A Morse function (and more generally a Morse-Bott function) on a compact manifold M has associated Morse inequalities. The aim of this
talk is to explain how we can associate Morse inequalities to any smooth function on M (reporting on work of/with G Penington).

For a smooth variety over a number field, one defines various different homology groups (Betti, de Rham, etale, log-crystalline), which carry various kinds of enriching structure and are thought of as a system of realisations for a putative underlying (mixed) motivic homology group. Following Deligne, one can study fundamental groups in the same way, and the study of specific realisations of the motivic fundamental group has already found Diophantine applications, for instance in the anabelian proof of Siegel's theorem by Kim.

It is hoped that study of fundamental groups should give one access to ``higher'' arithmetic information not visible in the first cohomology, for instance classical and p-adic heights. In this talk, we will discuss recent work making this hope concrete, by demonstrating how local components of canonical heights on abelian varieties admit a natural description in terms of fundamental groups.

Roelcke precompact groups are exactly the topological groups that can be realized as automorphism groups of omega-categorical structures (in continuous logic). In this talk, I will discuss a model-theoretic framework for the study of those groups and their dynamical systems as well as two concrete applications. The talk is based on joint work with Itaï Ben Yaacov and Tomás Ibarlucía.

(Joint with Gareth Boxall) In this talk I will introduce some properties of distal theories. I will remark that distality is preserved neither under reducts nor expansions of the language. I will then go on to discuss a recent result that the Shelah expansion of a theory is distal if and only if the theory itself is distal. 

We discuss the connections and differences between the ZFC set theory and univalent foundations and answer the above question in the negative.
 

The talk is based on my joint work with Anand Pillay and Tomasz Rzepecki.

I will describe some connections between various objects from topological dynamics associated with a given first order theory and various Galois groups of this theory. One of the main corollaries is a natural presentation of the closure of the neutral element of the Lascar Galois group of any given theory $T$ (this closure is a group sometimes denoted by $Gal_0(T)$) as a quotient of a compact Hausdorff group by a dense subgroup.

As an application, I will present a very general theorem concerning the complexity of bounded, invariant equivalence relations (whose classes are sometimes called strong types) in countable theories, generalizing a theorem of Kaplan, Miller and Simon concerning Borel cardinalities of Lascar strong types and also later extensions of this result to certain bounded, $F_\sigma$ equivalence relations (which were obtained in a paper of Kaplan and Miller and, independently, in a paper of Rzepecki and myself). The main point of our general theorem says that in a countable theory, any bounded, invariant equivalence relation defined
on the set of realizations of a single complete type over $\emptyset$ is type-definable if and only if it is smooth (in the sense of descriptive set theory). If time permits, I will very briefly mention more recent developments in this direction (also based on the results from the first paragraph) which will appear in my future paper with Rzepecki.
 

It is well known that the theory of differentially closed fields of characteristic 0 has prime models and that they are unique up to isomorphism. One can ask the same question for the theory ACFA of existentially closed difference fields (recall that a difference field is a field with an automorphism).

In this talk, I will first give the trivial reasons of why this question cannot have a positive answer. It could however be the case that over certain difference fields prime models (of the theory ACFA) exist and are unique. Such a prime model would be called a difference closure of the difference field K. I will show by an example that the obvious conditions on K do not suffice.

I will then consider the class of aleph-epsilon saturated models of ACFA, or of kappa-saturated models of ACFA. There are natural notions of aleph-epsilon prime model and kappa-prime model. It turns out that for these stronger notions, if K is an algebraically closed difference field of characteristic 0, with fixed subfield F aleph-epsilon saturated, then there is an aleph-epsilon prime model over K, and it is unique up to K-isomorphism. A similar result holds for kappa-prime when kappa is a regular cardinal.

None of this extends to positive characteristic.
 

Let k be a field of characteristic zero and K=k((t)). Semi-algebraic sets over K are boolean combinations of algebraic sets and sets defined by valuative inequalities. The associated Grothendieck ring has been studied by Hrushovski and Kazhdan who link it via motivic integration to the Grothendieck ring of varieties over k. I will present a morphism from the former to the Grothendieck ring of motives of rigid analytic varieties over K in the sense of Ayoub. This allows to refine the comparison by Ayoub, Ivorra and Sebag between motivic Milnor fibre and motivic nearby cycle functor.
 

The talk will focus on results of two related strands of research undertaken by the speaker. The first is a model of quantum mechanics based on the idea of 'structural approximation'. The earlier paper 'The semantics of the canonical commutation relations' established a method of calculation, essentially integration, for quantum mechanics with quadratic Hamiltonians. Currently, we worked out a (model-theoretic) formalism for the method, which allows us to
perform more subtle calculations, in particular, we prove that our path integral calculation produce correct formula for quadratic Hamiltonians avoiding non-conventional limits used by physicists. Then we focus on the model-theoretic analysis of the notion of structural approximation and show that it can be seen as a positive model theory version of the theory of measurable structures, compact domination and integration (p-adic and adelic).