Conference: Model theory and tame expansions of topological fields

An additively symmetric subset $X$ of a ring is an approximate subring if finitely many additive translates of $X$ cover $X\cdot X\cup (X+X)$. If $K$ translates are enough, we say that $X$ is a $K$-approximate subring.

In the first part of the talk, I will briefly discuss some fundamental issues around the notion of approximate subgroup. Then I will pass to approximate subrings and to my result on the existence of locally compact models for arbitrary approximate subrings. The rest of the talk will be devoted to applications of this theorem to structural results on approximate subrings, including my very recent (not circulated yet) theorem with Simon Machado describing the structure of finite $K$-approximate subrings. This can also be viewed as a unified general form of the so-called sum-product phenomenon, which will be briefly discussed. Another application of the aforementioned theorem on the structure of finite $K$-approximate subrings is a ring-theoretic counterpart of Gromov's theorem on groups of polynomial growth. In the same joint project with Simon Machado, we also applied locally compact models to obtain wide extensions of a classical theorem of Meyer classifying approximate lattices in $ℝ$ closed under multiplication (but this part of the project will not be discussed during the talk).

Basic model theory plays an essential role in this research. The construction of locally compact models is obtained via model-theoretic connected components of definable groups and rings. Structural results on approximate subrings are obtained either using the aforementioned components or locally compact models together with a pseudofinite context and some non-standard analysis.

Let $ℛ$ be an expansion of the real field such that every subset of $ℝ$ definable in $ℛ$ either has interior or is a finite union of discrete sets. Answering a question by Chris Miller, we show that for every $n\in \mathbb{N}$ and every definable subset $A\subseteq {ℝ}^{n+1}$ there is $N\in \mathbb{N}$ such that for all $x\in {ℝ}^n$ either $A_x$ has interior or is the union of $N$ discrete sets. This is joint work with Madie Farris.

The representation type of a finite-dimensional $k$-algebra is an algebraic measure of how hard it is to classify its finite-dimensional indecomposable modules.

Intuitively, a finite-dimensional $k$-algebra is of tame representation type if we can classify its finite-dimensional modules and wild representation type if its module category contains a copy of the category of finite-dimensional modules of all other finite-dimensional $k$-algebras. An archetypical (although not finite-dimensional) tame algebra is $k[x]$. The structure theorem for finitely generated modules over a PID describes its finite-dimensional modules. Drozd's famous dichotomy theorem states that all finite-dimensional algebras are either wild or tame.

The tame/wild dividing line is not seen by standard model theoretic invariants or even the more specialised invariants coming from Model Theory of Modules. A long-standing conjecture of Mike Prest claims that a finite-dimensional algebra has decidable theory of modules if and only if it is of tame representation type. More recently, I conjectured that a finite-dimensional algebra has decidable theory of (pseudo)finite dimensional modules if and only if it is of tame representation type. This talk will focus on recent work providing evidence for the second conjecture.

Since the work of Ax-Kochen/Ersov, the model theory of a valued field has always been understood in terms of its residue field and value group. For algebraically closed valued fields of equicharacteristic zero, work by Hrushovski-Kazhdan provides a deep analysis of definable sets, and leads to a classification of definable sets up to definable isomorphisms in terms of the residue field and value group. In joint work with Mathias Stout, we extend this framework of Hrushovski-Kazhdan to 1-h-minimal theories of equicharacteristic zero.

In this talk I will give an introduction to Hrushovski-Kazhdan style motivic integration and how to generalize it to the 1-h-minimal setting. Time permitting, I will give some applications and mention links with Cluckers-Loeser motivic integration.

I will review the main results of the last decade concerning the study of NIP (dp-finite/dp-minimal) rings and the main open questions, focussing on how algebra and model theory are intertwined in that context. I will start with division rings, commutative rings and end with presenting joint work with Halevi and Johnson concerning the classification of dp-minimal integral domains. If time permits, we will review some of the arguments and technics involved in the latter result.

We give a complete description of all the definable rings in an o-minimal expansion of a real closed field and, more generally, in arbitrary o-minimal structures. It turns out that even when the structure does not expand a field, an infinite field is definable in the ring structure alone, provided the multiplication is not trivial on the definably connected component of zero. Definability issues and problematic examples will be discussed.

While the model theory of henselian valued fields like the $p$-adics or formal Laurent series over the complex numbers has long been understood, there is much we do not know for henselian valued fields in positive characteristic. Most prominently, this affects Laurent series over finite fields, which are among the most interesting valued fields from a number-theoretic perspective. I will discuss ongoing work to at least understand the existential theories of various fields of this kind, allowing parameters from certain subfields. For instance, this leads to an understanding of the existential theories of almost all completions of a function field, with parameters from the function field. This is partly joint work with S. Anscombe and A. Fehm.

Let $W$ be an algebraic subvariety of the complex multiplicative group ${𝔾}_m^n$, and $H$ an algebraic subgroup of ${𝔾}_m^n$ such that $\dim W+\dim H$ is at least $n$. What can we say about intersections between $W$ and cosets of $H$? We show that if $W$ satisfies a condition known as geometrical non-degeneracy, there is a finite set $S$ of subgroups, depending only on $W$, such that if $H$ is not contained in any of the elements of $S$ then every coset of $H$ intersects $W$. The ingredients of the proof come from tropical geometry, equidistribution, and model theory. This is joint work with G. Dill.

In 2002 Zilber introduced the theory of a generic function on a field, coinciding with the limit theory of generic polynomials as developed by Koiran. Axiomatized in first-order logic by versions of Schanuel property and existential closedness, this theory turns out to be $\omega$-stable. As shown by Wilkie and Koiran, one can explicitly construct such a generic function on the complex plane in a form of a Taylor series, using the ideas behind Liouville numbers.

In this talk we look further into the properties of the theory of generic functions. As the main result, we show that adding any of these generic functions to the complex field gives an isomorphic structure, which ought to be quasiminimal, i.e. any definable subset has to be countable or cocountable. Thus we obtain a non-trivial example of an entire function which keeps the complex field quasiminimal.

This is joint work with Francesco Gallinaro.

Given an algebraically closed field $K$ of characteristic zero, we study the poset $\mathrm{Pos}(K)$ of irreducible proper algebraic subsets (points and curves) of the projective plane over $K$. Answering a question of Marcus Tressl, we prove that the poset interprets the field $K$ and the subring of integers, and it is in fact bi-interpretable with the two-sorted structure $(K,\mathrm{Fin}(K))$ consisting of the field $K$ and a sort for its finite subsets. We prove that the theory of the poset depends on the transcendence degree of $K$, so for instance the complex numbers and the algebraic numbers have non-elementary-equivalent posets. Taking for $K$ the complex numbers, we give a complete recursive axiomatization of $(K,\mathrm{Fin}(K))$ modulo the theory of the integers. We also show that the integers are stably embedded in this structure.

I will explain the Modular Existential Closedness with Derivatives conjecture which predicts when systems of equations, involving addition, multiplication, and the modular $j$-function and its derivatives, have solutions in the complex numbers. I will then report on recent joint work with Sebastian Eterovic and Vincenzo Mantova tackling a special case of the conjecture. Rather than discussing general results, I will focus on several examples illustrating the main ideas and tools used in our approach.

I will explain a differential transcendence result for Hardy fields, related to the model theoretic concept of co-analyzability. I will show how it can be used to prove a conjecture of Boshernitzan. I will also describe a cautionary example how this result cannot be generalized. This involves a theorem of Rosenlicht. (Joint work with Lou van den Dries and Joris van der Hoeven.)

Berarducci, Peterzil, and Pillay asked whether every abstract finite central extension of a definably connected definable group in an o-minimal structure is also definable. This question, which is related to a conjecture of Milnor for Lie groups, was solved in the affirmative by them in the abelian case. In this talk, I will explain how one can extend their result to solvable groups. This is joint work with Elías Baro.

I will report on joint work in progress with Omar León Sánchez and Amador Martin-Pizarro. Let ${\mathrm{DCF}}_p$ be the model companion of the theory of differential fields of characteristic $p>0$. Shelah proved in 1973 that the theory ${\mathrm{DCF}}_p$ is stable. We give an algebraic description of the forking independence relation in ${\mathrm{DCF}}_p$ by introducing a new differential algebraic notion of differential "transcendental imperfectness". We also show that types in ${\mathrm{DCF}}_p$ over algebraically closed sets (in the home sort) are stationary and give an example of a differential equation whose solution set is strictly disintegrated; that is, it has the induced structure of a pure set.

Definable groups in theories of fields can often be described in terms of algebraic groups. For example, by a result of Pillay, every definable group in a differentially closed field of characteristic 0 embeds into an algebraic group.

In this talk, we show that the same holds true for differentially closed fields of positive characteristic, following a strategy of Bouscaren and Delon. Before outlining the proof for differentially closed fields of positive characteristic, we will first introduce the theory and the properties that are used in the proof.

Strategies via differential algebra and o-minimality have been successful in proving functional transcendence results for important periodic functions like exponentiation and the $j$ function. However, there are functions of interest that are differentially transcendental and non-periodic, such as the Gamma function. In this talk, I will present a valuation-theoretic approach to proving cases of an Ax–Lindemann–Weierstrass-type result for the Gamma function.

I will report on work-in-progress with Moshe Kamensky on the quantifier-free fragment of $\mathrm{ACFA}$, as an approach to the birational geometry of algebraic dynamical systems. Motivated by the analogy with $\mathrm{DCF}$, I will discuss a result about the structure of dynamical systems that admit no proper fibrations, and show how it leads to a (so far non-optimal) bound on the degree of nonminimality.

Working with my students Ilgwon Seo and Carlos Cordoba, we are slowly piecing together what we hope will one day be a proof of Dulac's Problem. I will explain what we are doing and what still needs to be done.