Given two algebraic ODEs, is there a differential-algebraic relation between generic tuples of their solutions? In recent work with Freitag and Moosa, we produce a bound on the length of tuples one must look at to f ind a relation. Our proof relies on two ingredients. The first is differential Galois theory, combined with the recent proof by Freitag and Moosa of the Borovik-Cherlin conjecture in algebraically closed fields. The second is some general model theory result which allows us to factor any relation through some minimal ODE. I will give a precise statement of our result and sketch the proof. I will also explain why our bound is tight.[-]

A simple group is pseudofinite if and only if it is isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in 1995 and heavily relies on the classification of finite simple groups (CFSG). It easily follows that any simple pseudofinite group $G$ is finite-dimensional. In particular, if $\operatorname{dim}(G)=3$ then $G$ is isomorphic to $\operatorname{PSL}(2, F)$ for some pseudofinite field $F$. In this talk, we describe the structures of finite-dimensional pseudofinite groups with dimension $<4$, without using CFSG. In the case $\operatorname{dim}(G)=3$ we show that either $G$ is soluble-by-finite or has a finite normal subgroup $Z$ so that $G / Z$ is a finite extension of $\operatorname{PSL}(2, F)$. This in particular implies that the classification $G \cong \operatorname{PSL}(2, F)$ from the above does not require CFSG. This is joint work with Frank Wagner.[-]

Roth's theorem states that a subset $A$ of $\{1, \ldots, N\}$ of positive density contains a positive $N^2$-proportion of (non-trivial) three arithmetic progressions, given by pairs $(a, d)$ with $d \neq 0$ such that $a, a+d, a+2 d$ all lie in $A$. In recent breakthrough work by Kelley and Meka, the known bounds have been improved drastically. One of the core ingredients of the their proof is a version of the almost periodicity result due to Croot and Sisask. The latter has been obtained in a non-quantitative form by Conant and Pillay for amenable groups using continuous logic.
In joint work with Daniel Palacín, we will present a model-theoretic version (in classical first-order logic) of the almost-periodicity result for a general group equipped with a Keisler measure under some mild assumptions and show how to use this result to obtain a non-quantitative proof of Roth's result. One of the main ideas of the proof is an adaptation of a result of Pillay, Scanlon and Wagner on the behaviour of generic types in a definable group in a simple theory.[-]

A primitive permutation group $(X, G)$ is a group $G$ together with an action of $G$ on $X$ such that there are no nontrivial equivalence relations on $X$ preserved by $G$. An rough classification of primitive permutation groups of finite Morley rank, modeled on the O'Nan-Scott theorem for finite primitive permutation groups, has been carried out by Macpherson and Pillay and this classification was then used by Borovik and Cherlin to prove that if $(X, G)$ is a primitive permutation group of finite Morley rank, the rank of $G$ can be bounded in terms of the rank of $X$. We study the analogous situation for pseudo-finite primitive permutation groups of finite SU-rank, building both on supersimple group theory and classification results of Liebeck-Macpherson-Tent. This is joint work in progress with Ulla Karhumäki.[-]

In recent papers by Alon et al. and Fox et al. it is demonstrated that families of graphs with a semialgebraic edge relation of bounded complexity have strong regularity properties and can be decomposed into very homogeneous semialgebraic pieces up to a small error (typical example is the incidence relation between points and lines on a real plane, or higher dimensional analogues). We show that in fact the theory can be developed for families of graphs definable in a structure satisfying a certain model theoretic property called distality, with respect to a large class of measures (this applies in particular to graphs definable in arbitrary o-minimal theories and in p-adics). (Joint work with Sergei Starchenko.)[-]

If CCM denotes the theory of compact complex spaces in the langauge of complex-analytic sets, then the theory of models of CCM equipped with an automorphism has a model companion, denoted by CCMA. The relationship to meromorphic dynamical systems is the same as that of ACFA to rational dynamical systems. I will discuss recent joint work with Martin Bays and Martin Hils that begins a systematic study of CCMA as an expansion of ACFA. Particular topics we consider include: stable embeddedness, imaginaries, and the Zilber dichotomy.[-]

An atomic class $K$ is the class of atomic first order models of a countable first order theory (assuming there are such models). Under the weak $\mathrm{GCH}$ it had been proved that if such class is categorical in every $\aleph_n$ then it is categorical in every cardinal and is so called excellent. There are results when we assume categoricity for $\aleph_1, \ldots, \aleph_n$. The lecture is on a ZFC result in this direction for $n=1$. More specifically, if $K$ is categorical in $\aleph_1$ and has a model of cardinality $>2^{\aleph_0}$, then it is $\aleph_0$-stable, which implies having stable amalgamation, and is the first case of excellence.
This a work in preparation by J.T. Baldwin, M.C. Laskowski and S. Shelah.[-]

Which Noetherian integral domains are NIP? This is a natural question to ask, given the prominence of Noetherian rings in commutative algebra. We cannot hope to answer this question in full generality any time soon, as it includes other hard problems such as the conjectures on stable fields and NIP fields. Nevertheless, we present some interesting partial results which begin to paint a picture of NIP Noetherian rings. Let $R$ be a Noetherian domain which is NIP. Then either $R$ is a field, or $R$ is a semilocal domain of Krull dimension 1 and characteristic 0. Assuming the henselianity conjecture on NIP valued fields, $R$ is a henselian local ring. In the dp-minimal case, one can give a complete classification. Specifically, every dp-minimal Noetherian domain is a finite index subring of a dp-minimal discrete valuation ring. The situation in dp-rank 2 seems to be significantly worse, but a classification may still be possible in terms of differential valued fields.[-]

We extend the notion of beautiful pairs by Poizat to unstable theories via definable types, with a specific interest in such pairs of valued fields. In particular, we establish an analogue of Ax-Kochen-Ershov principles in for certain pairs of valued fields. In the specific case of ACVF, we classify all such pairs and deduce the strict pro-definability of various spaces of definable types, such as the stable completion introduced by Hrushovski-Loeser and a model theoretic analogue of the Huber analytification of an algebraic variety. This is joint with Pablo Cubides Kovacsics and Martin Hils.[-]

We give an arithmetic version of Tao's algebraic regularity lemma (which was itself an improved Szemerédi regularity lemma for graphs uniformly definable in finite fields). In the arithmetic regime the objects of study are pairs $(G, D)$ where $G$ is a group and $D$ an arbitrary subset, all uniformly definable in finite fields. We obtain optimal results, namely that the algebraic regularity lemma holds for the associated bipartite graph $(G, G, E)$ where $E(x, y)$ is $x y^{-1} \in D$, witnessed by a the decomposition of $G$ into cosets of a uniformly definable small index normal subgroup $H$ of $G$.[-]