# European Championship Model Theory Days Passau, 19.-21. June 2024

## Speakers:

- Matthias Aschenbrenner (University of Vienna)
- Lou van den Dries (University of Illinois at Urbana-Champaign)
- Pantelis Eleftheriou (University of Leeds)
- Immanuel Halupczok (University of Düsseldorf)
- Martin Hils (University of Münster)
- Franziska Jahnke (University of Münster)
- Piotr Kowalski (University of Wroclaw)
- Sebastian Krapp (University of Konstanz)
- Jana Marikova (University of Vienna)
- Patrick Speissegger (McMaster University
- Alex Wilkie (University of Oxford)
- Martin Ziegler (University of Freiburg)

## Programme:

**Wednesday, 19 June**

13:00 Welcome (in front of IM 242)

13:30 Martin Hils (University of Münster)

The Amalgamation Property for Definable Types (JUR 153)

14:30 Martin Ziegler (University of Freiburg)

Quantifier Elimination of Pairs of Algebraically Closed Fields (JUR 153)

15:30 Piotr Kowalski (University of Wroclaw)

Model Completeness and Semisimple Groups (JUR 153)

16.30 Short Break

16:45 Pantelis Eleftheriou (University of Leeds)

On the Global Zarankiewicz’s Problem (JUR 153)

18:00 Germany - Hungary (IM 242)

**Thursday, 20 June**

9:00 Coffee (IM 242)

9:30 Sebastian Krapp (University of Konstanz)

Embedding the Prime Model of Real Exponentiation (IM 242)

10:30 Jana Marikova (University of Vienna)

An O-minimal Følner Condition (IM 242)

11.30 Short Break

11:45 Patrick Speissegger (McMaster University)

Definability Questions for O-minimal Expansions of the real Field (IM 242)

12:45 Lunch (Mensa)

13:45 Lou van den Dries (University of Illinois at Urbana-Champaign)

What is ω-Freeness? (IM 242)

14:45 Coffee (IM 242)

15:00 Matthias Aschenbrenner (University of Vienna)

Remarks on Analytic Hardy Fields (IM 242)

19:00 Conference Party

**Friday, 21 June**

9:30 Coffee (IM 242)

10:00 Immanuel Halupczok (University of Düsseldorf)

Hensel Minimality (IM 242)

11:00 Franziska Jahnke (University of Amsterdam)

AKE Principles in Mixed Characteristic (IM 242)

12:00 Alex Wilkie (University of Oxford)

Analytic Continuation and Zilber’s Quasiminimality Conjecture (IM 242)

13:00 Lunch (Mensa)

## Title and Abstracts

**Matthias Aschenbrenner**

*Title: *Remarks on analytic Hardy fields

In this talk, after recalling some basic definitions around Hardy fields, I will focus on analytic Hardy fields, which are those that mainly arise in practice. I will speak about our result that every countable gap in an analytic Hardy field can be filled in an analytic Hardy field extension, and some of its consequences. Time permitting I will also discuss an ongoing project aimed at including information about the domains of convergence for holomorphic extensions. (Joint with Lou van den Dries.)

**Lou van den Dries**

*Title:* What is ω-freeness?*Abstract:* (Joint work with Aschenbrenner and Van der Hoeven) The notion of ω-freeness is crucial in our work on transseries and Hardy fields, but may seem a bit technical. Fortunately, it has a very attractive and natural definition in the setting of Hardy fields. This is what my talk is about. The historical origin is in theorems by Sturm (1836) on oscillating solutions of 2nd-order homogeneous linear differential equations. This gave rise to an enormous literature. In the setting of Hardy fields the first-order concept of ω-freeness yields a striking transparency, and clarifies issues around our quantifier elimination for maximal Hardy field.

**Pantelis Eleftheriou***Title:* On the global Zarankiewicz's problem

Abstract: The global Zarankiewicz's problem for hypergraphs asks for an upper bound on the number of edges of a hypergraph V, whose edge relation is induced by a fixed hypergraph E that has no sub-hypergraphs of a given size. In [Basit-Chernikov-Starchenko-Tao-Tran], linear Zarankiewicz bounds were found in the case of a semilinear E, namely E definable in a linear o-minimal structure. We establish versions of the same bounds in five new settings: when E is definable in (a) a semibounded o-minimal structure, (b) a model of Presburger Arithmetic, (c) the expansion (R, <, +, Z) of the real ordered group by the set of integers, (d) a stable 1-based structure without the finite cover property, and (e) a regular type in a stable theory, such as the locus of the generic type of the solution set of the Heat differential equation. This talk will explain some of the work involved in the above settings and in particular any methods they have in common.

This is joint work with Aris Papadopoulos.

**Immanuel Halupczok**

*Title:* Hensel minimality*Abstract:* Ever since o-minimality has become such a successful axiomatisation of "tame" real closed fields, there have been attempts to come up with an analogue for (different classes of) valued fields, notably P-minimality, C-minimality, and, more recently, V-minimality and b-minimality. However, each of those notions had some drawback (mostly being less general and/or less powerful than desired). I will present the latest attempt, 1-h-minimality, which on the one hand is quite general and on the other hand has powerful consequences. I will in particular explain in which sense this is a natural adaptation of o-minimality. This is joint work with Raf Cluckers, Silvain Rideau and Floris Vermeulen.

**Martin Hils**

*Title: *The amalgamation property for definable types*Abstract: *When generalizing Poizat's theory of belles paires to the unstable context, one needs to assume the amalgamation property for the class of global definable types. In the talk, we will highlight this connection, exhibit some important classes of theories in which definable types may be amalgamated, and exhibit examples of theories in which they don't. This talk is based on joint work with Pablo Cubides Kovacsics and Jinhe Ye, and with Rosario Mennuni.

**Franziska Jahnke**

*Title: AKE Principles in Mixed Characteristic*

*Abstract:* A classical theorem going back to Ax-Kochen and Ershov shows that the theory of an unramified henselian valued field with perfect residue field reduces to those of its residue field and value group. Notably, this gives rise to a model-theoretic understanding of the p-adic numbers. In this talk, we discuss a range of further such transfer principles in valued fields of mixed characteristic, on the one hand generalizing the classical case to finite ramification and imperfect residue fields, and on the other obtaining strengthenings of Kuhlmann's results for tame fields. A special focus lies on the role of the "core field", i.e., the rank-1 step where the characteristic shift happens. The talk contains results obtained in joint work with Anscombe, Dittmann, Kartas, and Ketelsen.

**Piotr Kowalski**

*Title:* Model completeness and semisimple groups*Abstract: *This joint work with Daniel Max Hoffmann, Chieu-Minh Tran, and Jinhe Ye. We aim to show that if G is a semisimple algebraic group over a model complete field K, then the (pure) group of K-rational points G(K) is model complete. I will report on progress in this project and some related questions.

**Lothar Sebastian Krapp**

*Title:* Embedding the prime model of real exponentiation*Abstract:* An ordered exponential field (K,E) is an expansion of an ordered field K by an order-preserving isomorphism E from the additive group of K to the multiplicative group of positive elements of K. Due Macintyre and Wilkie's groundbreaking work in the 1990s, the theory of real exponentiation, i.e. of the first-order theory of the real exponential field (R,exp), is o-minimal and – under the assumption of Schanuel's Conjecture – decidable.

Building on these results, the question arose whether the prime model of real exponentiation is an elementary substructre of any o-minimal ordered exponential field. In this talk, I will present the proof of the main result of [1] that, under the assumption of Schanuel's Conjecture, the prime model of real exponentiation embeds as a (not necessarily elementary) substructure into any o-minimal ordered exponential field. This is accomplished by an application of Kőnig's Infinity Lemma to a construction of the prime model of real exponentiation via solutions of Khovanskii Systems.

[1] L. S. Krapp, ‘Embedding the prime model of real exponentiation into o-minimal exponential fields’, Bull. Lond. Math. Soc. 56 (2024) 907–913.

**Jana Marikova**

*Title:* An o-minimal Følner condition*Abstract:* Suppose G is a pseudogroup acting on a set X. In [1], Ceccherini-Silberstein,

Grigorchuk and de la Harpe gave a Følner condition for (G, X), equivalent to theexistence of a G-invariant probability measure on P(X). In the o-minimal setting, when G is a pseudogroup of definable measure-preserving maps acting on a bounded definable set X with non-empty interior, we are interested in a Følner condition equivalent to the existence of a G-invariant probability measure on the definable subsets of X. We propose a candidate for such a Følner condition, discuss how much of the proof in [1] simplifies in our setting, and show how the desired equivalence would follow from a certain property of invariant measures in o-minimal structures. This is work in progress.

References

[1] T. Ceccherini-Silberstein, R. I. Grigorchuk, P. de la Harpe, Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. Trudy Mat. Inst. Steklova 2243: 68-111 (1999)

**Patrick Speissegger**

*Title:* Definability questions for o-minimal expansions of the real field*Abstract:* In a series of papers on the expansion of the real field by all restricted analytic functions and the exponential function, Van den Dries, Macintyre and Marker pioneered a method for deciding whether a given function is definable or not in this structure. Their arguments are based on an embedding theorem of the Hardy field of definable univariate germs into the field of transseries. In collaboration with Rolin and Servi, we obtain such an embedding theorem for the expansion of the real field by any generalized quasianalytic class and the exponential function. This helps us decide, for instance, the non-definability of Gamma in the expansion of the real field by Zeta, and vice-versa. Other types of definability, such as the definability of the complex Gamma function on suitable domains, have also recently become of interest. I will give an overview of what we know (joint work with Padgett) and what we are currently working on (joint project with Binyamini).

**Alex Wilkie**

*Title:* Analytic Continuation and Zilber's Quasiminimality Conjecture*Abstract:* This is the title of a paper accepted for the Zilber 75th birthday volume. The conjecture states that every set of complex numbers definable in the complex exponential field is either countable or co-countable. I shall discuss my approach to the problem as set out in that paper.

**Martin Ziegler**

*Title:* Quantifier elimination for pairs of algebraically closed fields*Abstract:* (joint work with Amador Martin-Pizarro)

Pairs of algebraically closed fields have QE with down to tame

formulas. We will show that tame definable sets have the DCC. Furthermore:

* The Morley rank of a definable set equals the Morley rank of its tame

closure.

* The Morley rank of an irreducible tame definable set equals its

foundation rank.

## Contact:

tobias.kaiser@uni-passau.de