The aim of this mini-course is to survey some of the recent interactions of model theory and combinatorics, including regularity lemmas for families of (hyper-)graphs definable in tame structures (algebraic, semi-algebraic, o-minimal, etc.), sum-product phenomena, polynomial expansion and generalizations of the Elekes-Szabó theorem. We will only assume knowledge of basic model theory (some familiarity with stability theory would be helpful), plus some basic measure theory. The tentative plan for the course is as follows:

1. Keisler measures, model theoretic classification and hypergraph regularity (stability, NIP, higher order generalizations).
2. Pseudo-finite dimension, distality and its combinatorial consequences (strong Erdős-Hajnal property, distal regularity lemma, generalized incidence bounds).
3. Recognizing groups and fields from generic data in stable and o-minimal structures.
4. Model theoretic approach to polynomial expansion, Elekes-Szabó theorem and related phenomena.

One of the main problems in algebraic differential equations is the classification of algebraic relations between solutions of differential equations. In this series of lectures, we plan to explain how recent model theoretic developments from groups of finite Morley rank and a new invariant related to forking in stable theories have played a role in strong classification results for algebraic relations between solutions of ordinary differential equations. We will introduce the basic model theory of differential fields, the history of its interaction with stability theory, the Borovik-Cherlin conjecture, and a number of other significant topics which play a role in the story.

This series of talks will cover some developments following the counting theorem proved by Pila and Wilkie. The theorem gives an upper bound on the number of rational points of bounded height lying on the transcendental part of a set definable in an o-minimal expansion of the real field. Since its proof in 2006, this result has been at the heart of many applications of o-minimality to Diophantine geometry.

We’ll start the lectures with a quick recap of the ingredients for the proof of the Pila-Wilkie Theorem, and then outline some of the ideas in the recent proof, by Binyamini, Novikov, and Zack, of Wilkie’s conjecture. This gives an improved bound in the case that the set is definable in the real exponential field, or in the expansion of the real field by restricted Pfaffian functions (which we’ll define).

We’ll then move on to some Diophantine applications. Here we’ll focus on those cases where we can say more using the improved counting results. The necessary definitions and results from Diophantine geometry will be covered (quite quickly) so no background in number theory is needed. Finally, if there is time, I’ll also mention some applications of ideas from number theory to problems in o-minimality.

In 2016, Habegger and Pila proved (using o-minimality) the Zilber-Pink conjecture for a curve in an abelian variety, both defined over the field of algebraic numbers; i.e., they proved that if the intersection of the curve with the union of all algebraic subgroups of codimension at least 2 is infinite, then the curve must be contained in a proper algebraic subgroup. In joint work with Fabrizio Barroero, we deduce from their theorem the same result in the case when the curve and the abelian variety are defined over the complex numbers, using a result of Gao. More generally, we reduce the full conjecture for abelian varieties to the case when everything is defined over the field of algebraic numbers, and we prove the full conjecture if the abelian variety contains no abelian subvariety of dimension > 4 that can be defined over the algebraic numbers. If time permits, I will sketch in some detail our proof of the full conjecture for a power of an elliptic curve with transcendental j-invariant and explain how o-minimality enters the picture.

A semigroup $S$ is a set together with an associative binary operation. Having an identity or not is usually not structurally important: the lack of inverses is. Algebraic semigroup theory is a rich and beautiful subject, which is coming into its own with results relating to structure, geometry, combinatorics, decidability and representation via $S$-acts.

Whereas the model theory of other algebraic structures (such as groups and rings, the latter often via their representation as modules) is well developed, there is no such coherent body of work for semigroups. The reasons behind this are varied (for example, congruences are not determined by a single class). Yet, bit by bit, a model theory of semigroups has emerged, intertwined with aspects listed above. Certainly model theory has influenced some of my work as a semigroup theorist. I will try and explain a few of the possible directions – specifically $\omega$-categoricity and homogeneity of semigroups, and stability of $S$-acts – a few results, and pose a number of open questions. The outlook is exciting.

In this talk we will motivate the study of the differential algebraic properties of power series, solutions of discrete functional equations. Some of these functions are for instance connected to counting problems in enumerative combinatorics as well as to Number theory. Understanding the differential algebraic complexity of these special functions allows to obtain some additional information on the sequence of coefficients of their power series expansion. When the discrete functional equation is linear, one can use difference Galois theory to tackle this question. In the peculiar case of dynamics the projective line, this Galoisian approach shows that any power series solution of a linear discrete equations is either rational or differentially transcendental. This is joint work with Boris Adamczewski and Thomas Dreyfus.

In joint work with Caroline Terry (Ohio State University), we showed that under the model-theoretic assumption of stability the conclusions of the arithmetic regularity lemma can be significantly strengthened. If time permits, I will also sketch more recent work in which we define a ternary notion of stability, with implications for higher-order regularity lemmas.

In this example driven course we will start with the Tarski-Seidenberg theorem and show how to derive from it various properties of the real field. Motivated by this example we will define o-minimal structures, and discuss their basic properties (the monotonicity theorem, cell decomposition, dimension) and some consequences (generic dfferentiability of definable functions, Taylor’s theorem, definable manifolds and the group topology, Steiniz exhange etc.).

We will then move on to discuss expansions of real closed fields that are not o-minimal. We introduce Shelah’s expansion (in the o-minimal context) and Shelah’s QE theorem for such expansions (in this specific context the result is due to Baizalov-Poizat). We will use this theorem to deduce weak o-minimality of RCVF. After a brief discussion of weak o-minimality and its basic properties, we will introduce the valuation group, the residue field and the sort of unit closed balls, K/O. This will give us examples of non-geometric weakly o-minimal structures.

We will then proceed to discuss ACVF where weak o-minimality is no longer relevant. This will motivate us to define dp-minimality, and dp-minimal uniform structures. We will then discuss some of the basic properties of such uniform structures as well as some consequences. If time allows we will discuss further examples, such as pCF (and Presburger arithmetic) as well as various ordered divisible abelian groups – introducing structures of finite dp-rank as well as strictly NIP examples.

In this course, I will introduce simple and NSOP1 theories. I will focus mainly on NSOP1 with the aim being to define Kim-independence, which is the analog of non-forking independence, and to prove some of its properties, especially symmetry over models. The tentative* plan is as follows:

1. Overview of simplicity, including examples and basic results.
2. Introduction to NSOP1: basic definitions and a Kim-Pillay style characterizations.
3. Kim-independence: definitions and basic properties including Kim’s lemma for Kim-independence.
4. Symmetry of Kim independence over models.

*Subject to change depending on the situation.

The goal of this introductory course to stability is to introduce some fundamental notions of geometric stability theory illustrated by many examples of particular theories. We assume a familiarity with an introductory course on model theory (including (the proof of) Morley’s theorem).

Contents (it is tentative and the contents may change according to the pace of the course):

1. Stability. Forking and Dividing and Kim-Pillay’s theorem.
2. The stability spectrum. Quantifier elimination and counting types.
3. \(\omega\)-stable theories and Morley rank. Erimbetov’s inequality.
4. Superstable theories and Lascar rank. Lascar inequalities. Orthogonality.
5. Local ranks.

Positive logic is first-order logic where formulas are built without negation, and with only existential quantification. Although it seems to be much more restrictive than full first-order logic, this is not the case and by clever choices of language (partial Morleyisations) one can show it is really a generalisation of full first-order logic, and even encompasses continuous model theory as well. While the basics of positive logic are well-established, doing stability theory and its generalisations in positive logic is relatively undeveloped.

In this talk I will explain some of the basics of positive logic, and how we can generalise concepts such as stability and simplicity from full first-order logic.

tba

We define the notion of mock hyperbolic reflection spaces and use it to study Frobenius groups, in particular in the context of groups of finite Morley rank including the so-called bad groups. We show that connected Frobenius groups of finite Morley rank and odd type with nilpotent complement split or interpret a bad field of characteristic zero. Furthermore, we show that mock hyperbolic reflection spaces of finite Morley rank satisfy certain rank inequalities, implying in particular that any connected Frobenius group of odd type and Morley rank at most ten either splits or is a simple non-split sharply 2-transitive group of characteristic 6 = 2 of Morley rank 8 or 10.