gave it some well needed care and attention, packaged it up a bit nicer and made it a little more accessible. I'll be giving a brief introduction to what the template involves, how to use it and how to change it. This is primarily aimed at final year PhDs who are starting to write up, but anyone might find it useful and could perhaps learn something about the inner workings of a latex class.

I will provide a practical overview of some computations that can be done with my Cadabra tensor computer algebra system. Emphasis will be on problems in (quantum) field theory, general relativity and related areas for which other computer algebra systems provide only little or no support. The aim is to provide interested newcomers an idea of the possibilities and a quick introduction.

Python is a general purpose programming language with an emphasis on readability and flexibility and has a well developed ecosystem of scientific computing packages. This makes it an excellent tool for mathematical research, where allowing for rapid iteration and experimentation can often be more important than maximising computational efficiency. I will discuss some of the general features of Python as well as examples of specific applications of interest to mathematicians including efficient matrix operations, symbolic algebra and interactive animations.

This talk will give a brief introduction to Matlab and the advantages it has over other interpretive languages (such as R). The discussion will focus on vectorised solutions, memory management, image processing, the Fast Fourier Transform (FFT) and GPU parallel computing.

I will talk about efficiencies which can be achieved and errors which can easily occur when performing various calculations in R.

What they are, some common types of patterns and why you might use them.

Anyone who has written c or c++ code will have used a compiler to turn their carefully crafted code into something a computer can run. But what do these magical black boxes do your code and how can you use these to make your programs super fast? We'll dive into the inner working of computers, look at some assembly code, compare benchmarks and hopefully won't end up more confused than when we started.

This talk is designed for general audiences of mathematics and statistics researchers.

A time series is a collection of observations which are recorded in time order. Due to their natural temporal characteristics, they arise in many walks of life, for example economics, medical sciences, and astronomy. One of the fundamental properties of statistical time series analysis is stationarity which means that the joint probability distribution does not depend on time. However, this assumption is easily violated for many time series datasets in practice where the underlying process changes their distribution over time. Time series segmentation (sometimes referred to as change-point detection) is a useful approach to remedy this issue as it divides a time series into a number of pieces corresponding to its own characteristic properties by identifying the boundaries of segments. Another type of segmentation is detecting change-points corresponding to linear or quadratic trend changes rather than distributional changes. I will introduce a range of topics and highlight some recent developments in the field of change point detection.

In the interior of many astrophysical objects such as stars and planets, systematic, large-scale flows that vary on long time scales exist alongside shorter-lived turbulent motions. Moreover, many of these objects harbour magnetic fields which also display remarkable signs of organisation against a backdrop of small-scale disorder. It remains an open problem to understand how such large-scale flows and magnetic fields are generated in fluid systems, particularly at extreme astrophysical parameters. In this talk I will review some of the major efforts to understand the generation of ordered magnetic fields and highlight some of the key open questions that remain. Time permitting, I will describe ongoing efforts to answer these questions using idealised mathematical models. This talk is aimed at a very general audience and no prior knowledge of fluid dynamics or magnetohydrodynamics (MHD) will be assumed.

(This is a talk for a very general audience of mathematics adjacent researchers.)

A hyperbolic surface is a surface, in the intuitive sense, with a geometry that is negatively curved at every point with the same curvature (-1) everywhere. These are not easy to visualize, but there are many of them.

Two interesting things to study on a hyperbolic surface are the dynamics of the geodesic flow (classical mechanics) and the Laplacian differential operator (quantum mechanics). The geodesic flow is chaotic and so the Laplacian there belongs to a field of study known as quantum chaos. Although these systems are far from being solvable in any sense, they are often the first place that we can see anticipated physical phenomenon rigorously. This is because for a hyperbolic surface, there is further structure (representation theory) that bridges the classical and quantum mechanics.

I will explain all this in simple terms, covering a range of paradigms that hyperbolic surfaces provide us to study.

If I have time, I'll then highlight some recent results in the field.

"The best thing about being a statistician, is that you get to play in everyone's backyard" is a famous quote attributed to John Tukey. This quote is most often taken to be a description of application areas of statistics, but is arguably also an accurate description of some statistical methodology too. In this accessible high-level talk, I will share my own adventures through other discipline's metaphorical backyards, both applied and methodological. This will include highlights from some of my research projects, including results of statistical machine learning applied to electronic health records of the majority of the Scottish population; a response to the Covid-19 pandemic focused on intensive care data; tailoring statistical methods to high performance computing architectures; and development of statistical methodology within the constraints of encrypted computation to preserve data (and possibly model) privacy.

I will discuss the problem of “conformally welding” a pair of disks, when a way to identify their boundaries (a homeomorphism) is specified. Unless the homeomorphism is very nice (quasisymmetric) it can be difficult to determine the existence and/or uniqueness of such a welding. I will focus on the problem when the homeomorphism is random, and arises from a so-called “Liouville quantum gravity” measure. This is a setting where existence/uniqueness of the welding seems hard to determine using complex analytic techniques, but thanks to some remarkable probabilistic relationships, it can actually be constructed directly. This results in a rich theory of “welding quantum surfaces” where the welding curves are given by Schramm-Loewner evolutions. I will try to explain some of this theory and its applications.

Connections between microscopic follow-the-leader and macroscopic fluid-dynamics traffic flow models are already well understood in the case of vehicles moving on a single road. Analogous connections in the case of road networks are instead lacking. This is probably due to the fact that macroscopic traffic models on networks are in general ill-posed, since the conservation of the mass is not sufficient alone to characterize a unique solution at junctions. This ambiguity makes more difficult to find the right limit of the microscopic model, which, in turn, can be defined in different ways near the junctions. In this paper we show that a natural extension of the first-order follow-the-leader model on networks corresponds, as the number of vehicles tends to infinity, to the LWR-based multi-path model introduced in [Bretti et al., Discrete Contin. Dyn. Syst. Ser. S, 7 (2014)] and [Briani and Cristiani, Netw. Heterog. Media, 9 (2014)].

Joint work with Emiliano Cristiani.

Deconvolution is a technique perhaps more usually applied to problems such as the de-noising of images, but is equally useful in the more esoteric problem of well test analysis in the world of petroleum engineering. Using only information on the pressure and the rate of production we can deconvolve a unique signature for the reservoir that gives insight into the geometry and geology of the system underground. Getting such rich information from a very simple data set is highly desirable, but reliable methods for deconvolution for this problem are not common.

In this talk, I will give a general introduction to the context in earth sciences, the mathematics underlying this problem (arising from the flow of fluids in porous media), and then focus on the statistical method to arrive at a solution. I will illustrate the complications that arise from working with progressively more complex and realistic data, and the adjustments required by the methodology. Ultimately, while we still arrive at a solution we find that (unsurprisingly to some) it might just have been better to be Bayesian from the start.

Computer algebra software is a standard tool for many scientists, engineers and mathematicians. Surprisingly, a lot of this software is still built around ideas which have not changed much at all since the first computer algebra systems were written in the early '60s. That could mean that those ideas are simply perfect, but evidence suggests otherwise. In this talk, I will try to convince you that there is plenty of room for improvement, and discuss recent attempts in this direction. I will in particular introduce you to my own computer algebra system for tensor fields "Cadabra", which tries to break with some of the tradition.

Khovanov homology is a combinatorial invariant of knots in 3-space. It has deep connections with representation theory, physics, and floer homology. I'll explore some of these connections.

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In this talk I will analyse what it means to have a mathematical theory of the real world, using the lens of Aristotelian/Thomist philosophy. I will begin by giving an overview of Thomist metaphysics with the help of cat pictures. I will then formulate a definition of a Mathematical theory of physics as a functor- like map, in one of two ways. First, as between a set of mathematical objects and the basic objects (or substances) in a conceptual model of the physical world. Second, as a map between certain numbers resulting from a mathematical model of a system, and certain observable properties (or accidents) of that system. This second map seems to be a kind of, 'probabilistic homomorphism' due to the error bars inherent in such a description. Finding the consistency of the first kind of map corresponds to a philosophical analysis of the physical theory in question, and I will argue that this motivates the conceptual analysis of a physical theory as a way of making progress in physics. Finally I will flesh out the abstract discussion of these issues by applying such an analysis to classical mechanics.

In this talk we present an introduction to an analogue between number fields and function fields, first discovered by Kenkichi Iwasawa.

A modular form f(z) for SL_2(Z) with rational coefficieints has a denominator, ie an integer D exists such that Df(z) has integral coefficients; this D is often arithmetically interesting - in particular, when f(z) is an Eisenstein series. Any modular form f(z) has an associated cohomology class for the space H/SL_2(Z), which will also have a 'de Rham denominator'; we hope that these two denominators are in fact the same! By an extension of the classical Shintani lift, I shall try to explain a method (so far only partially successful) in showing this conjecture. We give an overview of the methods used, as well as a couple of generalisations that I have been looking at this year. The talk should hopefully give an idea of the enormous arithmetic interest in the area of modular forms, as well as the way that geometric methods may be used in the pursuit of number-theoretic goals.

In recent work Lambropoulou and Antoniou characterised a number of natural phenomena in terms of topological surgery. To do so, they augmented the abstract notion of surgery in order to include dynamics and a notion of continuousness. I'll go through this augmented definition of surgery and some examples of natural processes in which it occurs.

The Monge-Ampere equation is a fully nonlinear partial differential equation strongly related to the Minkowski problem of hypersurfaces with prescribed Gauss curvature. Topological methods are used to state the existence of solutions by using a priori estimates. We will talk about these methods in the elliptic type case.

Cluster algebras are known to be closely linked to the study of the geometry of surfaces. Some recent work in this area accidentally gave rise to a number of diophantine equations along with a procedure to compute integer solutions. This might also have links in the geometric study of representations of surface fundamental groups, as well as quiver guage theories and cluster automorphisms. We will discuss ideas around this theme in whichever direction the audience prefer.

The polynomial ${X \choose m}$, where $m$ is a natural number is an example of a polynomial which takes integer values on $\mathbb{Z}$ even though its coefficients are not integer. A polynomial $f\in \mathbb{Q}[x]$ with the property $f(\matbb{Z}\subset \mathbb{Z})$ is called integer-valued. If $f$ is of degree at most $n$, then it is enough to check that $f({0,...n}) $ takes values in $\mathbb{Z}$ to know that $f$ is integer-valued. A finite set $A$ is called $n$-universal if $f(A)\subset \mathbb{Z}$ implies that $f$ is integer-valued for every $f$ of degree at most $n$. I will talk about $n$-universal sets when $\mathbb{Q}$ is replaced by a number field.

Mathieu Moonshine concerns a surprising observation relating string theory to the representation theory of a particular sporadic group, Mathieu 24. This is reminiscent of Monstrous Moonshine in which it was discovered that the coefficients of the modular j-function are related to the representation theory of the Monster group. In this talk we will introduce a topological invariant of string theories compactified on K3 surfaces, called the elliptic genus of K3, and see how Mathieu 24 appears in this context. To this date, the role of the large discrete symmetry M24 in String Theory is not properly understood. We will then discuss Umbral moonshine, which comprises of 23 examples of moonshine in which the Niemeier lattices are used to connect certain mock modular forms to finite groups.

Elliptic functions and modular forms are a common feature in certain calculations within string theory. I aim to give a light overview of some aspects of string theory to provide some context, before describing various elements of the calculations involved.

Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.

This will be an expository talk on sheaves-functions correspondence and its applications. I will start with finite Abelian groups and some involved arithmetic problems, and then turn to the characters and representations of Lie type groups.

Khovanov homology is a chain-complex valued invariant of links. Virtual knot theory is a generalisation of classical knot theory which considers embeddings of circles into thickened surfaces of genus g>0 (the g=0 case returns classical knot theory). We give an introduction to virtual knots and a quick overview of the definition of Khovanov homology, before going through the process of a generalising it to the virtual case in a picture-heavy and pedagogical way.

Both the Jones polynomial and the Alexander polynomial can be viewed as invariants arising from the representations of quantum (super)groups in type A. Skew Howe duality give these invariants particularly nice descriptions in terms of trivalent diagrams. This method is particularly powerful when defining knot homology theories categorifying these polynomials. I will discuss the relationship between representations of quantum groups and the trivalent diagrams appearing in calculations of knot invariants, and describe how this can be used to understand knot homology theories, and progress towards obtaining a 'quantum' categorification of the Alexander polynomial.

In 1965, Tom Willmore (Durham) first considered a curvature energy for immersed surfaces in R^3 which would become widely studied in differential geometry and the modelling of liquid membranes. I will briefly discuss Willmore's energy, an application in biology, and a diffuse interface approach to the minimisation problem. The diffuse model has computational advantages, but makes control of the topology of surfaces more involved. In the last part of the talk, I will indicate how we managed to prescribe connectedness for a limiting problem along with some computational evidence. This talk is based on joint work with Patrick Dondl and Antoine Lemenant.

Fermat observed that (except for p = 2) a prime p can be written as the sum of two squares if aond only if p = 1 (mod 4). This result motivates our basic question: which primes does a given quadratic form represent?

To begin to answer this, we will relate the question of primes represented by a quadratic form to questions about ideal classes in quadratic number fields. And we will then be able to study these questions using the powerful tools of class field theory.

The main goal of this talk will to give a complete answer to this question for a specific class of quadratic forms, the so-called principal forms x^2 + ny^2. In this case the answer has the following form: there exists a polynomial f_n(t) such that p = x^2 + ny^2 if and only if f_n(t) has a root modulo p. And for squarefree n, this polynomial f_n(t) has an explicit interpretation as the polynomial describing the `Hilbert class field' of Q(sqrt(n)).

A central property of quantum mechanical systems is entanglement: knowledge of the parts does not constitute knowledge of the whole. It turns out that in some circumstances, quantifying entanglement is equivalent to a classical problem, of finding minimal surfaces. I'll wave my arms a bit to motivate where this comes from, and then do some proper maths to prove a theorem in algebraic topology, which tells us which topological class of minimal surfaces to consider in the problem.

A quick introduction to cluster algebras from combinatorial and geometric view points.

The Alexander polynomial is a classical invariant of knots introduced in the 1920's with clear connections to the topology of knots and surfaces. The Reshetikhin-Turaev invariants are much more recent, and are in general much more poorly understood. These often arise from the representation theory of quantum groups. I will show how the Alexander polynomial can be interpreted as a Reshetikhin-Turaev invariant using representations of U_q(gl(1|1)), and show how this can be used to understand a category of representations of U_q(gl(1|1)). Finally, I will give some suggestions about how this should tie into categorifications of knot invariants, and particularly the connection between HOMFLY homology and Heegaard Floer knot homology.

Surreal numbers were invented by Conway, and used in his study of game theory. While the definition of a surreal number is surprisingly simple, it rapidly leads to a rich and deep structure encompassing not only the usual real numbers, but infinities, infinitesimals and more. In this talk I'll give an introduction to how surreal numbers work and an overview of the some of the weirdness that ensues.

In 2000, Khovanov introduced a knot invariant in the form of a homology theory now known as Khovanov Homology. I will introduce this invariant and go on to talk about a new knot invariant which takes the form of a suspension spectrum, and so is invariant up to stable homotopy. The Khovanov Spectrum was introduced by Lipshitz and Sarkar in 2011 and has been shown to be a stronger invariant than Khovanov Homology. I aim to discuss some questions I have been looking at related to this spectrum.

A graph-like manifold is a family of neighbourhoods of thickness ε>0 of a metric graph shrinking to the graph itself as ε approaches zero. In spectral geometry, graph-like manifolds were first introduced by Colin de Verdiere to prove that a manifold of dimension n greater or equal than 3 admits a metric with the first non-zero eigenvalue of the Laplacian having multiplicity n and since then, they have been used as a toy model to prove properties of manifolds or disprove conjectures. In physics, they are a model for nano and optical structure and metric graphs are believed to be a good approximation for them since the spectum of their Laplacian is a good approximation of the spectrum of the Laplacian of graph-like manifolds. In my talk I will explore the relation between the spectra of the Laplacian acting on 1-forms on the graph-like manifold and the Laplacian acting on 1-forms on the metric graph with some insights about higher degree forms.

A look at the classic Khintchine's theorem in diophantine approximation and extensions of it to function fields and number fields.

Modular forms of integral weight and half integral weight have many interesting applications in number theory. Shimura in 1973 defined a very important correspondence between the two which can be defined in the framework of theta lifts. More recently harmonic weak Maass forms (generalisations of classical modular forms) and their uses have been studied. In this talk I will discuss these objects and their properties and describe my work on a theta lift which links all of them together.

After the Jones polynomial was discovered in the 80's, Jones's methods were generalised repeatedly until Reshetikhin and Turaev described a very general method for constructing polynomial invariants of knots. For any representation of any simple Lie algebra, their method describes a procedure for constructing a knot invariant from it. This procedure involves the so-called 'quantum groups' described by physicists, which (for the purposes of this talk) are 1-parameter deformations of universal enveloping algebras. Even in the simplest cases, these knot invariants are quite poorly understood and are still very actively researched. In this talk I will try and describe (without too many messy details) quantum groups and how knot polynomials are obtained from them, with the construction of the Jones polynomial as an example.

Polylogarithms are a class of special functions which have applications throughout the mathematics and physics worlds. I will begin by introducing the basic properties of polylogarithms and some reasons for interest in them, such as their functional equations and the role they play in Zagier's polylogarithm conjecture. From here I will turn to a Aomoto polylogarithms, a more general class of functions and explain how they motivate a geometric view of polylogarithms as configurations of hyperplanes in Pn. This approach has been used by Goncharov to establish Zagier's conjecture for n = 3.

Two prototypes of algebraic groups are elliptic curves and GLn's, which stand for the projective side and the affine side respectively. Here the focus is on the latter one. Their importance and interests come from many sides, e.g. the deep relations with absolute Galois groups, and the mixture of explicit appearances and complicated structures. Here I will introduce some basic concepts on affine algebraic groups and talk about the parabolic induction approach to their representations.

Cluster algebras were first introduced by Fomin and Zelevinsky in an effort to provide concrete terms to describe "dual canonical bases" in different settings. Cluster algebras are special in that the final construction of the algebra is rarely interesting, rather it is the process of finding the generators of the algebra which yields fascinating results. Generators are found using an iterative process of mutation on labelled seeds, and I am particularly interested in those mutation classes of infinite size.

A heuristic introduction to my PhD topic. Some geometric measure theory, Gamma convergence of phase field models and pretty pictures.

The work on the Weil conjectures is one of the most exciting stories happened in the 20th century. These concrete statements on counting points over finite fields traced back to some of Gauss' work, and are amazingly encoded in what we now called étale cohomology, which is itself highly interesting (it generalizes Galois cohomology, gives a Hilbert Satz 90 for curves, and has applications to representation theory of algebraic groups, and etc). So this is also a good example on illustrating how an abstract machinery can be used to solve a very concrete problem. In this talk some of these smart ideas will be introduced.

Khovanov homology is an invariant of knots that 'categorifies' the Jones polynomial. In this talk, I will give a basic introduction to Khovanov homology for knots, and then show that by staying in the topological world of cobordisms we can obtain a chain complex that serves as an invariant of tangles, and specialises to Khovanov homology in the case of knots and links.

Fermat's observation about which primes can be written as the sum of two squares motivates the question: which primes does a given quadratic form represent? After relating quadratic forms with ideals in quadratic fields, we show how Class Field Theory can be applied to construct general criteria describing these primes.

We will first define half weight vector valued forms and then construct a twisted theta lift of weak harmonic Maass forms of weight 1/2 to automorphic forms on the upper half plane, as well as the relationship with the Shimura lift.

We will discuss the structure of a quasi-reflective group (sometimes known as a parabolic-reflection group), and give some examples from among the Bianchi groups. There are only finitely many of these groups in each dimension, and we present a classification of quasi-reflective Bianchi groups.

At a recent talk in St. Mary's College, I introduced a way of generalising chain spaces to an arbitrary vector space V, with an action of an arbitrary Lie group G. A lot is known for these chain spaces when V is real d-space and G = SO(d). In this talk, I will state some results in the real case and suggest a method of calculating cohomologies in the more general case. This method involves the Leray-Serre spectral sequence, and we will hopefully see a few particular examples. (If you haven't seen/heard of spectral sequences before, don't worry as I will explain what they are and why they are useful).

The congruent number problem is that of determining whether an given integer is the area of a right angled triangle with rational sides. This deceptively simple to state problem has a solution which leads us through some of the most intriguing theorems of modern number theory, including the Birch and Swinnerton-Dyer conjecture, one of the Clay Mathematical Institute's Millennium problems.

A series of 30 minute talks by James Allen (CPT), Luke Stanbra (Algebra), Benedict Powell (Stats), Ramon Vera (Topology), Sarah Chadburn (CPT), and John Mcleod (Geometry).

As a companion to the thrilling and stimulating offerings from the Graduate School about joining the postgraduate community here, I would like to invite you to a half day of talks from existing PhD students, who will be speaking about their research and the methods they have used during their work here.

The idea is to advertise those areas of expertise which the graduate community has, which may save you a month of struggling to learn a tool that is badly documented, or six months following the paper trail to discover some particular result, or many other examples. Mathematics is highly connected, and so it may be that some Pure student has the tools to integrate a particular Feynman diagrams (and may not realise it), or a Statistician has detailed knowledge of infinite-dimensional functional analysis. These are two examples which I am aware of in this department that surprised me!

It will not be possible to get very far in half a day, but I hope that a bit of "interdisciplinary mixing" will carry forward and bear fruit.

We outline how to extract the symbol of a multiple polylogarithm from a hook-arrow tree and then prove a simple result. It is the the end of term so talk will be 'easy' and include many tikz pictures. Disclaimer: previous exposure to this material from the speaker not needed.

Given a monic integral polynomial p, one may define its Mahler measure as the product of all its roots with absolute value at least 1. The smallest known Mahler measure is for a polynomial of degree 10, and Lehmer's problem is to find a smaller Mahler measure; the conjecture is that one cannot. The truth of the conjecture would imply another conjecture, known as the Short Geodesic Conjecture hyperbolic geometry. I will explain some of these ideas and how they relate to some of my own work (joint with M. Belolipetsky).

I will be discussing some introductory p-adic analysis and the p-adic weight space with the aim (time permitting) of defining a p-adic Riemann Zeta function.

Continuing last week's discussions, I will begin with a quick review of the analytic construction of the p-adic numbers, showing its link to the algebraic construction via inverse limits. I will then introduce the notion of "profinite completion" and show how the profinite completion of the integers relates to the p-adic integers.

An important part of the proof of the Riemann Existence Theorem involves the theory of Profinite Groups which are exactly those groups which arise from an inverse limit. We will delve a little deeper into the theory of inverse limits, starting from Category Theory and working up to the derivation of the p-adic numbers via an inverse limit.

We open this term with a couple of seminars on the Riemann Existence Theorem, which states the connection between the punctured Riemann sphere and Fuchsian groups. Along the way we will touch on a bit of Topology, a dash of algebraic geometry, and a smidgeon of group theory. The RET is an important ingredient in an approach to the Inverse Galois problem. By the end of the second seminar, we hope to have proven a lesser result, the Profinite Riemann Existence Theorem.

We present a survey of how the curvature tensor of all known homogeneous 6 dimensional nearly Kähler spaces (both in the definite and in the pseudo Riemannian case) can be expressed in an invariant way using the induced geometric structures on the 6 dimensional nearly Kähler space.

As an application we show how this can be used to study special classes of submanifolds in these spaces. In the latter case we will in particular focus on totally geodesic Lagrangian submanifolds and equivariant Lagrangian immersions.

In this talk, I will introduce diffusion geometry as a new framework for geometric and topological data analysis. Diffusion geometry uses the Bakry-Emery gamma calculus of Markov diffusion operators to define objects from Riemannian geometry on a wide range of probability spaces. We construct statistical estimators for these objects from a sample of data, and so introduce a whole family of new methods for geometric data analysis and computational geometry. This includes vector fields and differential forms on the data, and many of the important operators in exterior calculus. Unlike existing methods like persistent homology and local principal component analysis, diffusion geometry is explicitly related to Riemannian geometry, and is significantly more robust to noise, significantly faster to compute, provides a richer topological description (like the cup product on cohomology), and is naturally vectorised for statistics and machine learning. We find that diffusion geometry outperforms multiparameter persistent homology as a biomarker for real and simulated tumour histology data and can robustly measure the manifold hypothesis by detecting singularities in manifold-like data.

Knot/link Floer homology is a link invariant package, introduced independently by Ozsvath-Szabo and Rasmussen, has been shown to be quite useful to solve a number of questions in low dimensional topology in the last two decades. Although it is not a complete invariant of knots/links, a number of knots and links have been shown to be detected by this toolbox. The center of most of these results have been careful examination of certain essential surfaces in the knot/link exteriors and observing that operations on those surfaces can be kept track by the link/knot Floer homology of those knots/links. In this mostly self-contained talk, we will be talking about those results and some new ones. This is based on joint work with Fraser Binns, some of which is ongoing.

A slope p/q is non-characterising for a knot K in the 3-sphere if there exists a different knot K' in the 3-sphere such that Dehn surgery of slope p/q on each of K and K' produces orientation-preserving homeomorphic 3-manifolds. In this talk, we will explore 3 different approaches to constructing non-characterising slopes for satellite knots. For the |p|=1 case, I'll describe how to use JSJ decompositions to find suitable satellite knots of hyperbolic type (joint work with Patricia Sorya). For the |q|=1 case, I'll discuss how to use RBG links to address certain knots concordant to satellites of (2,k)-torus knots (joint work with Charles Stine). Finally, for the general p/q case, I'll explain how the Montesinos trick could potentially be used to show that every p/q can be realised as a non-characterising slope for some pair of satellite knots (joint work with Kyle Hayden and Lisa Piccirillo).

The Grove-Searle Maximal Symmetry Rank Theorem (MSRT) and Wilking Half-Maximal Symmetry Rank Theorem (1/2-MSRT) represent keystone results in the study of positively curved spaces with large isometry groups. In this talk, I will present work on extending the MSRT to a weaker curvature condition called positive intermediate Ricci curvature. I will focus on dimensions 4 and 6, where we so far are only able to establish a partial extension. If time permits, I will also describe current work-in-progress on extending the 1/2-MSRT to this weaker curvature condition. This talk will be based on joint work with Lee Kennard.

We would like to discuss the formality problem for compact manifolds with special holonomy, expose some recent results, its current status and open problems.

We will show that, in any homotopy class of mappings from complex projective space to a Riemannian manifold, the infimum of the energy is proportional to the infimal area in the class of mappings of the 2-sphere representing the induced homomorphism on the second homotopy group.  We will also give a related estimate for the infimum of the energy in a homotopy class of mappings of real projective space, and we will discuss several results and questions about energy-minimizing maps and their metric properties.

The Bakry-Émery Ricci tensor is a generalization of the classical Ricci tensor to the setting of weighted Riemannian manifolds, i.e. Riemannian manifolds whose Riemannian volume forms are weighted by a smooth function. In this talk we consider the question of which manifolds admit a weighted Riemannian metric of positive Bakry-Émery Ricci curvature. To obtain new examples we establish several surgery results for such manifolds, i.e. results that allow to perform certain cut and glue operations on the manifold while preserving the positivity of the Bakry-Émery Ricci curvature. In contrast to known surgery results for positive Ricci curvature these techniques are of local nature. Applications include connected sums and new examples in dimension 5. This is joint work with Francesca Tripaldi.

The Heisenberg group is a source of inspiration for many fields in mathematics and physics, including quantum theory, metric geometry, and harmonic analysis. I will discuss the sub-Finsler geometry of the Heisenberg group and explain how it is related to the isoperimetric problem in the non-Euclidean (Finsler) plane. We will then explore approaches to studying the curvature of the sub-Finsler Heisenberg group, focusing particularly on the measure contraction property that appears in the analysis of metric measure spaces. This is a joint work with Kenshiro Tashiro, Mattia Magnabosco, and Tommaso Rossi.

The 4-dimensional ultrahyperbolic equation arises as the defining equation for a function to be harmonic with respect to a metric of neutral signature (2,2).  In the case of the canonical neutral metric on the space of oriented geodesics of a 3-dimensional space form, it captures precisely the condition for a function on geodesic space to come from line integrals of a function on the space form. As this is the basis of modern tomography, including CT scans, one might expect that the equation has been extensively studied, but this turns out not to be the case. In this talk I will discuss the equation from the point of view of 3- and 4-manifold topology and how the equation allows one to X-ray a manifold from null boundary data.  The relationship between a conformal mean value theorem and doubly ruled surfaces in space forms will also be explored.

Alan Weinstein introduced the concept of "fat bundle" as a tool to understand when the total space of a fiber bundle with totally geodesic fibers allows a metric with positive sectional curvature.

In recent times, certain weaker notions than the condition of having a metric with positive sectional curvature have been studied due to the apparent scarcity of spaces that meet this condition. Positive kth-intermediate Ricci curvature (Ric_k >0) on a Riemannian manifold is a condition that bridges the gap between positive sectional curvature and positive Ricci curvature.

In this talk, I will discuss an ongoing project with Miguel Domínguez-Vázquez, David González-Álvaro, and Jason DeVito, which aims to construct new examples of compact Riemannian manifolds with Ric_2 > 0. We achieve this by employing a certain generalisation of the "fat bundle" concept.

Spaces with the Riemannian Curvature-Dimension condition (RCD spaces), are metric measure spaces that satisfy a synthetic notion of "having a lower Ricci curvature bound and an upper bound on the Hausdorff dimension." Examples of these include Riemannian manifolds and Alexandrov spaces but they also appear naturally as Gromov-Hausdorff limits.

In this talk we will first give a quick introduction to the Curvature-Dimension condition mentioning some structural properties and also some examples of RCD spaces.

Then we will look at some of the properties of the isometry group of such spaces and what happens when one takes the quotient by a compact Lie group acting by measure preserving isometries.

Lastly, we will focus on the case where the quotient space is one dimensional, that is, the cohomogeneity one case. Here we will find explicit examples of spaces that are not manifolds nor Alexandrov spaces. This last part is a joint collaboration with Diego Corro and Jesús Nuñez-Zimbrón.

Given a reduced plane curve $C_d$ of degree $d$ in $\mathbb{C}^2$, a classical question is to understand the singularities of it. Over the years many different measures of singularities have been explored, such as Multiplicity, Milnor number, Tjurina number to name a few. In this talk, I will focus on another invariant called the log canonical threshold, which has a long standing relation with the notion of K-stability. Firstly, for all curves of degree $d \leq 5$, I will explicitly show the exhaustive list of all possible log canonical threshold values that the curve $C_d$ can take at a singular point $p$ on it. Then, we will see how imposing restrictions on the multiplicity of the curve $C_d$ at the point $p$ can help us in saying more about this invariant. This is joint work with Erik Paemurru.

This paper delves into the study of a system of nonautonomous ordinary differential equations featuring a discontinuous right-hand side. We establish a comprehensive framework that covers the existence, uniqueness, and exponentially asymptotically stability of non-smooth periodic orbits. Additionally, we provide an explicit formula for determining the basin of attraction, all without the explicit need to compute a solution to a Fillipov equation. This theoretical foundation finds practical application in the stability analysis of green energy stock market trading.

In the ever-evolving landscape of green energy markets, recent regulatory changes have ushered in a new era of negative price trading. This transformative development necessitates a deeper examination of the transitions between positive and negative price domains. Such analysis offers a valuable and novel perspective for both market participants and analysts seeking to navigate this evolving market paradigm.

The Rasmussen invariant is a homomorphism from the knot concordance group to the integers that Jake Rasmussen defined in 2004 using Khovanov homology and which has interesting properties and applications.  In recent joint work, Lukas Lewark and I define another concordance homomorphism from Khovanov homology that is linearly independent of the Rasmussen invariant.  It plays a central role in a formula that we prove for the Rasmussen invariant of Whitehead doubles and other satellite knots.

We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. As a consequence, there are hyperbolic groups with finite-type subgroups that are not hyperbolic.

The main tool is Bestvina - Brady theory enriched with a combinatorial game recently introduced by Jankiewicz, Norin and Wise. This is a joint work with Italiano and Migliorini.

In this talk, by assuming a quadratic curvature pinching condition, we show that the submanifold evolving by mean curvature flow becomes approximately codimension one, in high curvature regions. This fundamental codimension estimate along with a cylindrical type estimate, at a singularity, allows us to establish the existence of a rescaling, which converges to a smooth codimension-one limiting flow in Euclidean space. This is possible using unique pointwise gradient estimates for the second fundamental form.

I present a an approach to Lorentzian geometry and General Relativity that does neither rely on smoothness nor on manifolds, thereby leaving the framework of classical differential geometry. This opens up the possibility to study curvature (bounds) for spacetimes of low regularity or even more general spaces. An analogous shift in perspective proved extremely fruitful in the Riemannian case (Alexandrov- and CAT(k)-spaces) and we provide examples and report on recent progress that suggest that our approach could have a similar impact on Lorentzian geometry and GR.

Given a Riemannian manifold (M, g), it is a fundamental problem to understand how the metric g and its curvature properties evolve under the Ricci flow. For instance, by the celebrated work of Hamilton, positive scalar curvature is preserved under the Ricci flow in every dimension. Moreover, both positive sectional and positive Ricci curvatures are preserved in dimension 3. It is then natural to ask whether any other curvature conditions are preserved in higher dimensions. In this talk, I will give some examples which admit metrics with different curvature conditions and discuss the evolution of their metrics under the Ricci flow. This is based on joint works with David González-Álvaro.

In MultiObjective Optimization (MOO), one analyzes tradeoffs between multiple objectives in search for optimal solutions. While a wide range of MMO solvers have been proposed, comparing the solvers have remained a significant challenge. For this, I introduce the benchmark problem suite called the Benchmark with Explicit Multimodality (BEM). The BEM was proposed by an interdisciplinary team combining researchers from evolutionary computation, mathematics and visualization. I start the talk by introducing central tools of our choice, the Reeb space and Reeb graph, which describe characteristics of functions using a topological construct. By employing them, we can design benchmark problems using a concise graph structure. Finally, I will show how we visualize the BEM and the solvers being benchmarked. This allows in-depth and/or statistical analysis on how the solvers are trapped in local optima or overcome them. In addition to our specialized visualization, particular advantages of the BEM include the high dimensionality of the design space and simplicity of the problem description.

Though Einstein's equation is well studied, relatively few Einstein metrics have been written in terms of explicit formulae via radicals.  In this talk we discuss many such examples that occur as anti-self dual Einstein metrics and describe their singularities.

The construction heavily relies upon the theory of isomonodromic deformation and related algebraic geometry developed by N.J. Hitchin in the 1990s and the equivalence of the anti-self dual Einstein equation to a certain Painlevé VI equation under some symmetry assumptions discovered by K.P. Tod. The solution to Painlevé VI is achieved through a relation of its solution to pairs of conics obeying the Poncelet's porism by exploiting Cayley's criterion.

In this talk we discuss some important cases that are not well fleshed out in the literature, such as the solution of Painlevé VI associated with the Poncelet porism where the inscribing-circumscribing polygons have an even number of sides.

Moreover, we provide some explicit metrics with unusual cone angle singularities along a singular real projective plane that were speculated about by Atiyah and LeBrun and discuss their sectional curvature.

Alexandrov spaces are complete, locally compact length spaces with finite (integer) Hausdorff dimension and curvature bounded below in the triangle comparison sense. They are metric generalizations of complete Riemannian manifolds with sectional curvature uniformly bounded below. In this talk, I will discuss extensions of classical results for 3-manifolds to the case of non-manifold Alexandrov spaces, including the prime decomposition theorem of Kneser and Milnor.

A filling curve  c  in a closed oriented surface  X  of genus g>1  determines a complex analytic structure on X in two different ways.  One is  via Grothendieck's  theory of dessins d'enfants. The other one arises as  the hyperbolic structure on X that minimises the length of the curve c.  We show that these two complex structures agree at least in the case in which the curve c admits a homotopy representative in minimal position  such  that all self-intersection points have the same self-intersection number and all  faces of the complement  X \ c  have the same degree. (This is joint work with E. Girondo and R. Hidalgo)

Among all left-invariant Riemannian metrics on a given Lie group, is there one whose isometry group contains that of all others?  We'll present the current state of knowledge on this question for solvable Lie groups along with some applications to the uniqueness of Ricci soliton metrics on solvmanifolds.

In this talk, I will present a curvature dependent lower bound for the filling radius of all closed Riemannian manifolds as well as an upper one for manifolds which are the total space of a Riemannian submersion. The latter applies also to the case of submetries. Moreover, I will give an introduction about the reach of a subset and show some results about its value for isometrically embedded manifolds into the space of bounded real valued functions and the Wasserstein space.

This talk will present a new characterisation of rectifiable subsets of a complete metric space in terms of local approximation, with respect to the Gromov--Hausdorff distance, by finite dimensional Banach spaces. This is a significant generalisation of a theorem of Marstrand and Mattila of classical geometric measure theory.

After a gentle introduction to analysis on metric spaces and geometric measure theory, this talk will present the main ideas and challenges behind the proof of the new theorem.

G_2 solitons are self-similar solutions to Bryant's Laplacian flow for closed G_2-structures on 7-manifolds, a relative of Ricci flow. I will describe examples of G_2 solitons that are asymptotically conical (of all three types: expanders, shrinkers and steady solitons) as well as a steady soliton with exponential volume growth. The solitons are defined on the anti-self-dual bundles of CP^2 and S^4 and have a cohomogeneity one action. This is joint work with Mark Haskins and Rowan Juneman.

In Riemannian geometry, it is a fundamental problem to study the existence of nonnegatively Ricci  curved metrics on a manifold. Moreover, if such a metric exists, it is interesting to describe the associated moduli space of nonnegatively Ricci curved metrics. In 2017, Tuschmann and Wiemeler published the first result on these moduli spaces' homotopy groups.

On the other hand, Lott, Sturm, and Villani proposed a synthetic definition of Ricci curvature lower bounds on singular spaces. This work gave birth to RCD(0,N) spaces (i.e. non-smooth spaces with Ric \( \geq \) 0 and dim \( \leq \) N).

In this talk, I will briefly introduce RCD(0,N) spaces and their associated moduli spaces. Then, I will state the non-smooth analogue of Tuschmann and Wiemeler's result (proved in collaboration with Andrea Mondino). I will then sketch the proof of that result, which relies on the topological properties of RCD(0,N) spaces.

A classic theorem of Tucker asserts that a finite group \( \Gamma \) acts on an oriented surface S if and only if \( \Gamma \) has a Cayley graph G that embeds in S equivariantly, i.e. the canonical action of \( \Gamma \) on G can be extended to an action of \( \Gamma \) on all of S. Following the trend for extending graph-theoretic results to higher-dimensional complexes, we prove the following 3-dimensional analogue of Tucker's Theorem: a finitely generated group \( \Gamma \) acts discretely on a simply connected 3-manifold M if and only if \( \Gamma \) has a "generalised Cayley complex" that embeds equivariantly in one of the following four 3-manifolds: (i) \( S^3 \) , (ii) \( R^3 \) , (iii) \( S^2 \times R \), and (iv) the complement of a tame Cantor set in \( S^3 \). In the process, we will see some recent theorems and lemmata concerning 2-complex embeddings and group actions over 2-complexes, and we will derive a combinatorial characterization of finitely generated groups acting discretely on simply connected 3-manifolds.

We ask to what extent the SL(2,C)-character variety of the fundamental group of the complement of a knot in S^3 determines the knot. Our methods use results from group theory, classical 3-manifold topology, but also geometric input in two ways: The geometrisation theorem for 3-manifolds, and instanton gauge theory. In particular this is connected to SU(2)-character varieties of two-component links, a topic where much less is known than in the case of knots. This is joint work with Michel Boileau, Teruaki Kitano and Steven Sivek.

Mirror symmetry conjecturally associates to a Fano orbifold a Laurent polynomial. Laurent inversion is a method for reversing this process, obtaining a Fano variety from a candidate Laurent polynomial. We apply this to find new Fano 3-folds with terminal quotient singularities and outline a program that implements these constructions systematically.

We understand this correspondence through toric degenerations. A Laurent polynomial f determines, through its Newton polytope P, a toric variety \( X_P \), which is in general highly singular. Laurent inversion constructs, from f and some auxiliary data, an embedding of \( X_P \) into an ambient toric variety Y. In many cases this embeds \( X_P \) as a complete intersection of line bundles on Y, and the general section of these line bundles is the Q-Fano 3-fold that want to construct, i.e. the mirror of f. This is joint work with T. Coates and Al. Kasprzyk.

In the quest for a combinatorial formula for the rational Pontrjagin classes, Goresky and MacPherson were led to study the space of oriented matroids as the classifying space for the bundle theory of a category of manifolds sitting somewhere between the smooth category and PL. This space is called the matroid grassmannian, and there is a map from the real grassmannian to the matroid grassmannian that pulls certain combinatorial cohomology classes back to the rational Pontrjagin classes. It has become known as Macpherson's conjecture that this comparison map should be a homotopy equivalence. By using ideas from tropical geometry, I'll provide some new information on the homotopy type.

It is known that when one has a sequence of closed n-dimensional Riemannian manifolds of Ricci curvature bounded below and diameter bounded above, one can always find a convergent subsequence (in the Gromov-Hausdorff sense). If the limit has dimension n, just like the elements of the sequence, we say the sequence doesn't collapse, otherwise we say the sequence collapses. A smooth manifold is said to be squishable if it admits a sequence of Riemannian metrics that makes it collapse. We study the relationship between the topology of a manifold with the property of being squishable and identify some possible limit spaces obtained after squishing.

In this talk I will review the known techniques to construct metrics of positive Ricci curvature via surgery by Sha-Yang and Wraith. I will then present a generalization of the surgery theorem of Wraith in which the surgery construction itself gets generalized. Finally, we will consider applications in dimension 6. Here we obtain a large class of new examples of closed, simply-connected 6-manifolds that admit a metric of positive Ricci curvature. These examples are constructed as boundaries of manifolds obtained by plumbings according to a simply-connected graph.

Mostow, constructed a family of lattices in PU(2,1) which is the holomorphic isometry group of complex hyperbolic 2-space. In this presentation, I use a description of these lattices given by Thurston in terms of cone metrics on the sphere to give an explicit fundamental domain for some of Mostow's lattices. The approach is along the lines of Parker's description of Livne's lattices in terms of cone metrics on the sphere. This presentation is based on published work by Parker and Boadi on the above title.

In the study of the dynamics of a transcendental entire function f, we aim to describe its locus of chaotic behaviour, known as its Julia set and denoted by J(f). For many such f, the Julia set is a collection of unbounded curves that escape to infinity under iteration and form a particular topological structure known as Cantor bouquet, i.e., a subset of the complex plane ambiently homeomorphic to a straight brush. We show that there exists f whose Julia set J(f) is a collection of escaping curves, but J(f) is not a Cantor bouquet. On the other hand, we prove for certain f that if J(f) contains an absorbing Cantor bouquet, that is, a Cantor bouquet to which all escaping points are eventually mapped, then J(f) must be a Cantor bouquet. This is joint work with L. Rempe.

We prove a conjecture of Toponogov on complete convex embedded planes, namely that such surfaces must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample. This is joint work with Brendan Guilfoyle.

In work with Carolyn Gordon, we have shown that certain nice metrics (Ricci solitons) on solvable Lie groups have the special property that their isometry algebras are as large as possible, in terms of containment. We discuss the algebraic consequences, on the underlying solvable group, of the existence of a maximal isometry algebra. [This talk has been postponed.]

In the first (longer half) I report for a more general audience about the results, which are parallel to the classical case of Riemann surfaces. Then I will explain the method of the proof using modified surgery.

We analyse a neighbourhood of the vanishing locus of the Cauchy-Riemann operator defined by a given domain-dependent complex structure. We extend the locus by elements induced by the cokernel of its linearization. We prove that the extended set is a smooth manifold. This is a modification of Fukaya's analysis of the moduli space of holomorphic curves in symplectic manifolds.

On a compact Riemannian manifold with smooth boundary, we define the biharmonic Steklov operator on the set of differential forms. This definition is motivated by an extension of the Serrin problem to differential forms. We then study the spectral properties of this operator and show that it has a discrete spectrum. In particular, we relate its first eigenvalue to different boundary problems, Dirichlet, Neumann and Robin defined on differential forms.

On a compact Riemannian manifold with smooth boundary, we define the biharmonic Steklov operator on the set of differential forms. This definition is motivated by an extension of the Serrin problem to differential forms. We then study the spectral properties of this operator and show that it has a discrete spectrum. In particular, we relate its first eigenvalue to different boundary problems, Dirichlet, Neumann and Robin defined on differential forms.

I will discuss the relationships among a variety of equivalence relations on 4-manifolds, such as diffeomorphism, homeomorphism, h-cobordism, and homotopy equivalence, with the goal of organising a zoo of counterexamples and discovering unanswered questions. There will be a flowchart, also available at http://tinyurl.com/4dcounterexamples. The talk is based on a partly survey paper joint with Daniel Kasprowski and Mark Powell.

Margulis showed that, for each dimension n, there is a positive constant epsilon so that for any hyperbolic manifold M of dimension n, the epsilon-thin part of M is a union of tubes around short, closed geodesics and cusp regions. In this talk I will focus on Margulis cusp regions in dimension n=4 associated to screw-parabolic maps with irrational rotational part. It turns out that the shape of the cusp region is closely connected to the continued fraction expansion of the rotational part. Using classical results from Diophantine approximation, I will show how to construct a slightly smaller region independent of the continued fraction.

A topological surface is finite type if its fundamental group is finitely generated. On any given finite type surface, there are (up to homeomorphism) finitely many types of essential arc. For infinite type surfaces there may be considerably more types of essential arcs; some more essential than others.

This talk will try to make those last five words less vague. We will also spend some time discussing big mapping class groups, arc complexes, and unicorns.

Wall classified smooth (n-1)-connected 2n-manifolds up to the action of homotopy spheres. We determine this action for 3-connected 8-manifolds, and therefore obtain a complete diffeomorphism classification. In dimension 8 there is a unique exotic sphere. We find that whether or not it acts trivially on a 3-connected M depends on the divisibility of the first Pontryagin class p_1(M). The proof is based on the Q-form conjecture, which provides a sufficient condition for two manifolds to be diffeomorphic. Joint work with Diarmuid Crowley.

Configuration spaces of manifolds are examples of spaces of embeddings, which can be employed for studying all other embedding spaces, via Goodwillie-Weiss-Klein calculus. We will discuss how certain classes in homotopy groups of configuration spaces give rise to nontrivial families of embeddings, that generalise lower central series of braid groups and Vassiliev-Gusarov-Habiro constructions in knot theory.

The space of persistence barcodes, equipped with the bottleneck metric, is a fundamental object of study in topological data analysis. There has been recent interest in describing this space as a topological space. In this talk we present a significant strengthening of these descriptions by studying the space of barcodes as a metric space. Namely, we show that the space of finite persistence barcodes is a bi-Lipschitz image of a convex subset of Euclidean space.

Time permitting, we will demonstrate how our geometric description naturally imposes a differential structure and allows approximations of the bottleneck distance, both of which are active topics of research.

This is joint work with David Bate.

In this talk we show that for 4-manifolds with residually finite fundamental group and non-spin universal cover if the macroscopic dimension of the universal cover is less than or equal to 3, then it has to be less than or equal to 2.​

The so-called spaces with the Riemannian curvature-dimension conditions (RCD spaces) are metric measure spaces which are not necessarily smooth but admit a notion of “Ricci curvature bounded below and dimension bounded above”. These spaces arise naturally as Gromov-Hausdorff limits of Riemannian manifolds with these conditions and, in contrast to manifolds, RCD spaces typically have topological or metric singularities. Nevertheless a considerable amount of Riemannian geometry can be recovered for these spaces. In this talk I will present recent work joint with Guido De Philippis, in which we show that the gradients of harmonic functions vanish at certain singular points of the space. I will mention two applications of this result which are new on smooth manifolds: there does not exist an a priori estimate on the modulus of continuity of the gradient of harmonic functions depending only on lower bounds of the sectional curvature and there is no a priori Calderón-Zygmund inequality for the Laplacian with bounds depending only on the sectional curvature.

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The ends of a space are the connected components of its ideal boundary. Under certain curvature conditions, it is possible to give uniform bounds for the number of ends of Riemannian manifolds. In this talk I will recall previous work by Z.-D. Liu in this direction and show a generalization of this result in the setting of metric measure spaces satisfying the curvature dimension condition CD(0,N) outside a compact set. This is joint work with Jesús Núñez-Zimbrón. Preprint: https://arxiv.org/abs/2108.10659

The mapping class group of a compact simply-connected 4-manifold is the set of self-diffeomorphisms (or self-homeomorphisms, in the topological category), up to isotopy. For a manifold with nonempty boundary, one assumes the self-automorphisms fix the boundary pointwise. In both the smooth and topological categories, I will describe sufficient conditions for two automorphisms to be pseudoisotopic. Pseudoisotopy is weaker than isotopy, but in the topological category we are able to use this theorem to compute the mapping class group in many cases. We use our theorem to prove new topological unknotting results for embedded 2-spheres in many classes of 4-manifold. This is joint work with Mark Powell.

The computation of the essential spectrum of the Laplacian requires the construction of a large class of test differential forms. On a general open manifold this is a difficult task, since there exists only a small collection of canonically defined differential forms to work with. In our work with Zhiqin Lu, we compute the essential k-form spectrum over asymptotically flat manifolds by combining two methods: First, we introduce a new version of the generalized Weyl criterion, which greatly reduces the regularity and smoothness of the test differential forms; second, we make use of Cheeger-Fukaya-Gromov theory and Cheeger-Colding theory to obtain a new type of test differential forms at the ends of the manifold. The generalized Weyl criterion can also be used to obtain other interesting facts about the k-form essential spectrum over an open manifold. Finally, we present some recent results on the form spectrum of negatively curved manifolds.

The Andrews-Curtis conjecture (1965) is one the most relevant open problems in low-dimensional topology, closely related to the Whitehead asphericity conjecture, the Zeeman conjecture and the smooth Poincaré conjecture. It states that any contractible 2-dimensional CW-complex 3-deforms to a point. Its algebraic equivalent formulation states that any balanced presentation of the trivial group can be transformed into the empty presentation through a sequence of a class of movements (called Q**-transformations) that do not change its deficiency. In this talk, I will introduce a new combinatorial method to study Q**-transformations of group presentations or, equivalently, 3-deformations of CW-complexes. The procedure is based on a refinement of discrete Morse theory in terms of Whitehead deformations. I will apply this technique to show that some known potential counterexamples to the Andrews-Curtis conjecture do satisfy the conjecture. Preprint: https://arxiv.org/abs/1912.00115

We define hyperbolicity of real polynomials as introduced by Garding in 1959. They occur in linear PDE, optimization, and differential geometry. We will then report results of Nuij on the space of such polynomials. We finally present our work on Nuij sequences in this space, namely a sufficient condition for certain families of linear operators on polynomials to preserve hyperbolicity.

There has been much interest in generalizing Kesten's criterion for amenability in terms of a random walk to other contexts, such as determining amenability of a deck covering group by the bottom of the spectrum of the Laplacian or entropy of the geodesic flow. A related problem in geometric group theory is that of word growth --- the free group $F_k$ on $k$ generators has exponential word growth $\log(2k-1)$. Grigorchuk gave a criterion for a quotient $F_k/N$ of a free group $F_k$ to be amenable in terms of the growth of the normal subgroup $N$; namely $F_k/N$ is amenable if and only if the exponential growth of $N$ (as a subset of $F_k$) is equal to that of $F_k$. I will discuss some of my work (joint with others) on dynamical versions of these problems.

A double disk-bundle is any manifold obtained by gluing the total spaces of two disk-bundles together by a diffeomorphism. While the definition may seem quite arbitrary, we will show that, in fact, double disk-bundles arise in diverse locations throughout geometry. We will also discuss the double soul conjecture, and its potential consequences, including the classification of Riemannian manifolds of non-negative sectional curvature under certain topological restrictions. This is partly joint work with Fernando Galaz-García and Martin Kerin.

Sub-Riemannian manifolds are metric spaces that model systems with non-holonomic constraints, and constitute a vast generalization of Riemannian geometry. They arise in several areas of mathematics, including control theory, subelliptic PDEs, harmonic and complex analysis, geometric measure theory and calculus of variations. In the last 10 years, a surge of interest in the study of geometric and functional inequalities on sub-Riemannian spaces revealed unexpected behaviours and intriguing phenomena that failed to fit into the classical schemes inspired by Riemannian geometry. In this talk I will review some recent developments on the subject, focusing on the topic of interpolation inequalities for optimal transport and comparison geometry of these structures.

This talk will concern embedded surfaces in 4-manifolds for which the fundamental group of the complement is infinite cyclic. Working in the topological category, necessary and sufficient conditions will be given for two such surfaces to be isotopic. This is based on joint work with Mark Powell.

Gelfand–Zeitlin systems are a well-known family of examples in symplectic geometry, singular Lagrangian torus fibrations whose total space is a coadjoint orbit of a unitary group and whose base space is a certain convex polytope. They are easily defined in terms of matrices but do not fit into the familiar framework of integrable systems with non-degenerate singularities, and hence are much studied as a sort of edge case.

Despite the prominence of Gelfand–Zeitlin systems, not much has been known about the topology of their fibers. In this talk, we discuss the combinatorics giving rise to them and compute their cohomology rings inductively using maps of Serre spectral sequences.

This represents joint work with Jeremy Lane.

Two hyperbolic surfaces are said to be (length) isospectral if they have the same collection of lengths of primitive closed geodesics, counted with multiplicity (i.e. if they have the same length spectrum). For closed surfaces, there is an upper bound on the size of isospectral hyperbolic structures depending only on the topology. We will show that the situation is very different for infinite-type surfaces, by constructing large families of isospectral hyperbolic structures on surfaces of infinite genus.

Let M_g be the moduli space of hyperbolic surfaces of genus g endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon>0$, as genus g goes to infinity, a generic surface $X\in M_g$ satisfies that the first eigenvalue $\lambda_1(X)>\frac{3}{16}-\epsilon$. This is a joint work with Yuhao Xue.

Robert Bryant showed that any closed immersed Willmore sphere in Euclidean three-space is the inversion of a complete minimal sphere with embedded planar ends. We proved that the Willmore Morse Index of the closed surface can be computed by using unbounded Area-Jacobi fields of the related minimal surface. As a consequence, we get that all immersed Willmore spheres are unstable except for the round sphere. This talk is based on work with Jonas Hirsch and Rob Kusner.

It is known that up to connected sum with a homotopy sphere, essentially all highly connected manifolds in dimensions 4k+3 admit a positive Ricci curvature metric. In this talk we consider the curvature of highly connected manifolds in dimensions 4k+1. It turns out that proving an analogous positive Ricci curvature result is out of range at present. However the problem becomes tractable if we consider curvatures which are intermediate between positive scalar and positive Ricci curvature. This is joint work with Diarmuid Crowley.

A natural notion of energy for a map is given by measuring how much the map stretches at each point and integrating that quantity over the domain. Harmonic maps are critical points for the energy and existence and compactness results for harmonic maps have played a major role in the advancement of geometric analysis. Gromov-Schoen and Korevaar-Schoen developed a theory of harmonic maps into metric spaces with non-positive curvature in order to address rigidity problems in geometric group theory. In this talk we discuss harmonic maps into CAT(k) spaces which are metric spaces with positive upper curvature bounds. By proving global existence and analyzing the local behavior of such maps, we determine a uniformization theorem for CAT(k) spheres. We highlight how this uniformization theorem relates to the Cannon Conjecture, a major open conjecture in geometric group theory.

I will report on recent joint work with Johannes Ebert. In this work we study the space $\mathcal{R}^+(M)$ of positive scalar curvature metrics on simply connected spin manifolds $M$ of dimension at least 5. We show that its homotopy type depends only on the dimension of $M$ and the question whether or not $M$ admits a metric of positive scalar curvature, i.e. whether or not $\mathcal{R}^+(M)$ is non-empty. I will also discuss a similar result for non-spin manifolds.

For a curve in projective space, the count of varieties parametrising its secant planes is among the most studied problems in classical enumerative geometry. We shall start with a gentle introduction to secant varieties and then explore the connection between their enumerative geometry and Virasoro algebras on one side, and tautological integrals on the other.

Holomorphic correspondences are polynomial relations P(z,w)=0, which can be regarded as multi-valued self-maps of the Riemann sphere (implicit maps sending z to w). The iteration of such multi-valued map generates a dynamical system on the Riemann sphere (dynamical system which generalises rational maps and finitely generated Kleinian groups). We consider a specific 1-(complex)parameter family of (2:2) correspondences F_a (introduced by S. Bullett and C. Penrose in 1994), which we describe dynamically. In particular, we show that for every a in the connectedness locus M_{\Gamma}, this family is a mating between the modular group and rational maps in the family Per_1(1); we develop for this family a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials; and we show that M_{\Gamma} is homeomorphic to the parabolic Mandelbrot set M_1. This is joint work with S. Bullett (QMUL).

The soap bubble theorem says that a closed, embedded surface of the Euclidean space with constant mean curvature must be a round sphere. Especially in real-life problems it is of importance whether and to what extent this phenomenon is stable, i.e. when a surface with almost constant mean curvature is close to a sphere. This problem has been receiving lots of attention until today, with satisfactory recent solutions due to Magnanini/Pogessi and Ciraolo/Vezzoni.

The purpose of this talk is to discuss further problems of this type and to provide two approaches to their solutions. The first one is a new general approach based on stability of the so-called "Nabelpunktsatz". The second one is of variational nature and employs the theory of curvature flows.

The Sullivan-conjecture claims that complex projective complete intersections are classified up to diffeomorphism by their total degree, Euler-characteristic and Pontryagin-classes. It follows from work of Kreck and Traving that the conjecture holds in complex dimension 4 if the total degree is divisible by 16. In this talk I will present the proof of the remaining cases. It is known that the conjecture holds up to connected sum with the exotic 8-sphere (this is a result of Fang and Klaus), so the essential part of our proof is understanding the effect of this operation on complete intersections. This is joint work with Diarmuid Crowley.

An open question in spectral geometry is to determine whether the Laplace spectrum detects the presence of orbifold singularities. In this talk I will show that the Laplace spectrum for functions, along with that for 1-forms, allows one to detect singular points in dimensions two and three. This is joint work with Katie Gittins, Carolyn Gordon, Magda Khalile, Mary Sandoval and Liz Stanhope.

Amorphic complexity is a conjugacy invariant which is particularly suitable to distinguish low complexity (specifically: zero entropy) dynamical systems. Here, a dynamical system is understood as a continuous action of a topological group on a compact space. We will introduce amorphic complexity (as well as the closely related concept of asymptotic separation numbers) and discuss some of its basic properties. We further take a closer look at its values for specific classes of examples including substitutive subshifts and, if time allows, regular cut and project schemes. This will allow us to observe surprisingly straight-forward connections to fractal geometry. I will provide definitions of all of the relevant non-standard notions so that the talk should be understandable by a broad audience.

This is joint work with Maik Gröger, Tobias Jäger and Dominik Kwietniak (carried out by three different subsets of the four of us).

A classical result due to Bochner says that for a compact, smooth and connected Riemannian manifold with non-negative Ricci curvature, the dimension of the first cohomology group is bounded from above by the dimension of the manifold. Moreover if these two dimensions are equal, then the manifold is a flat torus. In this talk I present a generalization of this result to the non-smooth setting of RCD spaces, by proving that if the dimension of the first cohomology group of a RCD*(0,N) space is N, then it is possible to construct an isomorphism between the space and the N-dimensional torus, equipped with its Riemannian distance and a constant multiple of the induced volume measure. This is a joint work with N. Gigli.

The 2-dimensional Lyness recurrence is a 5-periodic birational map (x, y) -> (y, (1+y)/x), which can be interpreted as a mutation between five open torus charts in a del Pezzo surface of degree 5, coming from a cluster algebra structure on the Grassmannian Gr(2,5). I will briefly recap this, and then explain the following 3-dimensional generalisation: the 8-periodic birational map (x, y, z) -> (y, z, (1+y+z)/x) can be used to exhibit a Laurent phenomenon for the orthogonal Grassmannian OGr(5,10). If time permits I will then explain some applications of this to mirror symmetry of Fano 3-folds.

The homotopy groups of the diffeomorphism group of a high dimensional manifold with infinite fundamental group can be infinitely generated. The simplest example of this sort is the solid torus $T=S^1\times D^{d-1}$. In fact, using Hatcher, Igusa, and Waldhausen's approach to pseudoisotopy theory, it is possible to show that in the range of degrees up to (roughly) $d/3$, the homotopy groups of $Diff(T)$ contain infinitely generated torsion subgroups.

In this talk, I will discuss an alternative point of view to study $Diff(T)$ which does not invoke pseudoisotopy theory: when $d=2n$, we interpret $Diff(T)$ as the "difference" between diffeomorphisms and certain self-embeddings of the manifold $X_g$ which is the connected sum of $T$ with the g-fold connected sum of $S^n \times S^n$.

We will see how infinitely generated torsion subgroups appear from this perspective, and that they can be found even up to degrees $d/2$. This is ongoing joint work with O. Randal-Williams.

In this talk, I will discuss a recent result concerning real forms of affine varieties. Given a real variety X, a real form of X is a real variety Y such that the complexifications of X and Y are isomorphic as complex varieties. I will show how to construct smooth rational affine algebraic varieties of dimension 4 or higher which admit infinitely many non-isomorphic real forms. This is joint work with A. Dubouloz and G. Freudenburg.

Let (V, ∥ · ∥) be a real Banach space. Fix n ≥ 2. Consider the following hypothesis:

Hn: all n-dimensional subspaces of V are isometric to each other.

In his 1932 book, Banach asked: Hn ⇒ (V, ∥ · ∥) is necessarily a Hilbert space?

This is the (real) 'isometric problem of Banach'. It is easy to see -I'll show it in the talk- that it really is a codimension 1 problem: if one knows that the question has a positive answer for a fixed n, for all (n + 1)-dimensional normed spaces, then it will have a positive answer, for that same n, for every (even infinite dimensional) Banach space (V, ∥ · ∥). Thus, one can restate Banach's question as:

If all hyperplanes Γ ⊂ (R^{n+1}, ∥ · ∥) are isometric to each other, is (R^{n+1}, ∥ · ∥) euclidean (n+1)-space?

In 1967, Gromov showed that the answer is yes for even n. In proving it, he found a way to relate this problem to the existence of certain G-structures on S^n, thus allowing some of the machinery of algebraic topology to come to aid.

Recently (in joint work with G. Bor (CIMAT), V. Jiménez and L. Montejano (UNAM)) we have shown that Banach's isometric problem has also a positive answer for every n ≡ 1 (mod 4), n ≠133.

In this talk I'll give a sketch of the proof of this result. I'll recall Gromov's key idea mentioned above, which together with some algebraic topology theorems, plus some basic representation theory, translates the problem to one in convex geometry: namely, a certain characterization for (n+1)-dimensional ellipsoids.

Most of the arguments should be accessible to graduate students.

In recent years a lot of effort has gone into studying spaces of Riemannian metrics with lower curvature bounds. In contrast to the case of positive scalar curvature, very little is known for positive Ricci curvature, especially when one is interested in higher homotopy or (co-)homology groups. In this talk I will demonstrate how to detect nontrivial higher rational cohomology groups of this space. The main new ingredient is the construction of bundles with base and fiber both products of spheres and non-vanishing A-hat-genus. This is joint work with Jens Reinhold.

Let P2(M) be the space of probability measures on a Riemannian manifold M. Using the solutions to Monge-Kantorovich's optimal transport problem it is possible to define a distance on P2(M), the so called L2−Wasserstein distance W2. This distance reflects many geometrical properties of the manifold such as: compactness, geodesics, and non-negative sectional curvature. In this talk we will discuss some intrinsic properties of Wasserstein spaces, more precisely we give a positive answer to the following question: If two closed Riemannian manifolds M, N are such that their corresponding Wasserstein spaces P2(M), P2(N) are isometric, does it follow then that M is isometric to N? Moreover, if we assume that the Riemannian manifold has positive sectional curvature we can also prove that the isometry groups of the manifold M and of the Wasserstein space P2(M) coincide.

The reach is an important geometric invariant of submanifolds of Euclidean space. It is a real-valued global invariant incorporating information about the second fundamental form of the embedding and the location of the first critical point of the distance from the submanifold. In the subject of geometric inference, the reach plays a crucial role. I will give a new method of estimating the reach of a submanifold, developed jointly with Clément Berenfeld, Marc Hoffmann and Krishnan Shankar.

A large body of research has been developed to address the following question: "Can we classify closed, positively curved manifolds that have large torus symmetries?" Essential tools in this area include Berger's Killing Field Zero-Set Theorem and Wilking's Connectedness Principle. In this talk, I will address the corresponding question for manifolds with positive k^th-intermediate Ricci curvature. On an n-manifold, this curvature condition interpolates between positive sectional curvature (k = 1) and positive Ricci curvature (k = n - 1). I will show how Berger's result and Wilking's result generalize to positive intermediate Ricci curvature. I will also demonstrate how these tools allow us to obtain topological information for manifolds of positive 2^nd-intermediate Ricci curvature with large torus symmetries.

In this talk we will discuss the following dynamical notion: two given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if f is a critical circle map with rotation number of bounded type. By contrast, in collaboration with Edson de Faria (Universidade de Säo Paulo), we recently proved that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in (0,1), then the number of equivalence classes is uncountable. The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. If there is enough time, we will show how, as a by-product of our techniques, we were able to construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and how we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints.

Homogeneous spaces are manifolds with many symmetries, and as such they are a fantastic playground for mathematical models ranging from general relativity to solid state physics. In this talk, I will give a non-technical approach to the different types of symmetries that one likes to consider - like reflections, special properties of geodesics, curvature, or differential operators - with many examples and applications. In the last part, I will present some recent classification results on certain classes of homogeneous spaces, and why they are interesting.

The talk is suitable as an introduction to the vast area of homogeneous spaces for non-experts.

For any smooth Jordan curve and rectangle in the plane, we show that there exist four points on the Jordan curve forming the vertices of a rectangle similar to the given one. Joint work with Josh Greene.

For every genus g larger than 1, there exists a 6g-6 dimensional deformation space of hyperbolic metrics (i.e. of constant curvature -1) on a closed orientable surface of genus g. In this space, one can find surfaces of arbitrarily large diameter. On the other hand, there is a lower bound on the diameter of a hyperbolic surface of genus g. In this talk I will speak about the asymptotic behavior of this bound as g tends to infinity. This is joint work with Thomas Budzinski and Nicolas Curien.

GKM manifolds are smooth manifolds endowed with an action of a torus generalizing in some sense toric manifolds. It is possible to assign to every GKM manifold a combinatorial datum, the so called GKM graph, which consists of an abstract graph with labeled edges. On the other side it is possible to define abstract GKM graphs and the realization problem asks if there exists a GKM manifold such that the GKM graph is the given one. We will examine this question in dimension 6 and for GKM graphs which are, in some sense, fiber bundles over other GKM graphs. Moreover we will exhibit some geometric properties about those GKM manifolds, in particular we will show that they are infinitely many Kähler manifolds such that the underlying symplectic form is invariant under a torus action in contrast to the Kähler metric.

Let SL(n,Z) be the special linear group over the integers. The topological Zimmer conjecture states that any action of SL(n,Z) on a compact manifold M factors through a finite group, when dim(M)< n -1. In this talk, we will show the following result: when the Euler characteristic of an orientable manifold M is not divisible by 6, any action of SL(n,Z) on M is actually trivial (not only finite), when dim(M) is strictly less than n-1.

Let S^3 be the unit 3-sphere with its standard Cauchy'“Riemann (CR) structure. We will consider the CR geometry of Legendrian curves in S^3 using the local CR invariants of S^3 thought of as a 3-dimensional CR manifold. More specifically, the focus is on the lower-order cr-invariant variational problem for Legendrian curves in S^3 and on its closed critical curves. The Liouville integrability of such a variational problem will be considered. We discuss the admissible contact isotopy classes of closed critical curves with constant bending. Subsequently, we characterize closed critical curves with non-constant bending in terms of three numerical invariants. In addition, we analyze the geometrical meaning of the numerical invariants in terms of the cr-symmetries of closed critical curves and of the their linking numbers with the symmetry axes.

Homological stability is a topological property that is satisfied by many families of groups, including the symmetric groups, braid groups, general linear groups, mapping class groups and more; it has been studied since the 1950's, with a lot of current activity and new techniques. In this talk I will explain a set of homological stability results from the past few years, on Coxeter groups, Artin groups, and Iwahori-Hecke algebras (some due to myself and others due to Rachael Boyd). I won't assume any knowledge of these things in advance, and I will try to introduce and motivate it all gently.

The Poincare conjectures roughly state that any closed n-manifold that looks like the n-sphere is the n-sphere. There are various versions of the conjecture: if a manifold is homotopy equivalent to the n-sphere, is it homeomorphic to the n-sphere? If it is homeomorphic to the n-sphere, is it diffeomorphic to the n-sphere? These are referred to as the topological and smooth Poincare Conjectures, respectively. It is claimed that they have been resolved in all cases except for the 4-dimensional smooth Poincare Conjecture, which remains shrouded in mystery.

In this talk, we will explore reasons for this gap and point to the incomplete understanding of Freedman's claimed resolution of the 4-dimensional topological case. The talk will centre on a series of unanswered MathOverflow questions:

https://mathoverflow.net/questions/87674/independent-evidence-for-the-classification-of-topological-4-manifolds https://mathoverflow.net/questions/108631/fake-versus-exotic https://mathoverflow.net/questions/252563/the-freedman-dichotomies

We will describe the background and motivation of these questions and explain why the claim of the Freedman Disk Theorem, being at the heart of the matter, is problematic. In addition, we will outline approaches to disproving Freedman's claim and the implications of such a disproof.

I will start with an introduction to classification of C*-algebras and Cartan subalgebras of C*-algebras. The main goal of the talk is then to explain how to construct Cartan subalgebras in all classifiable stably finite C*-algebras. Finally, I will discuss a concrete example, which reveals a surprising connection to topology and topological dynamics.

Aperiodic sequences over finite alphabets are ubiquitous in the study of topological dynamics, and as such, it's important that we have tools for studying such sequences. We're able to use methods from algebraic topology, such as Cech cohomology to provide invariants for these sequences, especially when the sequences have additional structure such as those generated by substitutions. One first builds a topological space associated with the sequence, called the tiling space for which cohomology can then be computed. These spaces are interesting in their own right and rather different to the standard beasts that a topologist might usually encounter. I will give a brief introduction to tiling spaces and explain how we are sometimes able to calculate cohomology for sequences over other (infinite) alphabets such as compact Lie groups.

The attractors of dissipative dynamics at the boundary of chaos often has universal geometry. The explanation for this universality comes from renormalization. There is a simple and powerful idea in related areas of physics: a change of coordinates leaves things essentially the same. This idea is at the heart of geometric universality at the boundary of chaos.

The tautological ring of smooth fibre bundles with fibre M is the subring of the cohomology of BDiff(M) generated by the generalised Miller-Morita-Mumford classes, which are defined as fibre integrals of characteristic classes of the vertical tangent bundle. The fibrewise Euler class can be defined more generally for fibrations with Poincaré fibre X so that there is an analogous definition of the tautological ring of fibrations as the subring of the cohomology of Bhaut(X) generated by fibre integrals of powers of the fibrewise Euler class. I will discuss how to compute it using the well-studied algebraic models of fibrations from rational homotopy theory. Furthermore, I will show how one can extract obstructions to smoothing fibrations for some rationally elliptic manifolds.

Let M be a spherical space form of dimension at least 5 which is not simply-connected. Then the moduli space of Riemannian metrics with positive scalar curvature on M has infinitely many path components as shown by Boris Botvinnik and Peter B. Gilkey in 1996. We will review this theorem which involves twisted spin structures, suitable bordism groups and eta invariants. We then show that it can be generalized to the class of topological spherical space forms, i.e. smooth manifolds whose universal cover is a homotopy sphere.

Many families of groups, such as braid groups, have a representation theory of wild type, in the sense that there is no known classification schema. Hence it is useful to shape constructions of linear representations for such families of groups to understand their representation theory. I will present a unified functorial construction of homological representations for these families of groups, which is a joint work in progress with Martin Palmer. For instance, this construction provides the family of Lawrence-Bigelow representations for braid groups. Under some additional assumptions, general notions of polynomiality on functors are a useful tool to classify these representations.

In 1983, Hirzebruch considered some arrangements of complex lines in the complex projective 2-space and showed that a suitable branched cover leads to a complex hyperbolic manifold, which turned out to be one of Deligne-Mostow lattices. In 2019, Dashyan constructed a Lefschetz fibration on this space and used it to build representations of 3-manifolds into PU(2,1), with image the lattice in question. I will explain his construction and tell you how we are planning to generalise this to all other Deligne-Mostow lattices and interpret this construction in terms of the fundamental domains I built for these lattices. This is a work (very much) in progress, joint with Elisha Falbel.

Automorphic Lie algebras are a class of infinite-dimensional Lie algebras that are closely related to a wide variety of algebraic structures that appear in integrable systems theory, mathematical physics and geometry. They can be viewed as a certain generalisation of the well-studied (twisted) loop algebras and current algebras. It can often be difficult to immediately gain an intuitive understanding of the algebraic structure behind an automorphic Lie algebra. However, this task can be made easier using techniques in representation theory. Associated to an automorphic Lie algebra is a commutative algebra of functions. Studying automorphic Lie algebras via evaluation maps parameterised by the representations of the associated commutative algebra provides a descending chain of ideals of the automorphic Lie algebra. A detailed study of this chain of ideals immediately shows that the representation theory of automorphic Lie algebras is wild, and enables us to describe the local Lie structure of the automorphic Lie algebra.

In the case where G is a group with periodic cohomology, there is a somewhat unusual correspondence between projective modules over the integral group ring Z G and homotopy types of certain CW-complexes over G. We exploit this connection to prove a special case of C. T. C Wall's conjecture on cohomologically 2-dimensional CW-complexes and also show how this leads to a supply of many interesting CW-complexes and manifolds in higher dimensions. I will assume no familiarity with any of the algebra involved.

(Joint with Hirotaka Akiyoshi, Ken'ichi Ohshika, Makoto Sakuma and Han Yoshida) In the the 1970s Riley gave a conjectural classification of Kleinian groups generated by two parabolic transformations. In particular, he identified a family of groups, called Heckoid groups, which are discrete and non free. These group generalise the classical Hecke groups. In 2002 Agol announced a strategy to show that any non-free Kleinian group with two generators is either a Heckoid group or else a two-bridge knot/link complement group. The goal of this project is to give a proof of this result by following Agol's announcement. I will give some general background and then talk about some aspects of the proof.

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We will discuss the nonorientable surfaces that torus knots bound. We use a surface construction introduced by Josh Batson together with tools from knot Floer homology to compute the nonorientable four-genus of infinite families of torus knots. Comparing this surface construction with the surfaces realizing torus knots' non-orientable three-genus, we show that the difference between nonorientable three- and four-genus can be arbitrarily large. This contrasts with the analogous situation in the orientable world. Kronheimer and Mrowka proved in 1993 that both the orientable three-genus and the orientable four-genus for T(p,q) are equal to (p-1)(q-1)/2. This is joint work with Stanislav Jabuka.

There are several different, but equivalent definitions of geodesics in a Riemannian manifold, They are generalized to sub-Riemannian manifolds, but become non-equivalent. H.R. Herz remarked that there are two main approaches for definition of geodesics: geodesics as shortest curves based on Mopertrui's principle of least action (variational approach) and geodesics as straightest curves based on d'Alembert's principle of virtual work (which leads to geometric descriptions based on the notion of parallel transport). We shortly discuss different definitions of sub-Riemannian geodesics and interrelations between them.

We recall the notion of an essential surface in a 3-manifold and explain how Culler-Shalen used curves in the representation variety of SL(2,C) to construct them. After generalizing this construction to SL(n,C) representations, we explain how all essential surfaces can be obtained from this construction.

This is joint work with Stefan Friedl and Takahiro Kitayama.

The Dai-Freed theorem provides a bridge between the theory of bordism and Quantum Field Theory (and more specifically, anomalies). I will review how these two areas are related, and then summarise some computations of bordism groups of classifying spaces of Lie groups and cyclic groups that we have performed recently, which are of particular interest for applications to four dimensional physics.

From the Morse theoretic point of view Zoll metrics are rather peculiar. All critical sets of the energy on the loop space are nondegenerate critical manifolds diffeomorphic to the unit tangent bundle. This especially implies that min-max values associated to certain homology classes coincide. In my talk I will explain that the coincidence of these min-max values characterises Zoll metrics in any dimension. A specially focus will lie on the case of the 2-sphere. This is work in collaboration with Marco Mazzucchelli (ENS Lyon).

Witt vectors are a generalization of p-adic numbers and show up in computations in topology. Motivated by those calculations, this talk will discuss a new structure on Witt vectors which is functoriality in certain polynomial maps. We will start by introducing Witt vectors and polynomial maps. Along the way we will focus on explicit examples. Then we will explain the main functoriality result. Finally, we will mention applications in algebra and topology. This is joint work with E. Dotto and K. Moi.

A knot in the 3 sphere is called (smoothly) slice if it bounds a properly (smoothly) embedded disk in the 4 ball. Nowadays there are many computable invariants that help us tackle the in general difficult question of whether or not a given knot is slice. In this talk we will discuss the well known family of pretzel knots from the perspective of this question. We will discuss some classification results and some intriguing open questions.

Lipshitz and Sarkar associated to any link L in the 3-sphere a certain prespectrum X_{Kh}(L) whose stable homotopy type is a link invariant. Moreover, the reduced cohomology of X_{Kh}(L) is isomorphic to the Khovanov homology of L. A link L may admit nontrivial symmetries, hence a natural question to ask is whether these symmetries can be lifted to X_{Kh}(L). It turns out that in the case of rotational symmetries such a lift exists and equips X_{Kh}(L) with a group action. The purpose of this talk is to discuss consequences of the existence of the group action on X_{Kh}(L) induced by the symmetry of L. In particular, by studying the fixed points of the action we will obtain a nontrivial relation between the Khovanov homology of a periodic link and the Khovanov homology of the quotient link. If time permits, we will also sketch the construction of the group action.

Two n-strand braids close to isotopic links in the solid torus if and only if they represent conjugate elements of the braid group B_n. This is a textbook theorem, which is proved in the books of Burde-Zieschang and Kassel-Turaev, as well as a paper of Morton.

In joint work with Agata Smoktunowicz (who was supported by an LMS undergraduate bursary at the University of Aberdeen) we prove an analogue of this result for closed surface braids. Let S be a closed orientable surface of genus at least 2. Then two surface braids close to isotopic links in S times S^1 if and only if they represent conjugate elements in the surface braid group B_n(S).

The mapping class group Γ(g) of a surface #ᵍ(S¹ x S¹) of genus g shares many features with its higher dimensional analogue Γ(g,n)'”the group of isotopy classes of diffeomorphisms of #ᵍ(Sⁿ x Sⁿ). Some aspects, however, become easier to analyse in high dimensions, for instance the so-called Torelli subgroup. This enabled Kreck in the 70's to describe Γ(g,n) for n>2 in terms of an arithmetic group and the group of exotic spheres, but his answer left open two extension problems, which were later understood in some particular dimensions, but remained unsettled in general. In this talk, I will recall Kreck's description of Γ(g,n) and explain how to resolve these extension problems in the case of n being odd.

We consider the set of connected surfaces in the 4-ball that bound a fixed knot in the 3-sphere. We define the stabilization distance between two surfaces as the minimal g such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most g. Similarly, we obtain the double point distance between two surfaces of the same genus by minimizing the maximal number of double points appearing in a regular homotopy connecting them. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces that give lower bounds on the stabilization distance and the double point distance. This is joint work with Ian Zemke.

Motivated by a recent work of Delgove and Retailleau on right-angled hexagons in hyperbolic 5-space, we will discuss right-angled polygons in hyperbolic spaces of arbitrary dimension. Clifford algebras will be used to model hyperbolic space with its isometries and to exploit cross-ratios in higher dimensional hyperbolic spaces. This will allow the generalisation of the aforementioned work in order to present an algorithm to construct a p-gon given by p-3 Clifford vectors. This is joint work with Edoardo Dotti.

The world of quantum invariants started with the discovery of the Jones polynomial. Then, Reshitikhin-Turaev introduced a purely algebraic construction that having as input a quantum group produces link invariants. The coloured Jones polynomials {J_N(L,q)}_N are sequences of link invariants constructed in this way using the quantum group U_q(sl(2)), whose first term is the original Jones polynomial. R. Lawrence introduced a sequence of topological braid group representations based on the homology of coverings of configuration spaces. Using that, Bigelow and Lawrence gave a homological model for the Jones polynomial, using its Skein relation nature. We will present a topological model for all coloured Jones polynomials. We will show that J_N(L,q) can be described as graded intersection pairings between two homology classes in a covering of the configuration space in the punctured disc. This shows that the Lawrence representations are rich objects that contain enough information to encode all coloured Jones polynomials and possibly more. In the last part of the talk we will present some directions towards a geometrical categorification for J_N(L,q) that can be defined out of this topological model.

We present a construction which associates to a KdV equation the lamplighter group. In order to establish this relation we use automata and random walks on ultra discrete limits. It is also related to the L2 Betti numbers introduced by Atiyah which are homotopy invariants of closed manifolds.

Minimizers of nonlinear variational problems can have vortex-like point singularities. To know what type of information is encoded in the singularities in a given problem, one needs to check which topological invariants are preserved under the weak convergence which makes the sub-levelsets of the energy precompact. In the classical setting of harmonic maps from R^3 to S^2, defects have a "turning number" which measures the multiplicity with which the image of a small sphere around a singularity covers the target. In other problems, the Hopf degree and other topological invariants play the analogous role. If we are studying problems on vector bundles, "how much the the bundle is turning in the vicinity of a defect" can be quantified via appropriate Chern numbers. In all the above cases, the energy "contained in a configuration of defects" can be re-expressed as an optimal-transport-type problem. We will see some examples of how these auxiliary optimal transport problems allow to better understand/control the nonlinear variational problems one started with.

ordered (sutured) Heegaard Floer homology is an invariant for three-manifolds with boundary: one such manifold is the complement of a link in the three-sphere. I will talk about some older Heegaard Floer invariants of links in the three-sphere, and try to give some idea of the relationship between these and the corresponding bordered invariants of their complements.

For any 3-manifold M with torus boundary, we find finitely generated subgroups of Diff_0(\partial M) whose actions do not extend to actions on M; in many cases there is even no action by homeomorphisms. The obstructions are both dynamical and cohomological in nature. We also show that, if \partial M = S^2, there is no section of the map Diff_0(M) \to \Diff_0(\partial M). This answers a question of Ghys for particular manifolds; and gives tools for progress on the the general program of bordism of group actions. This is joint work with Kathryn Mann.

We explain how Bar-Natan's algorithm for calculating Khovanov cohomology can be adapted to calculating s-invariants of knots in a way that will noticably speed up the process.

In low dimensional topology, several invariants can be obtained as signatures of Hermitian matrices. For instance, in knot theory, such an example is given by the Levine-Tristram signature, whose origin traces back to the 60's. After reviewing several properties of this knot invariant, we will describe a new signature function that takes as input both a knot and a representation of the knot group. I will argue that this "twisted signature" is a natural generalisation of the Levine-Tristram signature and provides obstructions to a knot being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

With the advent of knot homology theories such as Khovanov and knot Floer homologies, new invariants to study knot concordance have been developed. For example, Rasmussen's s-invariant, the Ozsvath-Szabo tau-invariant and the s_N invariants due to Lobb and Wu, independently, and another family of invariants introduced by Lewark and Lobb. All these invariants share three fundamental properties, which were identified by Livingston, and were called ''slice-torus invariants'' by Lewark. Some among the slice-torus invariants, namely s, tau and the s_N's admit a generalisation to strong concordance invariants for links. Motivated by the properties shared by these extensions, together with A. Cavallo, I gave the definition of slice-torus link invariants and studied their properties.

In this seminar I will give the definition of slice-torus link invariants, and describe some of their properties. Finally, I will give some applications and define some new strong concordance invariants using Whitehead doubling.

We introduce a new method to obtain Harnack inequalities for extrinsic curvature flows such as the mean curvature flow and more general fully nonlinear flows. For example, this method allows us to deduce Harnack inequalities for the mean cur- vature flow in locally symmetric (Riemannian or Lorentzian) Einstein spaces, for flows with convex speeds in the De Sitter space and for the Gauss curvature flow in Minkowski space.

Let d>0 be a square-free integer and L_d be the Hilbert lattice, i.e. the even lattice of signature (2,2), corresponding to the ring of integers of the real quadratic field Q(\sqrt(d)). Consider the group \Gamma which is a finite index subgroup of O^+(L_d) generated by reflections and containing -id, and let A(\Gamma) be the algebra of \Gamma-automorphic forms. We study for which values of d the algebra A(\Gamma) can be free.

The Bonnet-Myers theorem is a classical theorem which gives an estimate of the diameter in term of the positive Ricci curvature bound of a manifold. In the discrete setting of graphs, Ollivier's notion of Ricci curvature provides a discrete analogue of Bonnet-Myers theorem. In view of Cheng's rigidity result, it is natural to ask for which graphs the Bonnet-Myers estimates is sharp. We call such graphs Bonnet-Myers sharp graphs.

We prove that, under an extra condition of antipodalness (i.e. each vertex has a vertex whose distance between them is equal to the diameter), a Bonnet-Myers sharp graph must be ``strongly spherical', which is a combinatorial property that has been completely classified. The proof includes a new method of transport geodesic, which I will explain in my talk.

This is joint work with Bourne, Cushing, Koolen, Liu, M\'{u}nch, and Peyerimhoff.

Many sequences of groups satisfy a phenomenon known as homological stability. In my talk, I will report on recent work proving a homological stability result for sequences of Artin monoids, which are monoids related to Artin and Coxeter groups. From this, one can conclude homological stability for the corresponding sequences of Artin groups, assuming a well-known conjecture in geometric group theory called the K(\pi,1)-conjecture. This extends the known cases of homological stability for the braid groups and other classical examples. No familiarity with Coxeter and Artin groups, homological stability or the K(\pi,1)-conjecture will be assumed.

In this talk we consider transformations of the unit interval of the form βx + α mod 1 where 1< β<2 and 0≤ α ≤ 2 - β. These transformations are called intermediate β-transformations. We will discuss some old and new results concerning these transformations, for instance, their kneading sequences, their absolutely continuous invariant measures and dynamical properties such as topological transitivity and the sub-shift of finite type property. Moreover, we address how the kneading sequences and absolutely continuous invariant measures change as we let (β,α) converge to (1,θ), for some θ ∈ [0, 1]. Finally, some open problems and applications of these results to one-dimensional Lorenz maps and quasicrystals will be alluded to.

The Hochschild cohomology of a differential graded algebra or more generally of a differential graded category admits a natural map to the graded centre of its derived category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the possible failure of the characteristic homomorphism to be injective or surjective answering a question by Bernstein. To illustrate this, we will discuss several illuminating examples from geometry and topology, like coherent sheaves over algebraic curves, as well as examples related to free loop spaces and string topology. This is joint work with Markus Szymik (NTNU Trondheim).

We will recall different versions of the Brunn-Minkowski inequality (interpolation of sets) and their connection to the isoperimetric problem and the theory of optimal transportation (interpolation of measures). We will examine the interpolation of sets on some sub-Riemannian manifolds, including the Heisenberg group and the Grushin plane.

Isothermic surfaces are surfaces which have a conformal curvature line parametrisation and surfaces of revolution, minimal surfaces and CMC surfaces are examples. Since the latter two surface classes are given by a harmonic map, one can introduce a spectral parameter and derive new surfaces from methods of integrable systems, such as the Darboux transform and the simple factor dressing. CMC surfaces are isothermic surfaces which are constrained Willmore; we will discuss how the different associated families are linked.

In this talk we introduce a geometrical model of continued fractions and discuss its appearance in rather distant areas of Mathematics: -- values of quadratic forms (Perron Identity for Markov spectrum) -- the 2nd Kepler law on planetary motions -- Global relation on singularities of toric varieties

I will explain how the topographical representation of binary quadratic forms introduced by Conway is related to some classical results and geometric constructions in number theory. The talk is based on joint work with A.P. Veselov.

We prove that compact toric Kaehler manifolds do not admit (invariant) metrics that are critical for the first (equivariant) eigenvalue as a function on the moduli space of (invariant) metrics. This is joint work with Zuoqin Wang (USTC).

We will present a detailed proof for the existence of a closed convex hypersurface in the hyperbolic ball with prescribed 1 \le k < n - Weingarten curvature. Specifically, we deal with the equivariant problem for a sufficiently large group of hyperbolic automorhphisms. The proof proceeds by establishing (nonlinear) strict ellipticity of the associated PDE. Then we obtain existence in C^{1,a} for an auxiliary problem by Schauder theory, C^2 smoothness using ellipticity and a Lemma by Cheng-Yau, and C^{2,a} - regularity by Evans-Krylov. Finally, existence of a solution is established by degree theory in the equivariant setting. The results presented are part of the speakers PhD thesis.

This is joint work with Doron Puder (Tel Aviv University). For a positive integer r, fix a word w in the free group on r generators. Let G be any group. The word w gives a `word map' from G^r to G: we simply replace the generators in w by the corresponding elements of G. We again call this map w. The push forward of Haar measure under w is called the w-measure on G. We are interested in the case G = U(n), the compact Lie group of n-dimensional unitary matrices. A motivating question is: to what extent do the w-measures on U(n) determine algebraic properties of the word w?

For example, we have proved that one can detect the 'stable commutator length' of w from the w-measures on U(n). Our main tool is a formula for the Fourier coefficients of w-measures; the coefficients are rational functions of the dimension n, for reasons coming from representation theory.

We can now explain all the Laurent coefficients of these rational functions in terms of Euler characteristics of certain mapping class groups. I'll explain all this in my talk, which should be broadly accessible and of general interest. Time permitting, I'll also invite the audience to consider some remaining open questions.

Knots and their groups are a traditional topic of geometric topology. In this talk I will explain how the subject can be approached by an algebraic topologist, using ideas from Quillen's homotopical algebra, rephrasing old results and leading to new ones.

The diffeomorphism classification of 4-manifolds is a very hard problem. But it gets considerably easier when one allows connected sums with complex projective planes. In this talk I will show that the stable diffeomorphism type in this sense is often determined by the Postnikov 2-type of the manifold. This is joint work with Mark Powell and Peter Teichner.

Following the work of Artin and Series, continued fractions can be viewed as geodesics intersecting the Farey triangulation in the upper half plane. One can use this approach to construct geometric multiplication maps of continued fractions, by constructing maps between triangulations of the upper half plane. For specific primes we have been able to show that these triangulations have a common tiling. As a result, one can construct a punctured surface with two triangulations, such that for any geodesic, prime multiplication of a continued fraction can be represented by the map between the cutting sequences of these triangulations. This work has been motivated by a reformulation of the p-adic Littlewood Conjecture; an open problem in Diophantine approximation.

I will present a 1-parameter family of affine interval exchange transformations (AIETs) that display various dynamical behaviours. We will see that a fruitful viewpoint from which to study such a family is to associate to it what we call a dilation surface, which should be thought of as the analogue of a translation surface in this setting. The study of this example is a good motivation for several conjectures on the dynamics of AIETs that we will try to explain.

We prove that the null-homotopic class in every 3'“manifold other than the 3-sphere contains an infinite family of knots, all topologically concordant, but not piecewise linear concordant to one another. This is joint work with Matthias Nagel, Patrick Orson and Mark Powell.

Periodic patterns of Euclidean space are decorations by motifs, such as point patterns or tiles, which have full-rank global translational symmetry. This means that they can be described from just a fundamental domain and their symmetry group. An aperiodically ordered pattern is one which can frequently repeat itself on finite patches but without being globally periodic. These are far more complicated to analyse and a variety of abstract tools has been developed to understand them. In this talk I shall explain how one studies them topologically, via associated moduli spaces of locally indistinguishable patterns. Topological invariants are applied, such as K-theory or Cech cohomology. I shall briefly outline how one goes about computing these invariants and how one may visualise what they say about the original pattern. At present most attention is dedicated to studying these patterns translationally. Bringing in rotations introduces some interesting challenges; a 3-dimensional periodic pattern, for example, has associated translational moduli space simply the 3-torus, but the rotational version is a 6-manifold whose topology depends crucially on the rotational symmetries of the pattern. I shall explain some recent progress with John Hunton in computing topological invariants for these spaces.

In order to implement knotted configurations in physical systems it is often very useful to have an explicit function, ideally a polynomial, f:R^3 -> C with a zero level set of given knot type. In this talk I will introduce an algorithm that for every link L constructs a polynomial f:R^4 -> R^2 whose zero level set on the unit three-sphere is L. Applying stereographic projection then makes these functions applicable to physical systems.

This constructive approach allows us to prove several results about the functions and their knotted zero level sets, for example under which conditions f can be taken to have an isolated singularity or when arg f is a fibration of the link complement over S^1.

Virtual knot theory is an extension of classical knot theory which considers knots and links in equivalence classes of thickened orientable surfaces. Khovanov homology is a powerful invariant of classical links, and it can be applied to virtual links using Z_2 coefficients. However, a number of problems arise when one attempts to use other coefficient rings. In this talk we describe doubled Khovanov homology: an extension of Khovanov homology to virtual links with arbitrary coefficients. Unlike other extensions of Khovanov homology, doubled Khovanov homology requires no new diagrammatics, as all the work is done algebraically. We shall describe the construction of the invariant as well as some of its applications, in particular to virtual knot concordance.

In this talk I will discuss the utilisation of neutral metrics - metrics of signature (n,n) - to investigate the topology of manifolds of various dimensions. Such metrics, while relatively neglected in comparison to their Riemannian and Lorentzian counterparts, arise in a surprising number of natural settings. In particular, embedding problems between manifolds often inherit such a metric, primarily because the Coddazi-Mainardi equations form an under-determined hyperbolic system. To illustrate this a series of canonical isometric embeddings will be presented along with their links with classical surface theory, the X-ray transform, Legendrian knot invariants, quasi-linear Navier-Stokes equations and, ultimately, a framework for a Grand Unification Theory of the fundamental forces in physics.

This will be a wrap-up of the study group on optimal transport led by Norbert Peyerimhoff with participants from the probability, applied, and pure groups. We will give an accessible introduction to the question in the plane : move a pile of sand into a hole with the same volume by minimizing the transportation cost. First we introduce the measure-theoretic formulation of Leonid Kantorovich. This allows for duality of the variational problem and the Kantorovich potential and, by the Direct Method of the Calculus of Variations, results in existence of a weak solution in the space of measures. Secondly we proceed with Yann Brenier's representation of the minimizer for quadratic cost. This is based on the Legendre transform used in the passage from Lagrangian mechanics to Hamiltonian mechanics. Thirdly, time allowing, we describe Alessio Figalli's C(1,\alpha) regularity of the Brenier potential via the fully nonlinear elliptic Monge-Ampere equation.

We consider the geometry of hyperbolic 4-manifolds which have an end modelled on a parabolic screw motion whose rotation angle is an irrational multiple of pi. There is a very close relationship between the shape of the associated cusp region (where the manifold looks like a product) and the Diophantine properties of the rotation angle. This work is based on earlier results of Erlandsson and Zakeri and of Susskind.

We discuss an aglorithm to produce Harnack inequalities for various parabolic equations. As an application, we obtain a Harnack inequality for the curve shortening flow and one for the parabolic Allen Cahn equation on a closed n-dimensional manifold.

In this talk we describe work in collaboration with WK exploring Weingarten relations for surfaces in euclidean 3-space using parabolic methods. Even in the case of a linear Weingarten relation such flows exhibit a variety of behaviours. We will discuss closed solutions as well as qualitative aspects of the flow.

Ricci curvature plays a very important role in the study of Riemannian manifolds. In the discrete setting of graphs, there is very active recent research on various types of Ricci curvature notions and their applications. We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most 3 linear parts, with at most 2 linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartesian product of two regular graphs is completely determined by the idleness functions of the factors.

We prove injectivity and a support theorem for the X-ray transform on 2-step nilpotent Lie groups with many totally geodesic 2-dimensional flats. The result follows from a general reduction principle for manifolds with uniformly escaping geodesics.

A recent class of composite materials, known as Metamaterials, have gained much attention and interest in the Mathematics and Physics community over the last decade or so. These composites can roughly be characterised as exhibiting much more pronounced physical properties than their constituent components. These responses are due to scale-interaction effects.Mathematically, such metamaterial type effects could be rigorously justified and explained due to 'partial degeneracies' in underlying multi-scale continuum models.

In this talk, we shall introduce a notion of a partial degeneracy in parameter-dependent variational systems, motivated by examples from classical and semi-classical homogenisation theory, and present an approach to study the leading-order asymptotics of such systems. The determined asymptotics of the variational system can serve as effective models for phenomena due to multi-scale interactions and are given with order-sharp error estimates in the uniform operator topology.

This is joint work with Dr Ilia Kamotski(UCL) and Prof. Valery Smyshlyaev(UCL).

The isoperimetric inequalities are mathematical responses to centuries-old problems. For example, the Ancients already knew that to construct a surface-maximal city while having a limited amount of resources for the ramparts, they had to give a circular shape to the fortifications. Mathematically well-formulated, this is the essence of all isoperimetric inequalities, see https://en.wikipedia.org/wiki/Isoperimetric_inequality

In this talk, we will explore the famous Levy-Gromov isoperimetric inequality, which is a generalisation of this problem to a compact Riemannian manifold with some curvature assumptions. We will follow Gromov's proof dating back to 1986 and give some physical and mathematical applications.

This talk will be accessible to a wide audience and to students.

The classical Severi inequality for complex surfaces dates back to a paper of Severi himself in 1932, in which a gap was found afterwards. In 2005, Pardini gave a complete proof of this inequality based on a clever covering trick and the slope inequality of Xiao. In 2009, Mendes Lopes and Pardini proposed a question about generalizing this inequality to arbitrary dimension. In this talk, I will first introduce the classical Severi inequality and explain the above two ingredients in Pardini's proof. Then I will introduce the generalized Severi inequality which answers the aforementioned open question.

This talk may be viewed as a continuation of the previous one I gave in the same seminar two years ago.

The programm is available here : http://maths.dur.ac.uk/~dma0wk/YDGD2017.html

We consider the space $\mathcal{ M}(n,m)$ of ordered $m$-tuples of distinct points in the boundary of quaternionic hyperbolic $n$-space, ${\bf H}_{\bh}^n$, up to its holomorphic isometry group $PSp(n,1)$. We obtain the moduli space for $\mathcal{ M}(n,m)$.

The isoperimetric inequality for Steiner symmetrization of any codimension is investigated and the equality cases are characterized. Moreover, a quantitative version of this inequality is proven for convex sets. The importance of the Steiner symmetrization relies upon the fact that it acts monotonically on many geometric and analytic quantities associated with subsets of R^n, e.g. the perimeter. A characterization of the sets whose perimeter is preserved under the Steiner symmetrization of codimension 1 was given by Chlebík, Cianchi and Fusco, see "The perimeter inequality under Steiner symmetrization: cases of equality", Ann. of Math. 162, 525'“555 (2005).

In a this talk I will present some results in the general case of codimension k, with 1 \leq k \leq n-1, which have been obtained in collaboration with Marco Barchiesi and Nicola Fusco. We introduce a different approach, based on the regularity properties of the barycenter of the vertical sections of a set. The advantage of this approach is twofold. Firstly, we recover and extend the result proved by Chlebík, Cianchi and Fusco for k = 1 to any codimension, with a new and simpler proof. Secondly, we are able to obtain a quantitative isoperimetric estimate for convex sets which, to the best of our knowledge, is the first result of this kind in the framework of Steiner symmetrization.

An ovaloid is a closed, unparametrized surface of positive curvature in Euclidean 3 - space. The collection \S of *all* C^{2,a}-regular ovaloids is equipped with a natural submanifold topology. Then \S admits Euclidean motions, and, less trivially, parallelism (resulting from pairs of ovaloids of constant ambient distance), where both act continuously on \S. In this talk we consider the quotient space \L := \S modulo parallelism, which inherits the quotient topology from \S. We then report a result, obtained jointly with B. Guilfoyle, that details a regularity property of the topological space \L. This is proved using the extrinsic geometry of ovaloids, namely properties of the principal curvature foliation that are invariant under parallelism, and thereby descend to \L. Our talk will be self-contained, and in particular we will develop the required elements of classical differential geometry in an elementary and conceptual way.

Ricci solitons and quasi-Einstein metrics are two natural and related generalisations of the Einstein condition. I will report on some work with Thomas Murphy and Wafaa Batat where we investigate the geometry of these metrics on some special 4-dimensional manifolds (the toric surfaces of the title). I'll also detail some numerical work with Thomas Murphy giving explicit approximations to such metrics.

Homological invariants in low-dimensional topology (like Heegaard-Floer homology or Khovanov homology) often admit several filtrations giving rise to numerical invariants that say something directly about topology. If you take a couple of these filtrations and blend them artfully, you can sometimes get much more information than you expected. The first example of this is the so-called "upsilon" invariant in Heegaard-Floer homology. Lukas Lewark and I came up a while ago with an analogous invariant in quantum knot cohomologies, but it's not yet written up. We decided to call it "gimel" but I can't remember why. Anyways, I'll explain some of this story.

We will state the Theorem of Sard-Smale on regular values of Fredholm operators. Then we will apply this to the Cauchy-Riemann equation on the disc with boundary condition. The boundary condition takes the form of a two real dimensional surface in two complex dimensional space, which the graph of the holomorphic function inhabits. We finally show how Sard-Smale implies that the boundary problem under consideration has a solution for a dense open set of real surfaces in the complex surface.

I will give a short introduction into how and why chemists get to the crystals, how the structures are solved and analysed. It will give some insight into practical use of maths in material sciences.

In this talk I will introduce basic concepts in connection with the magnetic Laplacian on a manifold and will then discuss various eigenvalue estimates for this operator. These estimates are analogues of well known results for the classical Laplacian on functions: Cheeger and higher order Cheeger inequalities, Lichnerowicz type inequalities, as well as higher order Buser inequalities on manifolds with lower Ricci curvature bounds. This material is based on joint work with Michela Egidi, Carsten Lange, Shiping Liu, Florentin Muench, and Olaf Post.

An SL_2-tiling is an infinite grid of positive integers such that each adjacent 2x2-submatrix has determinant 1. These tilings were introduced by Assem, Reutenauer, and Smith for combinatorial purposes.

We will show that each SL_2-tiling can be obtained by a procedure called Conway--Coxeter counting from certain infinite triangulations of the circle with four accumulation points. We will see how properties of the tilings are reflected in the triangulations. For instance, the entry 1 of a tiling always gives an arc of the corresponding triangulation, and 1 can occur infinitely often in a tiling. On the other hand, if a tiling has no entry equal to 1, then the minimal entry of the tiling is unique, and the minimal entry can be seen as a more complex pattern in the triangulation.

The infinite triangulations also give rise to cluster tilting subcategories in a certain cluster category with infinite clusters related to the continuous cluster categories of Igusa and Todorov. The SL_2-tilings can be viewed as the corresponding cluster characters.

A quiver is a weighted oriented graph, a mutation of a quiver is a simple combinatorial transformation arising in the theory of cluster algebras. In this talk we connect mutations of quivers to reflection groups acting on linear spaces and to groups generated by point symmetries in the hyperbolic plane. We show that any mutation class of rank 3 quivers admits a geometric presentation via such a group and that the properties of this presentation are controlled by the Markov constant p^2+q^2+r^2-pqr, where p,q,r are the weights of the arrows in the quiver. This is a joint work with Pavel Tumarkin.

When we want to understand a geometric picture, finding the zero set of a polynomial hiding in it can be very helpful: it can reveal structure and allow computations. This technique is known as the polynomial method, and was first used to count point-line incidences in 2008 by Dvir, for the solution of the Kakeya problem in finite fields. Since then, the polynomial method has revolutionised discrete incidence geometry, largely thanks to the fact that interaction of lines with varieties is, to an extent, well-understood. Recently, Guth discovered agreeable interaction between varieties and tubes as well, opening up the exciting possibility that many problems of point-tube incidence flavour could also have algebraic structure; and such problems are of interest in harmonic analysis. In this talk, we will present the polynomial method via simple discrete analogues of the Kakeya problem, and discuss its potential to be extensively used in harmonic analysis.

I will discuss the geometry of incompressible and, more generally, \pi_1-injective surfaces in closed hyperbolic 3-manifolds. By a result of Kahn-Markovic we know that such surfaces are always present, and that there are plenty of them. We investigate the relation between the genus and the area of \pi_1-injective surfaces and geometric invariants of the ambient manifold such as its volume, Heegard genus and systole. As an application, we prove that the free-rank of the fundamental groups of the congruence covers of an arithmetic hyperbolic 3-manifold grows polynomially with volume.

The identification of projective algebras and projective closed ideals of Banach algebras, besides being of independent interest, is closely connected to continuous Hochschild cohomology. One of the main methods for computing cohomology groups is to construct projective or injective resolutions of the corresponding module and the algebra. In this talk we consider the question of the left projectivity and biprojectivity of some Banach algebras A and we give applications to the second continuous Hochschild cohomology group H^2(A,X) of A and to the strong splittability of singular extensions of A.

We study when a C^2 - smooth function K on the upper half plane occurs as the relation on the curvatures of a closed convex classical surface S. If K gives rise to a (nonlinear) elliptic relation at the umbilic points, then S is known to be a round sphere (Hopf). We prove that there exist *non-round* surfaces S in case the relation K is non-degenerate hyperbolic at the umbilics. The proof is by (nonlinear) curvature flow with speed K, which is shown to converge by establishing certain a-priori estimates.

Scalar topology in the form of Morse theory has provided computational tools that analyze and visualize data from sci-entific and engineering tasks. Contracting isocontours to single points encapsulates variations in isocontour connectivity in the Reeb graph. For multivariate data, isocontours generalize to fibers'”inverse images of points in the range, and this area is therefore known as fiber topology. However, fiber topology is less fully developed than Morse theory, and current efforts rely on manual visualiza-tions. This paper presents how to accelerate and semi-automate this task through an interface for visualizing fiber singularities of multivariate functions R3 R2. This interface exploits existing conventions of fiber topology, but also introduces a 3D view based on the extension of Reeb graphs to Reeb spaces. Using the Joint Contour Net, a quantized approximation of the Reeb space, this accelerates topological visualization and permits online perturbation to reduce or remove degeneracies in functions under study. Val-idation of the interface is performed by assessing whether the interface supports the mathematical workflow both of experts and of less experienced mathematicians.

The aim of this talk is to show that a locally compact geodesic metric space has non-positive curvature in the sense of Alexandrov (i.e. is a CAT(0)-space) if and only if it admits a quadratic isoperimetric inequality for curves with sharp Euclidean constant, that is, if every closed curve of length $l$ bounds a disc of area at most $(4\pi)^{-1} l^2$.

The proof of this result is based on (1) a solution of the classical problem of Plateau in the general setting of proper metric spaces and (2) properties of the intrinsic structure of minimal discs in metric spaces. Based on joint work with A. Lytchak.

TBA.

I will look at a connection between the topology, dynamics and ergodic theory of a wide class of laminations (aka matchbox manifolds). This is joint work with Alex Clark.

The classification of surfaces with given curvature conditions is a fundamental question in differential geometry. The Minkowski problem and its solution (Minkowski, Niremberg, Pogorelov, Cheng-Yau and others) by means of analytic methods led to the development of the theory of Monge-Ampere equations. This has inspired others to address the same question for mean and scalar curvatures, which are particular cases of k-symmetric curvatures. We give an overview of investigations on the existence of the solution of this problem initiated by the work of B. Guan and P. Guan.

The modern development of contact geometry in 3 dimensions has seen several (due to Giroux, Wendl, Latschev and Wendl, Hutchings, and others) invariants of contact structures meant in some sense to measure non-(Stein/symplectic)-fillability of the structure. We will describe ongoing work to approach this issue via a refinement of the `contact class' of Heegaard-Floer homology, inspired by the `algebraic torsion' of Latschev and Wendl, and Hutchings (this is joint with Kutluhan, Matic, and Van Horn-Morris).

It is well known that the group generated by reflections in the sides of a hyperbolic triangle is rigid, even when embedded in the isometry group of higher dimensional hyperbolic space. However it is possible to deform such a group when it is embedded in the isometry group of complex hyperbolic space. In his ICM talk, Rich Schwartz gave a series of conjectures about such groups. In particular, he conjectured that discreteness of these representations is controlled by a particular element. In this talk I will give a survey of the topic and then discuss certain cases where Schwartz's conjecture is true. This is joint work with Jieyan Wang and Baohua Xie and with Pierre Will.

Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

In this talk we will see how, in a joint work with John Parker, we extended this construction of fundamental polyhedra to all Deligne-Mostow lattices with three fold symmetry.

In many areas of mathematics and its application in other sciences one is confronted with the task of estimating or recosntruction a function based on partial data. Of course, this will not work for all functions well. Thus one needs an restriction to an adequate class of functions. This can be mathematically modeled in many ways. Spacial statistics or complex function theory are relevant areas of mathematics which come to ones mind.

We present several results on reconstrucion and estimation of functions which are solutions of elliptic partial differential equations on some subset of Euclidean space. We comment also on analogous statements for solutions of finite difference equations on graphs.

One of the biggest challenges in low-dimensional topology is to understand Dehn surgery. Relatively recently defined Heegaard Floer homology has been used successfully in answering many questions about Dehn surgery. I will describe the basics of Heegaard Floer homology, exemplify some applications and outline why Heegaard Floer homology is so suitable when dealing with surgery. I will sketch the proof of the fact that only finitely many alternating knots can give a fixed space by surgery.

Harmonic and asymptotically harmonic spaces are Riemannian manifolds where the Laplace-Beltrami operator assumes a particularly simple form in polar coordinates and horocyclic coordinates, respectively. Known noncompact examples are, besides the Euclidean spaces, all rank one symmetric spaces and Damek-Ricci spaces. Useful tools in the treatment of these spaces come from Riemannian and spectral geometry. In this talk, we will discuss recent results based on spherical and horocyclic averages which were obtained in collaboration with Evangelia Samiou (in the case of harmonic spaces) and Gerhard Knieper (in the case of asymtptotically harmonic spaces). It should be mentioned that the most challenging problem in this area is the classification of these spaces in the noncompact case which is still widely open. (The classification problem in the compact case, known as "Lichnerowicz conjecture", was settled in 1990 by Z.I. Szabo.).

I will start by explaining how deformations help to answer two important questions about the family of (colored) sl(N) link homology theories: What relations exist between them? What geometric information about links do they contain? I will recall Bar-Natan and Morrison's version of Lee's deformation of Khovanov homology and sketch how it generalizes to the case of colored sl(N) link homology. Finally, I will state a decomposition theorem for deformed colored sl(N) link homologies which leads to new spectral sequences between various type A link homologies and concordance invariants in the spirit of Rasmussen's s-invariant. Joint work with David Rose.

This is a report on joint work with Lukas Lewark (once of Durham, now of Bern). No prior knowledge assumed. I'll talk about the smooth "4-ball genus" of a knot - the history of its study, where "quantum" invariants fit into that story, and the weird and wonderful phenomena that we have recently observed.

We will present an approach for the definition and investigation of Schrödinger operators with delta-potentials on manifolds. In particular we will consider the case when the manifold is a closed curve in R^3.

Starting with the Pestov's Identity for the tangent bundle and the geodesic flow, I will derive an analogous formula for the principal bundle of orthonormal frames and for the frame flows and I will show that under integration over the principal bundle this formula gives a hint of possible dynamical applications.

Associahedra were defined independently by Tamari and Stasheff more than 50 years ago and have been rediscovered and studied in many different contexts since then. Fomin and Zelevinsky observed their relation to finite cluster algebras of type A and obtained generalized associahedra by extending their description to cluster algebras of other finite types. I will present a unified construction that yields many polytopal realizations of these objects and uses geometry and combinatorics of finite Coxeter groups.

In this talk, I will give an introduction to aperiodic tilings. Usually, one studies a topological space associated to these tilings rather than one specific tiling (this is the analogue to studying a subshift rather that one single word in symbolic dynamics). It is a natural question to ask what happens to the underlying tilings when there is a homeomorphism between tiling spaces. I will try to give some leads to answer this question. One of the consequences of such a homeomorphism is that the complexity function of the tiling changes in a very controlled way.

A lattice in a Lie group is a discrete group with finite volume quotient. An arithmetic group is a group that is discrete essentially because the integers are discrete in the real numbers. Arithmetic groups are lattices, but not all lattices are arithmetic. The only Lie groups possibly containing non-arithmetic lattices are SO(n,1) and SU(n,1) and in the latter case it is an open question for n at least 4. From work of Deligne and Mostow in 1986 there are nine examples in SU(2,1) and one in SU(3,1). This talk is the third of three where I will describe joint work with Deraux and Paupert where we construct ten new examples in SU(2,1). These are the first new examples to be found since the work of Deligne and Mostow. In the first talk I will give the background and outline the results. In the second and third I will describe the construction of five of the examples.

A lattice in a Lie group is a discrete group with finite volume quotient. An arithmetic group is a group that is discrete essentially because the integers are discrete in the real numbers. Arithmetic groups are lattices, but not all lattices are arithmetic. The only Lie groups possibly containing non-arithmetic lattices are SO(n,1) and SU(n,1) and in the latter case it is an open question for n at least 4. From work of Deligne and Mostow in 1986 there are nine examples in SU(2,1) and one in SU(3,1). This talk is the first of two where I will describe joint work with Deraux and Paupert where we construct ten new examples in SU(2,1). These are the first new examples to be found since the work of Deligne and Mostow. In the first talk I will give the background and outline the results. In the second I will describe the construction of five of the examples.

A lattice in a Lie group is a discrete group with finite volume quotient. An arithmetic group is a group that is discrete essentially because the integers are discrete in the real numbers. Arithmetic groups are lattices, but not all lattices are arithmetic. The only Lie groups possibly containing non-arithmetic lattices are SO(n,1) and SU(n,1) and in the latter case it is an open question for n at least 4. From work of Deligne and Mostow in 1986 there are nine examples in SU(2,1) and one in SU(3,1). This talk is the first of two where I will describe joint work with Deraux and Paupert where we construct ten new examples in SU(2,1). These are the first new examples to be found since the work of Deligne and Mostow. In the first talk I will give the background and outline the results. In the second I will describe the construction of five of the examples.

Since the double helical structure of DNA was discovered in 1953, decades of research have revealed many other fascinating phenomena about of the molecule of life. In 1965, Jerome Vinograd discovered that DNA in the polyoma virus is naturally found in a circular form. This work opened the gates to a new interdisciplinary field that studies the topology of DNA and its biological implications for the functionality of the molecule.

In this talk I will discuss topological questions (and answers) that arise when considering the naturally occurring DNA of three different unicellular organisms: bacteria, ciliates and trypanosomes.

This is a joint work with Bobo Hua and Chao Xia. After introducing the definition of generalized harmonic maps from weighted Riemannian manifolds into Hadamard spaces in the sense of Korvaar-Schoen, I will explain a Kendall type theorem (reducing validity of Liouville theorem of harmonic maps to that of harmonic functions) based on a lemma of Jost and method of Li-Wang. Finally, I will explain that this is a key ingredient for the scheme of proving Liouville type theorems for harmonic maps with finite energy without using Bochner techniques (which is typically not available in a general setting).

In the spring some of you may have seen me give some elementary 'colloquium-style' talks on aperiodic patterns and tilings. Here I would like to give some more detailed treatment of how such geometric patterns can be analysed using techniques from algebraic topology.

One of the outstanding challenges in 3-manifold theory is to relate the modern Heegaard Floer invariants to the fundamental group. Recently, a conjectural picture has emerged from the work of Boyer-Gordon-Watson: a closed, irreducible rational homology sphere M is an L-space (i.e. it has the simplest possible Heegaard Floer homology) if and only if its fundamental group is not left-orderable. Whereas there has been encouraging evidence supporting the truth of the conjecture, the problem remains poorly understood for key classes of 3-manifolds.

In this talk, we focus on negative-definite graph manifolds (one of these poorly understood classes): for these, Nemethi constructs a lattice cohomology, an invariant inspired in ideas from singularity theory and conjecturally isomorphic to Heegaard Floer homology. Using the combinatorial tractability of lattice cohomology, we produce several comprehensive families of manifolds against which to test the Boyer-Gordon-Watson conjecture. Then, using either horizontal foliation arguments or direct manipulation of the fundamental group, we prove that they do indeed satisfy the conjecture.

I shall talk at a basic level about a gauge-theoretic invariant of knots and 3-manifolds, and progress towards the first non-trivial calculation.

Hurwitz numbers enumerate ramified coverings of the 2-sphere of a fixed topological type. Amazingly, the same numbers occur in many different contexts, in particular, they are related to geometry of the moduli spaces of curves via so-called ELSV formula. I am going to make a short overview of different interpretations of Hurwitz numbers and methods of computation. I'll explain their relation to the moduli spaces of curves, and another relation to the random matrix models (in fact, I'll use a mathematical replacement for the random matrix theory given by topological recursion for some multi-differentials on Riemann surfaces. The latter one I'll explain from the very beginning).

We will speak about generalizations of classical Kleinian subgroups of PSL(2,C) to the case of PSL(n+1,C) and discuss their geometry and dynamics.

I will discuss the geometry of heterotic string compactifications with fluxes. The compactifications on 6 dimensional manifolds which preserve N=1 supersymmetry in 4 dimensions must be complex conformally balanced manifolds which admit a no-where vanishing holomorphic (3,0)-form, together with a holomorphic vector bundle on the manifold which must admit a Hermitian Yang-Mills connection. The flux, which can be viewed as a torsion, is the obstruction to the manifold being Kahler. I will describe how these compactifications are connected to the more traditional compactifications on Calabi-Yau manifolds through geometric transitions like flops and conifold transitions. For instance, one can construct solutions by flopping rational curves in a Calabi-Yau manifold in such a way that the resulting manifold is no longer Kahler. Time permitting, I will discuss open problems, for example the understanding of the the moduli space of heterotic compactifications and the related problem of determining the massless spectrum in the effective 4 dimensional supersymmetric field theory. The study of these compactifications is interesting on its own right both in string theory, in order to understand more generally the degrees of freedom of these theories, and also in mathematics. For example, the connectedness between the solutions is related to problems in mathematics, for instance Reid's fantasy, that complex manifolds with trivial canonical bundle are all connected through geometric transitions.

We first describe the canonical neutral Kahler structure of the space of oriented geodesics in hyperbolic 3-space and then describe its area-stationary surfaces. Furthermore, we investigate the Hamiltonian stability of the minimal Lagrangian surfaces.

A Mobius group is a finitely generated discrete group of orientation-preserving isometries acting on hyperbolic n-space. The deformation space of the Mobius group is the set of all discrete, faithful and type-preserving representation into the full group of orientation-preserving isometries factored by the conjugation action.

Mostow-Prasad rigidity states that for n>2 the deformation space of a torsion-free cofinite volume Mobius group acting on hyperbolic n-space is trivial. Thus there is no deformation theory for such a Mobius group. For a gemetrically finite Mobius group, we have Marden quasiconformal stability in H^2 and H^3. That is, for a geometrically finite Kleinian group all deformations sufficiently near the identity deformation are quasiconformally conjugate to the identity.

We prove that this quasiconformal stability cannot be generalized in 4-dimensional hyperbolic space. This is due to the presence of so called screw parabolic isometries in dimension 4. In particular, a thrice-punctured sphere group has a large deformation space of quasiconformally distinct representations.

Recently M. Belolipetsky and myself proved that for any \epsilon > 0 and n >= 2 there exists a hyperbolic n-manifold with a closed geodesic of length less than \epsilon. The proof is by construction and generalises one for the n=4 case by I. Agol. We also showed that these manifolds are non-arithmetic if \epsilon is small enough, thus providing another example to complement Gromov and Piatetski-Shapiro's construction of non-arithmetic lattices in PO(n,1). The volume of the constructed manifolds is seen to grow at least as \epsilon^-(n-2) when \epsilon --> 0. I will make some remarks on the so-called non-coherence of the manifolds' fundamental groups.

he groups F, T and V were defined by Richard Thompson in 1965. Since then there has been a lot of interest in these groups. We will present an informal introduction into Thompson's Group F and Belk and Matucci's more geometric solution to the conjugacy problem.

Let Jordan arc $\gamma$ be the invariant set for a digraph system S of contraction similarities. Then, we show that either the arc $\gamma$ is an invariant set for some multizipper, and admits non-trivial deformations, or $\gamma$ is a straight line segment, the system S does not satisfy weak separation property and the self-similar structure $(\gamma,S)$ is rigid. All required definitions and ideas of the proofs will be explained on the talk.

I will discuss a number of concrete problems which come from my previous work on geometry and arithmetic of lattices in semisimple Lie groups.

Coloured posets arise in a number of areas of mathematics. The ones that will chiefly concern us in this talk are: Khovanov's categorification of the Jones polynomial, the geometry of certain varieties associated to Weyl groups, and the Hochschild homology of associative algebras.

The bulk of the talk will run through one of the motivating examples in some detail (the Khovanov homology of a knot). We will then give a couple of the fundamental theorems of coloured poset homology and some applications if time permits.

We consider orthogonal polynomials on the unit circle with random coefficients and study the statistical distribution of their zeros. For slowly decreasing random coefficients, we show that the zeros are distributed according to a Poisson process. For rapidly decreasing coefficients, the zeros have rigid spacing (clock distribution). For a certain critical rate of decay we obtain the circular beta distribution.

In the 1920s Salomon Lefschetz and Jakob Nielsen presented groundbreaking work on fixed points of continuous maps.This inspired much topological research in the subsequent decades.We will review some of the classical results and then turn to very recent developments concerning fixed points and, more generally, coincidences. Given two maps between manifolds,we study the geometry of their coincidence locus (using nonstabilized normal bordism theory and pathspaces). We extract an invariant which must necessarily be trivial if the two maps can be deformed away from one another. Often this is also sufficient. Surprisingly however, in certain cases the full answer involves also the Kervaire invariant (which was originally introduced and used in an entirely different area of topology, namely: manifolds without smooth structures and exotic spheres). Similarly other central notions of topology turn out to play a crucial role here, e.g. various versions of Hopf invariants (a la James, Hilton, Ganea...).

When a vector field in the plane has a complicated singularity at a point, its behavior near that point can be studied by 'blowing-up' the singular point, that is by making a singular co-ordinate change that maps the point to a curve, along which the transformed vector field may have a number of simpler singularities. In this talk I will describe the technique, and illustrate its use in the study of Binary Differential Equations, that is implicit differential equations that define at most two directions in the plane. BDEs appear frequently in differential geometry, for example, the principal, asymptotic and characteristic directions on a surface in 3-space are all solutions of BDEs.

A manifold is called aspherical if its universal covering space is contractible. This is the case, for example, if the universal covering is homeomorphic to an Euclidean space. Given an abstract group Gamma, there is the basic question if it is possible to construct compact aspherical smooth manifolds with fundamental group Gamma, and also to understand the geometric properties of such manifolds. Ideally, one would like to classify them up to homeomorphism or up to diffeomorphism. For example, 'most' polycyclic groups Gamma appear as fundamental groups of so called solvmanifolds. Another type of examples which appear in geometry are the fundamental groups of locally symmetric spaces. We would like to discuss a method which allows to build 'mixed' examples from these basic building blocks. This construction corresponds to the notion of group extension on the level of the fundamental group, and it has many interesting geometric properties.

Note: This lecture is sponsored by the LONDON MATHEMATICAL SOCIETY via a Scheme 2 Grant

"A triangle group is the group generated by reflections in the sides of a triangle in Euclidean, spherical or hyperbolic geometry. It is a lattice when the triangle has certain special internal angles. This idea can be generalised to groups generated by three complex reflections in complex hyperbolic space. We know of rather few complex hyperbolic lattices; most of them are (related to) triangle groups with certain special angles. By studying triangle groups we can find more groups that are candidates for being lattices. This talk will be a survey of how this may be done and will cover recent progress."

"A geodesic in a Riemannian homogeneous manifold $(M=G/K, g)$ is called a homogeneous geodesic if it is an orbit of an one-parameter subgroup of the Lie group $G$. We investigate $G$-invariant metrics with homogeneous geodesics (i.e. such that all geodesics are homogeneous) when $M=G/K$ is a flag manifold, that is an adjoint orbit of a compact semisimple Lie group $G$. We use an important invariant of a flag manifold $M=G/K$, its $T$-root system, to give a simple necessary condition that $M$ admits a non-standard $G$-invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds $M=G/K$ of a simple Lie group $G$, only the manifold $\operatorname{Com}(\Bbb R^{2\ell +2})= SO(2\ell +1)/U(\ell)$ of complex structures in $\Bbb R^{2\ell + 2}$, and the complex projective space $\Bbb C P^{2\ell -1}= Sp(\ell)/U(1)\cdot Sp(\ell -1)$ admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only $G$-invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e. the metric associated to the negative of the Killing form of the Lie algebra $\gg$ of $G$). According to F. Podest\'a and G.Thorbergsson, these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer."

"We study the family of Hamiltonians which corresponds to the adjacency operators on a percolation graph. We characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions. This set of energies is a dense subset of the algebraic integers. The integrated density of states has discontinuities precisely at this set of energies. We show that the convergence of the integrated densities of states of finite box Hamiltonians to the one on the whole space holds even at the points of discontinuity. For this we use an equicontinuity-from-the-right argument. The same statements hold for the restriction of the Hamiltonian to the infinite cluster. In this case we prove that the integrated density of states can be constructed using local data only. Finally we study some mixed Anderson-Quantum percolation models and establish results in the spirit of Wegner, and Delyon and Souillard. (See http://arxiv.org/math-ph/0405006) "

"We will discuss two examples of situations where interesting representations of the symmetric group arise in topology. The two examples share the feature that an evident action of the symmetric group $\Sigma_n$ extends to a more hidden action of $\Sigma_{n+1}$. In the first example, the symmetric group acts on the cohomology of a certain space of trees and the representation afforded is closely related to the free Lie algebra and the representation $Lie_n$. The second example gives a $\Sigma_{n+1}$ action on an $n$-fold tensor product of a suitable Hopf algebra $H$ and this has a connection with solutions of the Yang-Baxter equation. "

"This work is concerned with generalising the ideas used by Dupont and Sah in their algebraic proof of Sydler's theorem, by interpreting them in terms of the Hochschild homology of Clifford algebras. Thus, we are able to compute the Eilenberg-Mac Lane homology in degree 1 and 2 of the special linear group of rank two over the real and complex numbers, with coefficients in the adjoint representation. In the complex case, the groups were previously known thanks to the works of Cathelineau and Elbaz-Vincent. In the real case, they were known only up to a quotient. One of the original motivations for computing that kind of homology group is the problem of scissors congruences, also known as Hilbert's third problem. This consists in detecting the equivalence classes of polytopes for the following relation: given two polytopes in one of the three classical geometric spaces, we say that they are scissors congruent if it is possible to cut the first one into finitely many pieces (i.e. subpolytopes), and, by moving these around, reassemble them into the second one. The theorem of Sydler states that, for Euclidean three-space, volume and the so-called Dehn invariant suffice to detect scissors congruence classes. By the work of Dupont, this is known to be equivalent to the purely algebraic statement that the second homology group of the special orthogonal group of rank 3 with coefficients in its natural geometric representation is trivial. One of the greatest achievements in the area is the direct algebraic proof of that last fact by Dupont and Sah, using the Hochschild homology of quaternions. By reformulating it in terms of Clifford algebras, one can actually extend their constructions to other algebras than quaternions, and thus in particular carry out the computations mentioned above. Finally, let us conclude by observing that the general construction we use makes sense in higher dimensions as well, although the computations seem to quickly become much more complicated."

"We present a theorem of Bogomolov which gives restrictions on the Chern classes of a semistable holomorphic vector bundle over a complex algebraic surface, and sketch some applications."

"The problem of scissors congruences consists in detecting equivalence classes for the following relation: fixing a geometric space (spherical, Euclidean or hyperbolic space), we will say that two polytopes are scissors congruent if one can cut the first into finitely many pieces (i.e subpolytopes), and, by moving these around, reassemble them into the second one. The problem is trivial for polygons, and area is an invariant separating the classes. It turns out, however, that in higher dimensions volume no longer suffices to decide whether two polytopes are scissors congruent. Another family of scissors congruence invariants of a more arithmetical nature, the so-called Dehn invariants, appear, which are distinct from volume. The obvious question is whether the Dehn invariants, together with volume, form a complete set of invariants for the problem. The only complete result to this day is Sydler's theorem, which states that volume and the classical Dehn invariant suffice to detect scissors congruences classes for Euclidean three-space (and four-space, as an easy corollary). The original argument of Sydler was extremely complicated, however. J. Dupont linked the problem to the homology of the orthogonal group (considered as discrete group) at the end of the '70s, and in 1990 gave in a joint work with C.-H. Sah an algebraic proof of Sydler's theorem. I will explain, in this talk, how the algebraic computation of Dupont and Sah fits in a geometrical setting, by considering Euclidean geometry as an infinitesimal version of either spherical or hyperbolic geometry. This allows to link Sydler's theorem to computations of certain homology groups by Cathelineau and others."

"The moduli space M of semistable principal Sp(2)-bundles on a complex algebraic curve X of genus g can be identified, via the standard representation of Sp(2) on \C^4, with the moduli space of vector bundles of rank 4 on X which carry a bilinear antisymmetric form. Using this identification, we describe the singular locus and semistable boundary of M."

" Given an abelian extension of global fields, one may combine the Artin L-functions for each Galois character into an equivariant L-function. The general aim of the Stark-type conjectures is to relate the special values of this, particularly at 0, to the units and class numbers of the fields, in the spirit of the analytic class number formula. Stark's abelian conjecture deals only with the first derivative, and was extended by Rubin to higher derivatives, giving the conjectural integrality properties of a generalised "Stark unit" mapping to the special value under the logarithmic regulator map. On the other hand, Gross made a conjecture about the zeroth derivative, relating it to the class number of the base field and a group-ring valued "regulator". Gross's conjecture is stronger than Rubin's for this derivative but becomes trivial for higher derivatives. I will discuss a common refinement of these conjectures formulated by David Burns, which refines Rubin's in the spirit of Gross's. Some special cases and behavioural properties will be presented, as well as a discussion of its relation to some of the numerous other conjectures in the field. In particular, a somwehat different-looking conjecture of Henri Darmon about circular units can be shown to give a "base-chage"-type property."