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Online Talks

Durham Cathedral

London Mathematical Society Durham Symposium
New Directions in the Model Theory of Fields
Monday 20th July - Thursday 30th July 2009
List of abstracts
Tibor Beke (University of Massachusetts)Tuesday 28th July 09:00
Finitely additive measures on o-minimal sets
We discuss the space of finitely additive measures on o-minimal sets and its relationship to classical work in convex geometry.

Alessandro Berarducci (Università di Pisa)Saturday 25th July 09:00
Equivariant homotopy of definable groups
(Joint work with M. Mamino)
In the light of the work of several authors on Pillay's conjectures, we consider natural transformations between homotopy functors that arise naturally in the study of the topological properties of definable groups in o-minimal structures.

Daniel Bertrand (Institut de Mathematiques de Jussieu)Tuesday 21st July 11:45
C1: Galois descent in Kummer theory and diophantine geometry.
By Galois descent, we mean: if something comes from there and becomes trivial here, then, it was already trivial there.

In this talk, we will illustrate this principle by surveying results, mainly due to Bashmakov and Ribet, on the Galois groups attached to division points on abelian varieties. As a possible application, we will mention the recent work of Zannier on Hilbert irreducibility over elliptic curves.

Daniel Bertrand (Institut de Mathematiques de Jussieu)Wednesday 22nd July 11:45
C2: Galois descent in differential Galois theory and Schanuel problems
By Galois descent, we mean : if something comes from there and becomes trivial here, then, it was already trivial there.

In this talk, we will illustrate the principle by surveying the results of Andre, respectively Pillay and the speaker, on the differential Galois groups attached to logarithms, resp. exponentials, on Abelian schemes over a curve. As a possible application, we will mention the recent work of Masser and Zannier on torsion anomalous points.

Raf Cluckers (K.U. Leuven)Monday 27th July 10:00
Approximation results in p-adic geometry
Partly in joint work with G. Comte and F. Loeser, and partly with I. Halupczok several new results in p-adic geometry, like stratifications with certain properties, or like preparation theorems of functions simultaneously with their derivatives, have been obtained. In general, approximations of p-adic functions and sets have been important in several theories of motivic integration, which I will explain in the first part of the talk as an introduction, before turning to some of the new results.

Philipp Habegger (ETH Zurich)Monday 27th July 11:45
Intersecting varieties with algebraic subgroups
In this talk we work with a fixed algebraic subvariety of a semiabelian variety defined over a field of characteristic zero. Zilber and Pink and Bombieri, Masser, Zannier stated conjectures on the set of points on the subvariety which are contained in the union of all "sufficiently small" algebraic subgroups of the ambient group variety. For example, if the fixed subvariety is a curve, then "sufficiently small" means of codimension at least two. In this case one expects the set described above to be finite unless there is a major geometrical obstruction.

This talk revolves around evidence in favor of such conjectures when working over the field of algebraic numbers. In the case of curves, finiteness is known by a recent result of Maurin if the group variety is an algebraic torus. We will discuss a new proof, based on joint work with Bombieri, Masser, and Zannier, which is effective. So at least in theory, it allows one to find all points in the finite set. The proof involves methods from diophantine geometry and in particular a height function. We shall replace the curve by a seemingly more complicated surface. But for this surface we can prove a result related to the conjectures mentioned above which in itself generalizes to a subvariety of any dimension.

Ivo Herzog (Yonsei University/Ohio State University)Thursday 23rd July 09:00
A3: Homology of an Abelian Category
We will describe the formal calculus of positive-primitive formulae in the language of left modules over a ring R. This leads to the definition of a complex whose homology in dimensions 0 and 1 will be related to the K-theory of Ab(R), the free abelian category over R. In fact, one obtains in dimension 0 an isomorphism between the homology and the Grothendieck group. In order to show that this isomorphism is natural, one needs to set up a rather general framework. This involves generalizing the above notions to the context of an abelian category with a distinguished object.

Martin Hils (CNRS-Université Paris 7)Wednesday 29th July 10:00
Generic Automorphisms in Green and Bad Fields
We show that the generic automorphism is axiomatisable in Poizat's green fields, as well as in the bad fields obtained by collapsing Poizat's green fields to finite rank. As a corollary we obtain "bad pseudo-finite fields" in characteristic 0. Recall that a bad field is an algebraically closed field with an infinite proper subgroup of its multiplicative group (the green points) such that the field structure, augmented by a predicate for the green points, is of finite Morley rank. Poizat's green fields are infinite rank analogues of bad fields in characteristic 0. In order to deal with multiplicity issues occurring in the proof of these results, we use a constructibility result for varieties in characteristic 0: An irreducible subvariety of the standard n-dimensional torus is called simple if its generic point is simple in the sense of Zilber. We show that simplicity of a variety is a definable property (in uniformly definable families, the set of parameters giving rise to simple varieties is constructible).

Ehud Hrushovski (Hebrew University, )Tuesday 28th July 17:00
Some examples of (de)categorification in model theory
In a number of instances, categorification and quantification form alternative solutions to the same problem.

(1) the Denef-Loeser isomorphism between rational motives for finite group actions, and definable sets over pseudo-finite fields.
(2) a conjectural extension to a connection between motives, and definable sets over ACFA.
(3) definable groupoids, covers, and elimination of imaginaries.
(4) different approaches to motivic integration.

I will describe some of these, in a Grothendieck group setting.

Martin Hyland (University of Cambridge)Wednesday 22nd July 17:00
Categorical Model Theory
This is a catch-all title. I am liasing with colleagues with the aim of there being a useful coverage to relevant topics. I hope to make the talk very accessible.

David Kazhdan (Hebrew University)Thursday 23rd July 11:45
B3: Motivic integration
I will define the basic concepts of the theory of motivic integration and explain some connetion with the algebraic geometry and the theory of representations.

David Kazhdan (Hebrew University)Friday 24th July 09:00
B4: Motivic integration
I will define the basic concepts of the theory of motivic integration and explain some connetion with the algebraic geometry and the theory of representations.

Minhyong Kim (University College London)Wednesday 22nd July 16:00
Diophantine geometry and Galois theory

Jochen Koenigsmann (University of Oxford)Wednesday 29th July 16:00
Defining Z in Q
We will present a universal definition of Z in Q and show, via Model Theory, that, under a very mild arithmo-geometric conjecture, Z is not existentially definable in Q.

Piotr Kowalski (Uniwersytet Wrocŀawski)Monday 27th July 17:00
Algebraic independence in positive characteristic
We discuss versions of Schanuel Conjecture over a non-Archimedean field of positive characteristic. The role of the exponential map is played by an additive power series, for example the exponential map of a Drinfeld module.

Tom Leinster (University of Glasgow)Tuesday 28th July 10:00
For many types of mathematical object there is a canonical notion of "size". Sets have cardinality, topological spaces have Euler characteristic, vector spaces have dimension, probability spaces have entropy, and so on. I will draw together some common threads, and explain the central role played by the notion of the cardinality (or Euler characteristic) of a category.

Leonard Lipshitz (Purdue University)Wednesday 29th July 09:00
Fields with analytic structure.
(Joint work with Raf Cluckers)
We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o-minimality holds. For Henselian valued fields, both the model theory and the analytic theory are developed. b-minimality is established, as well as other properties useful for motivic integration on valued fields. We give a list of examples that comprises, to our knowledge, all principal previously studied, analytic structures on Henselian valued fields, as well as some new ones.

Francois Loeser (Ecole Normale Superieure)Monday 27th July 09:00
Tameness in non-archimedean geometry
It is by now well known that o-minimal structures are an instance of tame geometry in the sense of Grothendieck. In this talk I will present some recent work in collaboration with Ehud Hrushovski on tameness in non-archimedean geometry. The basic object of study are spaces of stably dominated types which are shown to be pro-definable. These are closely related to Berkovich spaces and can in turn be used to obtain new results on Berkovich spaces.

Angus Macintyre (Queen Mary, University of London)Wednesday 29th July 17:00
Revisiting the model theory of the adeles
(Joint with J. Derakhshan)
We consider the structure of definable sets in the rings of adeles over global fields K, with emphasis on uniformity. Natural quantifier eliminations are given, with applications to decidability and to computations of measures of definable sets.

Michael Makkai (McGill University)Saturday 25th July 11:45
Algebraic model theory for simplicial homotopy
First Order Logic with Dependent Sorts (FOLDS) was developed in around 1995, for the purposes of the foundations of mathematics. The main feature of FOLDS is a type-dependent concept of identity, FOLDS identity, replacing the Fregean unique identity meant by Frege to be valid for all entities regardless of their types. The statements formulated in the FOLDS language are invariant under FOLDS identity, and there is a strong converse to this fact. (See: M. Makkai: Towards a Categorical Foundation of Mathematics, in: Logic Colloquium '95, Lecture Notes in Logic no. 11, Springer 1998; and "First Order Logic with Dependent Sorts, with Applications to Category Theory" at www.math.mcgill.ca/makkai/.) The first half of the talk will give a general introduction to the model theory of FOLDS. The second half will specialize FOLDS to simplicial sets. The usual concept of homotopy equivalence, for Kan complexes and also weak Kan complexes (quasi categories), is an instance of the general concept of FOLDS identity. Other aspects of the Quillen model category structure of simplicial sets will also be related to FOLDS.

Amador Martin-Pizarro (CNRS Universite Claude Bernard Lyon 1)Wednesday 29th July 11:45
Relative Geometric complexity and definable groups
(Joint work with T. Blossier and F.O. Wagner)
Characterizations of the geometry of certain structures, such as 1-basedness, CM-triviality and flatness, played a crucial role in understanding partially the nature of definable groups and fields. We will introduce relative notions of the above to the set up of a theory with respect to a reduct that will generalize known results of definable groups in differentially closed fields as well as present a structural theorem for definable groups obtained in Hrushovski-Fraisse's amalgams.

Ieke Moerdijk (Sheffield/Utrecht)Tuesday 28th July 11:45
Introduction to the theory of classifying toposes
The notion of "classifying topos" is formally similar to that of "classifying space" in algebraic topology, and has close connections to first order logic. For example, the universal structures over classifying toposes, similar to the universal bundles in topology, are closely related to generic models in set theory. The theory of classifying toposes is a main tool to construct toposes and deduce structure theorems. It also has applications in logic. I aim to give a gentle introduction to some aspects of this beautiful theory.

Rahim Moosa (University of Waterloo)Tuesday 21st July 16:00
Generalised Hasse varieties
(Joint work with T. Scanlon)
We introduce and begin to develop a theory that simultaneously generalises various notions of "closed subvariety" in enriched geometric settings, including differential, Hasse-differential, difference, and difference-differential geometry. This is done by studying abstract prolongations defined using Weil restrictions of scalars.

Matthew Morrow (University of Nottingham)Monday 27th July 16:00
Two-dimensional integration
I will discuss the problem of translation-invariant integration on two-dimensional local fields (= a complete discrete valuation field whose residue field is a local field), especially the case of finite residue characteristic, where model theory currently offers little help and there are relations to ramification theory.

Amnon Neeman (Australian National University)Wednesday 22nd July 09:00
A2: An introduction to well generated triangulated categories
We will begin with a review of compactly generated triangulated categories, which will be elementary and will highlight some applications. Then we will discuss the high-cardinal version, that is the notion of well generated triangulated categories.

Amnon Neeman (Australian National University)Friday 24th July 10:00
A4: A modern approach to dualizing complexes
We will review some recent work, which provides a surprising new angle on Grothendieck dualizing complexes. The talk will begin with a reminder of the classical theory, then it will proceed to describe three articles that appeared in 2005-06, due to Iyengar, Jorgensen and Krause, and it will end with very recent work by the speaker and by his student Daniel Murfet. Together, the articles which began in 2005 offer an unexpected new angle on dualizing complexes. In the course of the talk I will explain how the techniques of well generated categories provide much sharper results than the older methods of compactly generated categories.

Jonathan Pila (University of Bristol)Tuesday 28th July 16:00
Rational points of definable sets and diophantine problems

Mike Prest (University of Manchester)Tuesday 21st July 09:00
A1: Model theory in categories
I will discuss and illustrate the relationship between the model theory of sets-with-structure and that of objects which take all their structure from the category in which they lie.

Jiri Rosicky (Masaryk University)Saturday 25th July 10:00
Facets of accessibility
Accessible categories were introduced by Makkai and Pare leaning on earlier works of Grothendieck, Gabriel and Ulmer. They were meant as a categorical counterpart of infinitary first order logic but it has turned out that they are useful elsewhere, notably in homotopy theory. We are going to survey recent developments in this area.

Damian Rossler (Universite Paris-Sud)Friday 24th July 17:00
The Mordell conjecture for curves over function field in positive characteristic
This is a report on an article by Buium and Voloch ( Lang's conjecture in characteristic $p$: an explicit bound. Compositio Math. 103 (1996), no. 1, 1--6. ). The interest of this proof is that it makes no assumption of ordinarity on the jacobian of the curve and might thus point to an algebraic proof of the Mordell-Lang conjecture in general.

In our presentation, we shall replace the language of Buium's p-jet schemes by the older language of Weil restrictions, to make the proof understandable by a wider audience.

Thomas Scanlon (Univ. of California, Berkeley)Tuesday 21st July 10:00
B1: Motivic integration
In these two lectures, I will present an interested outsider's perspective on motivic integration.

Thomas Scanlon (Univ. of California, Berkeley)Wednesday 22nd July 10:00
B2: Motivic integration
In these two lectures, I will present an interested outsider's perspective on motivic integration.

Catharina Stroppel (Bonn)Friday 24th July 16:00
Fusion algebras and quantum cohomology

Boris Zilber (University of Oxford)Thursday 23rd July 10:00
C3: Intersections in semi-abelian varieties

Boris Zilber (University of Oxford)Friday 24th July 11:45
C4: Intersections in semi-abelian varieties