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.

Venue: zoom

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.

Venue: MCS3070

The talk discusses generalisations of Liouville's theorem to nonlocal translation-invariant operators. It is based on a joint work with D. Berger and R.L. Schilling, and a further joint work with the same co-authors and T. Sharia. We consider operators with continuous but not necessarily infinitely smooth symbols.

It follows from our results that if $\left\{\eta \in \mathbb{R}^n \mid m(\eta) = 0\right\} \subseteq \{0\}$, then, under suitable conditions, every polynomially bounded weak solution $f$ of the equation $m(D)f=0$ is in fact a polynomial, while sub-exponentially growing solutions admit analytic continuation to entire functions on $\mathbb{C}^n$.

Venue: MCS3070

Mathematical models of non-Newtonian fluids play an important role in science and engineering, and their analysis has been an active field of research over the past decade. This talk is concerned with the mathematical analysis of numerical methods for the approximate solution of systems of nonlinear elliptic partial differential equations that arise in models of chemically reacting viscous incompressible non-Newtonian fluids, such as the synovial fluid found in the cavities of synovial joints. The synovial fluid consists of an ultra filtrate of blood plasma that contains hyaluronic acid, whose concentration influences the shear-thinning property and helps to maintain a high viscosity; its function is to reduce friction during movement. The shear-stress appearing in the model involves a power-law type nonlinearity, where, instead of being a fixed constant, the power law-exponent is a function of a spatially varying nonnegative concentration function, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove the convergence of the sequence of numerical approximations to a solution of this coupled system of nonlinear partial differential equations, a uniform Hölder norm bound needs to be derived for the sequence of numerical approximations to the concentration in a setting, where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely an L^∞ function. This necessitates the development of a discrete counterpart of the De Giorgi–Nash–Moser theory. Motivated by an early paper by Aguilera and Caffarelli (1986) in the simpler setting of Laplace’s equation, we derive such uniform Hölder norm bounds on the sequence of continuous piecewise linear finite element approximations to the concentration. We then use these to deduce the convergence of the sequence of approximations to a weak solution of the coupled system of nonlinear partial differential equations under consideration.

Venue: MCS0001

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.

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Wilson lines are a prototypical example of defect in quantum field theory. After reviewing the superconformal case - in which the one-dimensional defect CFT that they define is particularly interesting - I will discuss some analytic tools that may prove useful in this context, but are developed for generic 1d CFTs. Among them, a representation of the four-point correlator as a Mellin amplitude and via a recently derived dispersion relation.

Venue: MCS2068

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Venue: MCS0001