# Danube Rhine Model Theory Seminar

## Danube Rhine Model Theory Seminar

The Danube Rhine Model Theory seminar was founded on Salma Kuhlmann's initiative from the University of Konstanz and is organized by Philipp Habegger (University of Basel), Amador Martin-Pizarro (University of Freiburg), Salma Kuhlmann (University of Konstanz), and Tobias Kaiser (University of Passau). It gives a joint institution for the reserachers in model theory and its applications from the upper-german language area. It happens once or twice a year at changing places. Talks are delivered by international guests and young scientists.

The next seminar will take place at Passau.**Termin:** Tuesday, February 14th 2023, and Wednesday, February 15th 2023**Programm:**

Tuesday, February 14th:

19:00: Dinner

Wednesday, February 15th:

Ort: University of Passau, SR 034

09:30 - 10:20: Martin Hils (Münster)

10:30 - 11:20: Francesco Gallinaro (Freiburg)

11:30 - 12:30: Mittagspause (Mensa)

12:30 - 13:20: Vincent Bagayako (Konstanz)

13:30 - 14:20: Matthias Aschenbrenner (Vienna)**Titels und Abstracts:**

**Vincent Bagayoko (Konstanz)**

Title: Some ordered groups of generalized power series

Abstract: I will introduce an elementary class of ordered groups, of which o-minimal geometry and ordered differential algebra provide natural examples. I will show how to use certain generalized power series, similar to transseries, in order to answer an open question regarding orderable groups.

**Francesco Gallinaro (Freiburg)**

Title: Quasiminimality of Complex Powers

Abstract: A conjecture due to Zilber predicts that the complex exponential field is quasiminimal: that is, that all subsets of the complex numbers that are definable in the language of rings expanded by a symbol for the complex exponential function are countable or cocountable. Zilber showed that this conjecture would follow from Schanuel's Conjecture and an existential closedness-type property asserting that certain systems of exponential-polynomial equations can be solved in the complex numbers; later on, Bays and Kirby were able to remove the dependence on Schanuel's Conjecture, shifting all the focus to the existence of solutions. In this talk, I will discuss recent work about the quasiminimality of a reduct of the complex exponential field, that is, the complex numbers expanded by multivalued power functions. This is joint work with Jonathan Kirby.

**Martin Hils (Münster)**

Title: Lang-Weil type bounds in finite difference fields

Abstract: We establish Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that, working in any non-principle ultraproduct K of finite difference fields, the normalized pseudofinite dimension of a quantifier free partial type p is equal to its transformal dimension, i.e., to the maximal transformal transcendence degree over K of a realization of p.

The proof uses a strong form of the Lang-Weil estimates (due to Cafure and Matera) and, as key ingredient to obtain equidimensional Frobenius specializations, the recent work of Dor and Hrushovski on the non-standard Frobenius acting on an algebraically closed non-trivially valued field, in particular the pure stable embeddedness of the residue difference field in this context.

This is joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou.**Matthias Aschenbrenner (Vienna)**

Titel: A uniform framework for analytic Nullstellensätze

Abstract: I shall present a technique for establishing Nullstellensätze in rings of power series using model theory. Our approach, in contrast to others, relies on quantifier elimination theorems for valued fields.

This leads to new Nullstellensätze and Hilbert’s 17th Problem-type results for p-adic power series, both formal, convergent, and restricted. (Joint work with A. Srhir.)

In the winter 2020 the seminar took place at the University of Konstanz (24. January).

http://www.math.uni-konstanz.de/~krapp/DRMTA5/index.html

In the winter 2018 the seminar took place at the University of Freiburg (30. November).

https://sites.google.com/view/drmta-4th-meeting/

In the summer 2018 the seminar took placeat the University of Basel (13. July).

https://sites.google.com/site/drmta3/

In the Winter 2017/2018 it was hosted by the University of Passau.**Date:** Friday, December 1st 2017, and Saturday, December 2nd 2017.**Programme:**

Friday, December 1st:

19:00: Dinner (Heilig Geist Stiftschenke)

Saturday, December 2nd

Ort: University of Passau, HS 12

09:00 - 09:50: Christian Urech (Basel)

10:00 - 10:50: Zaniar Ghadernezhad (Freiburg)

11:00 - 11:50: Merlin Carl (Konstanz)

12:00 - 13:00: Lunch hour (Restaurant Akropolis Athen)

13:00 - 13:50: Piotr Kowalski (Wroclaw)

14:00 - 14:50: Gabriel Dill (Basel)**Titles und Abstracts:**

Merlin Carl (Konstanz)

Titel: Real closed fields, models of arithmetic and exponentiation

Abstract: Roughly, an integer part of a real closed field K is a subring that relates to K in a similar way that **Z** relates to** R**. By a classical theorem of Shepherdson, integer parts of real closed fields coincide with models of open induction, a weak fragment of arithmetic. This motivates the question how models of stronger fragments of arithmetic relate to real closed fields of which they are integer parts, which was started by D'Aquino, Knight and Starchenko and then carried on by Marker, Schmerl, Steinhorn, Kuhlmann and others. In this talk, we will show that real closed fields with an integer part that is a model of Peano arithmetic allow a weak form of exponentiation, called "left exponentiation" (which was proved in joint work with S. Kuhlmann and P. D'Aquino, building on work of D. Marker) and that, conversely, models of true arithmetic are integer parts of real closed fields that are elementary equivalent to the reals with exponentiation.

Gabriel Dill (Basel)

Titel: Unlikely intersections between isogeny orbits and curves

Abstract: In the spirit of the Mordell-Lang conjecture, we consider the intersection of a curve in a family of abelian varieties with the images of a finite-rank subgroup of a fixed abelian variety A_0 under all isogenies between A_0 and some member of the family. After excluding certain degenerate cases, we can prove that this intersection is finite by applying the Pila-Zannier method with a variant of the Pila-Wilkie theorem due to Habegger and Pila. This proves a conjecture of André-Pink-Zannier in the case of curves. We can even allow translates of the finite-rank subgroup by abelian subvarieties of controlled dimension if we strengthen the degeneracy hypotheses suitably.

Zaniar Ghadernezhad (Freiburg)

Title: Non-amenablility of automorphism groups of generic structures

Abstract:

The study of amenable groups is originated in the works of von Neumann in his analysis of Banach-Tarski paradox. Since then amenability, non-amenability and paradoxicality has been studied for various groups appearing in different parts of mathematics. A topological group G is amenable if every G-flow has an invariant Borel probability measure. Well-known examples of amenable groups are finite groups, solvable groups and locally compact abelian groups. The study of amenability of topological groups benefit from various viewpoints that ranges from analytic approach to combinatorial. Kechris, Pestov, and Todorcevic established a very general correspondence which equates a stronger form of amenability, called extreme amenability, of the automorphism group of an ordered Fraisse structure with the Ramsey property of its finite substructures. In the same spirit Moore showed a correspondence between the automorphism groups of countable structure and a a structural Ramsey property, which englobes Fölner's existing treatment. In this talk we will consider automorphism groups of certain Hrushovski's generic structures. We will show that they are not amenable by exhibiting a combinatorial/geometric criterion which forbids amenability.

Piotr Kowalski (Wroclaw)

Title: Virtually free groups and Galois actions

Abstract:

I will talk about joint work with Özlem Beyarslan. We show that for a finitely generated virtually free group G, the theory of actions of G on fields has a model companion, which we call G-TCF. We also give an algebraic condition on G, which is equivalent to simplicity of the theory G-TCF.

Christian Urech (Basel)

Titel: Subgroups of elliptic elements of the Cremona group

Abstract: The Cremona group is the group of birational transformations of the complex projective plane. This classical topic has recently again turned into a popular subject of research and in the last decade many classicalproblems were solved with modern tools from birational geometry and dynamics. In this talk I will explain how a basic application of the compactness theorem allows the classification of groups of elliptic elements of the Cremona group - a class of groups that hasn't been understood well so far. In particular, one can deduce from this classification the Tits alternative for the Cremona group. We will also see that every simple subgroup of the Cremona group is isomorphic to a simple subgroup of PGL(3,C).

In the Summer 2017 it took place at the University of Konstanz.