# Vorträge im Wintersemester 2023/24

### 12 Oct - Rosario Mennuni. Some definable types in the wild

While definable types are usually studied in "tame" contexts, their usefulness and amenability to model-theoretic investigation even "in the wild" is, historically, not a surprise: for instance, Lascar defined the tensor product of definable types by generalising the existing notion on ultrafilters, which may be viewed as (trivially) definable types in the richest possible language on a given set.

By pushing the tensor product forward along the addition, one shows that the usual sum of integers may be extended to the space of ultrafilters over Z, yielding a compact right topological semigroup. The analogous construction also goes through for the product, and these facts had important applications in additive combinatorics and Ramsey theory.

Recently, B. Šobot introduced two (ternary) notions of congruence on the space above. I will talk about joint work with M. Di Nasso, L. Luperi Baglini, M. Pierobon and M. Ragosta, in which the study of these congruences led us to isolate a class of ultrafilters enjoying characterisations in terms of tensor products, directed sets, profinite groups, and more.### 19 Oct - Alessandro Codenotti. Ranks for tame dynamical systems and model theory

By slightly generalizing a notion of rank originally introduced in the context of metric tame dynamical systems by Glasner and Megrelishvili, we define an ordinal valued rank for the action of Aut(M) on the space of types over M, where M is a model of some theory T. We then investigate the relationship between this rank and the dividing lines in the model theoretic hierarchy, in particular we characterize NIP theories as those with non-infinite rank and stable theories as those with rank 0. This is joint work with Daniel Max Hoffmann.

### 2 Nov - Benjamin Brück. Top-degree cohomology in the symplectic group of a number ring

I will indicate how one can use Tits buildings to show that the "top-degree" cohomology of Sp2n(R), the symplectic group over a number ring R, depends on number theoretic properties of R:

In recent work with Himes, we proved that Sp2n(R) has non-trivial rational cohomology in its virtual cohomological dimension if R is not a principal ideal domain. We gave a lower bound for the dimension of these cohomology groups in terms of the class number of R. This contrasts results joint with Santos-Rego-Sroka that show that the top-degree cohomology group of Sp2n(R) is trivial if R is Euclidean.

Both of these results have counterparts in the setting of SLn(R) that were established by Church-Farb-Putman. A key ingredient in all of this is the action of these groups on associated Tits buildings.### 9 Nov - Marco Amelio. Non-split sharply 2-transitive groups in odd characteristic

Until recently, the existence of non-split sharply 2-transitive groups (i.e., sharply 2-transitive groups without a normal abelian subgroup) was an open problem. The first examples of such groups were exhibited by Rips, Segev and Tent in 2017 and by Rips and Tent in 2019. It is possible to associate to every sharply 2-transitive group a characteristic that is either 0 or a positive prime number. The first of these examples were in characteristic 2, while the others were in characteristic 0, leaving the problem open for odd characteristics. In this talk, I will outline recent progress made in adapting the construction in characteristic 0 to build examples of non-split sharply 2-transitive groups in odd characteristic using methods of geometric small cancellation. I will also give a rough explanation of how these methods relate to the usual small cancellation conditions for group presentations. This is joint work with Simon André and Katrin Tent.

### 23 Nov - Yuri Santos Rego. Reflection groups and their finite quotients

There has been growing interest in the following question: to what extent do finite quotients of a given group determine its structure? For instance, do profinite completions detect specific properties, isomorphism types, or elementary theories?

In this talk we shall review some concepts and known results and problems around the above mentioned question, and then shift focus to the state of knowledge for the family of Coxeter groups. Based on joint work with Petra Schwer.### 30 Nov - Simone Ramello. Definable henselian valuations in positive residue characteristic

**Takes place in 120.029 / 120.030**(second floor Orleansring 10)!Jahnke and Koenigsmann started a classification of when a henselian valuation is definable on a henselian field, providing a full characterization for the case where the canonical henselian valuation has residue characteristic zero. We extend their work to the case where the residue characteristic is positive, drawing on the toolkit of independent defect to summon a definable henselian valuation out of certain defect extensions. This is joint work with Margarete Ketelsen (Münster) and Piotr Szewczyk (Dresden).

### 7 Dec - Katrin Tent. On the model theory of free and open generalized polygons

We show that for any \(n\geq 3\) the theory of free generalized \(n\)-gons is complete and strictly stable yielding a new class of examples in the zoo of stable theories exhibiting many of the properties of free groups with very elementary proofs.

Joint work with A.-M. Ammer.

### 14 Dec - Martin Bays. TBA

TBA