Geometric and combinatorial configurations in Model theory

Model theory studies structures from the point of view of first-order logic. It isolates combinatorial properties of definable sets and uses these to obtain algebraic consequences. A key example is the group configuration theorem which asserts that certain combinatorial patterns are necessarily induced by the existence of a group. This has proven a powerful tool in applications, e.g. in diophantine geometry and more recently to combinatorics.

Valued fields are another example of the confluence of stability theory and algebraic model theory. While algebraically closed valued fields were already studied by Robinson in 1959, the importance of the tools from geometric stability only became apparent in work of Haskell-Hrushovski-Macpherson on elimination of imaginaries, and have in turn triggered the development of the abstract theory of stable domination, leading to applications in non-archimedean geometry.

In the project, we aim to strengthen the recent relations between model theory and combinatorics, develop the model theory of valued fields using tools from geometric stability and carry out an abstract study of the configurations which are a fundamental tool in these two areas.

Upcoming meetings

Past Meetings

1. Geomod half-day Meeting on model theory and combinatorics, June 18 2021.
   Programme

   9h30 - 10h10: Frank Wagner (Université Lyon 1), A survey of model-theoretic methods in combinatorics
   I shall survey the main applications of model theory in combinatorics (of which I am aware!): pseudofinite dimension and measure, the stabilizer theorem, approximate subgroups and Erdös geometry. slides

   10h20 - 11h: Tingxiang Zou (WWU), Coarse approximate subgroups in weak general position and Elekes-Szabó problems for nilpotent groups
   The Elekes-Szabó's theorem says very roughly that if a complex irreducible subvariety V of X*Y*Z has ''too many'' intersections with cartesian products of finite sets, then V is in correspondence with the graph of multiplication of an algebraic group G. It was noticed by Breuillard and Wang that the algebraic group G must be abelian. There is a condition for the finite sets witnessing ''many'' intersections with V, called in general position, which plays a key role in forcing the group to be abelian. In this talk, I will present a result which shows that in the case of graphs of multiplication in complex algebraic groups, with a weaker general position assumption, nilpotent groups will appear. More precisely, for a connected complex algebraic group G the following are equivalent:
   1. The graph of multiplication of G has ''many'' intersections with finite sets in weak general position;
   2. G is nilpotent;
   3. The ultrapower of G has a pseudofinite coarse approximate subgroup in weak general position.
   Surprisingly, the proof of the direction from 2 to 3 invokes some form of generic Mordell-Lang theorem for commutative complex algebraic groups. This is joint work with Martin Bays and Jan Dobrowolski.

   11h - 11h40: Coffee Break.

   11h40 - 12h20: Daniel Palacín (Albert-Ludwigs-Universität Freiburg), Robustly stable relations and squares
   The main goal of this talk is to introduce the notion of a robustly stable relation, which is a generalization of stability in the presence of a measure, and to discuss some structural theorems. Furthermore, I shall explain how to obtain certain desired combinatorial patterns within (robustly) stable sets, using model theoretic techniques. For example, given an abelian group G and a (robustly) stable binary relation S with positive density in GxG, we will prove the existence of a square in S, that is, a quadruple of points of the form (x,y),(x,y+g),(x+g,y) and (x+g,y+g). This is work in progress with Amador Martin-Pizarro and Julia Wolf.

2. GeoMod "Journée d'études", Monday 22 March.
   Programme

   9h - 9h50: Jinhe Vincent Ye (Sorbonne Université), The étale open topology and the stable fields conjecture
   For any field K, we introduce natural topologies on K-points of varieties over K, which is defined to be the weakest topology such that étale morphisms are open. This topology turns out to be natural in a lot of settings. For example, when K is algebraically closed, it is easy to see that we have the Zariski topology, and the procedure picks up the valuation topology in many henselian valued fields. Moreover, many topological properties correspond to the algebraic properties of the field. As an application of this correspondence, we will show that large stable fields are separably closed. (Joint work with Will Johnson, Chieu-Minh Tran, and Erik Walsberg.)

   10h - 10h40: Blaise Boissonneau (WWU), Artin-Schreier extensions & combinatorial complexity
   Model-theoretic combinatorial complexity and algebraic properties are interlinked in sometimes surprising ways. An example of that is a well-known result by Kaplan, Scanlon and Wagner, which states that infinite NIP fields of characteristic p have no Artin-Schreier extension. Since then, the model-theoretic complexity induced by Artin-Schreier extensions has been studied in many other settings, most notably in NTP2 fields. We will see what implications this has in the study of NTP2 henselian fields, especially in the mixed characteristic case.

   10h40 - 11h: Coffee Break.

   11h - 12h: Short talks (in this order) by Franziska Jahnke, Sylvy Anscombe, Rosario Mennuni, Akash Hossain, Martin Hils, Silvain Rideau-Kikuchi, Martin Bays.

   12h - 13h: Time and space for mathematical discussions via "gather town", in groups of variable sizes.

3. First GeoMod virtual meeting, Friday 29 January.
   Programme

   10h - 10h10: Elisabeth Bouscaren (CNRS - Université Paris-Saclay), Introduction.

   10h15 - 11h: Rosario Mennuni (Westfälische Wilhelms-Universität Mänster), The domination monoid
   This talk is concerned with the interaction between the semigroup of invariant types over a monster model U and the preorder of domination, i.e. small-type semi-isolation. In the superstable case, the induced quotient semigroup, which goes under the name of "domination monoid", parameterises "finitely generated saturated extensions of U" and how they can be amalgamated independently. In general, the situation is much wilder, and the domination monoid need not even be well-defined. Nevertheless, this object has been used to formulate AKE-type results, can be computed in various natural examples, and there is heuristic evidence that well-definedness may hold under NIP. I will give an overview of the subject, present some results on these objects from my thesis, and talk about ongoing work with Martin Hils on the domination monoid in theories of valued fields.

   11h15 - 12h: Léo Jimenez (Université Claude Bernard Lyon 1), Towards new structures without the CBP
   The canonical base property (CBP) is a fine structural concept, appropriate for structures of finite Lascar rank. It has striking consequences, for example a simplified proof of the Zilber trichotomy in differentially closed fields of characteristic zero. There is currently only one known example of a theory without the canonical base property, due to Palacín, Pillay and Hrushovski. In this talk, we will motivate the search for more counterexamples, describe some sufficient conditions for a theory to not have the CBP, and show how this could be used to obtain new counterexamples.

   12h15 - 13h: Michael Lösch (Albert-Ludwigs-Universität Freiburg), Additive covers and the canonical base property
   Iternality is a fundamental notion in geometric model theory in order to understand a theory in terms of its building blocks, its minimal types of rank 1. The Canonical Base Property (CBP) isolates a key notion behind various descent arguments for differential and difference varieties and states the following: Over a realization of a stationary type, its canonical base is almost internal to the family of all non-locally modular minimal types. Though many relevant examples of theories satisfy the CBP, Hrushovski, Palacín and Pillay produced an example of an uncountably categorical theory without the CBP.
   Without assuming a strong familiarity with stability theory, I will present an alternative description of their counterexample in terms of additive covers of an algebraically closed field and give a new proof of the failure of the CBP, exploiting a notion which already appeared in Chatzidakis’ work on the UCBP. In the end I will sketch new additive covers without the CBP and connect this failure with the aforementioned notion.