Globally Valued Fields, studied jointly with E. Hrushovski, are a formalism for fields in which the sum formula for valuations holds, such as number fields or function fields of curves. They form an elementary class (in continuous first order logic), and model-theoretic questions regarding this class give rise to difficult yet fascinating geometric questions.

I intend to present « Lyon school » approach to studying GVFs. This consists of reducing as much as possible to local considerations, among other things via the "fullness" axiom.

# Past Logic Seminar

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).

Groups which are "attached" to theories of fields, appearing in models of the theory

as the automorphism groups of intermediate fields fixing an elementary submodel are called geometrically represented.

We will discuss the concept ``geometric representation" in the case of pseudo finite fields. Then will show that any group which is geometrically represented in a complete theory of a pseudo-finite field must be abelian.

This result also generalizes to bounded PAC fields. This is joint work with Zoe Chatzidakis.

Motivated by the Dixmier-Moeglin equivalence, which belongs to the realm of algebra representations, we look at a differential version of this equivalence for algebraic D-groups, which belong to the realm of finite Morley rank groups in differentially closed fields. We will see how the proof of this equivalence reduces to a standard model-theoretic fact (on binding groups). Time permitting we will present an application to Hopf-Ore extensions. This is joint work with J. Bell and R. Moosa.

We consider the differentiability of definable functions in tame expansions

of algebraically closed valued fields.

As the Frobenius inverse shows such a function may be nowhere

differentiable.

We prove differentiability almost everywhere in valued fields of

characteristic 0

that are C-minimal, definably complete and such that, in the valuation

group,

definable functions are strongly eventually linear.

This is joint work with Pablo Cubides-Kovacsics.

I will review the notion of classifying topos of a first-order (geometric) theory and explain the central role enjoyed by theories of presheaf type (i.e. classified by a presheaf topos) in the context of the topos-theoretic investigation of the model theory of geometric theories. After presenting a few main results and characterizations for theories of presheaf type, I will illustrate the generality of the point of view provided by this class of theories by discussing a topos-theoretic framework unifying and generalizing Fraissé’s construction in model theory and topological Galois theory and leading to an approach to the problem of the independence from l of l-adic cohomology.

I will explain the model theoretic notion of ampleness

and present the geometric context of recent constructions.

This concerns recent work with P. Corvaja in which we relate the Hilbert Property for an algebraic variety (a kind of axiom linked with Hilbert Irreducibility, relevant e.g. for the Inverse Galois Problem) with the fundamental group of the variety.

In particular, this leads to new examples (of surfaces) of failure of the Hilbert Property. We also prove the Hilbert Property for a non-rational surface (whereas all previous examples involved rational varieties).

Transseries arise naturally when solving differential equations around essential singularities. Just like most Taylor series are not convergent, most transseries do not converge to real functions, even when using advanced summation techniques.

On the other hand, we can show that all classical transseries induce analytic functions on the surreal line. In fact, this holds for an even larger (proper) class of series which we call "omega-series".

Omega-series can be composed and differentiated, like LE-series, and they form a differential subfield of surreal numbers equipped with the simplest derivation. This raises once again the question whether all surreal numbers can be also interpreted as functions. Unfortunately, it turns out that the simplest derivation is in fact incompatible with this goal.

This is joint work with A. Berarducci.

We discuss how one can define transseries in several variables. The idea is

to combine the construction of the univariate transseries with a blow up procedure. The

latter allows to normalize transseries in an arbitrary number of variables which makes

them manageable as usual transseries.