Calendar of activities

Gros toposes in classical algebraic geometry provide categories of affine schemes an enveloping context with much better properties, and where operations that are familiar among sets, such as taking parts, may also be performed for arbitrary objects. The techniques to construct those toposes were abstracted by Lawvere so that they become applicable not just to rings, but also to other categories of algebras such as C-infinity rings and varieties of semirings with idempotent addition. We show that these techniques are also applicable to exponential rings.

The behavior of certain "special" varieties of arithmetic interest is expected to be governed by the Zilber-Pink conjecture. Although this conjecture remains very much open, there have been some crucial developments. Very recently, Klingler and Tayou announced a result which proves the conjecture when the varieties are geometrically generic. In joint work with V. Aslanyan and G. Fowler, we prove a similar result for subvarieties of powers of the modular curve. Our methods differ from the Klingler-Tayou proof, and build upon the joint work of Pila and Scanlon, which use model theory and differential algebraic methods

O-minimality, a model-theoretic tool that helps with certain arguments concerning analytic functions, has become an extremely successful tool for proving important results in arithmetic geometry, especially in the realm of so-called "unlikely intersections". Various attempts have been made to produce a non-archimedean model theoretic analogue of o-minimality. In this talk I will focus on the notion of Hensel minimality, developed in recent works of Cluckers, Halupczok, Rideau-Kikuchi and Vermeulen, and I will discuss various parallel results one can obtain in a similar fashion to the o-minimal case. In particular, we will look at the Tate uniformization map for elliptic curves. This is joint work with F. Vermeulen.

We will discuss a general framework of fields with operators which allows certain commutativity relations between the operators. The main observation is model-companionability, and then also some model-theoretic results in characteristic zero. Joint work with J. Dobrowolski.

We will discuss fields in characteristic $p>0$ equipped with a derivation and an automorphism that commutes with the derivation. We prove that this has a model-companion, in model-theoretic parlance ${\mathrm{DCF}}_p\mathrm{A}$ exists (the case of characteristic zero has been known for over 20 years). Joint work with K. Ino.

Valued difference fields are fields equipped with a valuation and a compatible automorphism. I will talk about joint work with Jan Dobrowolski and Francesco Gallinaro on existentially closed valued difference fields. We prove that they admit an equivariant section of the valuation, characterise amalgamation bases for $\mathrm{ac}$-valued fields of residue characteristic $0$, and deduce that they are ${\mathrm{NTP}}_2$ in the sense of positive logic.

Generalized power series can be used to represent germs of some functions definable in polynomially bounded o-minimal expansions of the reals. In fact, most polynomially-bounded o-minimal expansions of the reals are constructed from some algebra of smooth functions whose germs admit faithful representations as generalized power series (GQAs). This can be used to try to expand some Hahn fields to elementary extensions of these o-minimal structures, because generalized power series can be naturally evaluated at positive infinitesimal elements in these Hahn fields.

In this talk, after a somewhat detailed discussion of its motivation and applications we will outline the proof of the following fact: expansions of Hahn fields by algebras of generalized power series have the property that sets closed under truncations generate substructures closed under truncations, provided that the algebra of generalized power series used in the expansion is itself closed under partial derivations and truncations. The presented applications will be concerned with the problem of constructing well-behaved truncation closed embeddings of o-minimal fields into Hahn fields.

We study generic derivation on algebraically bounded structures. Given an algebraically bounded theory $T$ (e.g. the theory of algebraically closed fields, or of real closed fields, or of pseudo-finite fields…), we consider the expansion of $T$ with an extra function $\delta (x)$ and an axiom saying that $\delta$ is a derivation. This new theory has a model completion with interesting geometric and model-theoretic properties.

We continue the study of generic derivations, but examine the case of a theory $T$ which is not algebraically bounded. We characterize the situation when $T$ + "$\delta (x)$ is a derivation" has a model completion (with some examples and counterexamples) and relate it to geometric properties of $T$.

Transseries generalise power series by including exponential and logarithmic terms. I'll give a very down-to-earth introduction to transseries (or rather the 'omega-series' variant), building up on what you already know about Taylor series, and mention the key results in the area. I'll then discuss some missing bits and pieces about the properties of composition, some of which were presented by Vincent Bagayoko at the INdAM miniworkshop in May, and some corresponding applications to normalisation of hyperbolic series (joint with D. Peran, J.-P. Rolin, T. Servi).

Local stability has been used in the recent years to treat problems in additive combinatorics. Whilst many of the techniques of geometric stability theory have been generalised to simple theories, there is no local treatment of simplicity. Kaplan and Shelah showed that the theory of the additive group of the integers together with a predicate for the prime integers is supersimple of rank 1, assuming Dickson's conjecture. In joint work with Daniel Palacín (UC Madrid), we will see how to use their result to deduce that all but finitely many integers belongs to infinitely many arithmetic progressions in the primes, which resonates with previous unconditional work (without assuming Dickson's conjecture) of van der Corput and of Green. If times permits, we will discuss analogous results asymptotically for finite fields.

The 'existential closedness' problem for analytic functions is an idea derived from Zilber's work about the model theory of complex exponentiation, and it predicts that many systems of equations involving special analytic functions have complex solutions. I will focus on the techniques being used for attacking this problem, and discuss in detail the 'one variable' cases which in principle require little more than undergraduate complex analysis. I will show the well known case of exponential polynomials and hint at how we adapted it to work for the $j$-function and its derivatives (joint with V. Aslanyan and S. Eterović).