11 Apr - Kevin Iván Piterman. Title tba
18 Apr - Thomas Koberda. Title tba
25 Apr - Jerónimo García-Mejía. Title tba
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. A group action version of the Elekes-Szabó Theorem
I will show how to strengthen the Elekes-Szabó result, that any ternary algebraic relation in characteristic 0 having large intersections with (certain) finite grids must essentially be the graph of a group law, by weakening the combinatorial hypothesis to an asymmetric situation modelled on a group action; the proof will go via first obtaining then disposing of an algebraic group action. This is recent work with Tingxiang Zou.
18 Jan - Anna de Mase. Value groups of finitely ramified henselian valued fields and model completeness
A result obtained by J. Derakhshan and A. Macintyre states that the theory of a mixed characteristic henselian valued field with finite ramification, and whose value group is a Z-group, is model complete in the language of rings if the theory of the residue field is model complete in the language of rings. In this talk, we will see how this result can be generalized to mixed characteristic henselian valued fields with finite ramification, but with different value groups. We will address the case in which the value group is an ordered abelian group with finite spines, and (if time permits) the case in which it is elementarily equivalent to the lexicographic sum of Z with a minimal positive element. In both cases, we give a one-sorted language (expansion of the language of rings) in which the theory of the valued field is model complete if the theory of the residue field is model complete in the language of rings.
25 Jan - Zahra Mohammadi. Model theoretic aspects of the free factor complex
Free Factor Complexes are important objects in Geometric Group Theory. We are studying the model theoretic aspects of these complexes. Bestvina and Bridson showed that the automorphism group of the free factor complex is naturally isomorphic to the automorphism group of the free group, respectively Out(Fn) for the complex of conjugacy classes of free factors. We obtained a model theoretic proof for this result and proved that the free factor complexes are homogeneous in the sense of model theory. Our next aim is to construct saturated models of these complexes in order to study them from the point of view of stability.
A list of talks that took place between 2014 and Summer 2023 can be found here.