Talks winter 2024/25
10 Oct - Sam Shepherd. One-ended halfspaces in group splittings
I will introduce the notion of halfspaces in group splittings and discuss the problem of when these halfspaces are one-ended. I will also discuss connections to JSJ splittings of groups, and to determining whether groups are simply connected at infinity. This is joint work with Michael Mihalik.
17 Oct - Silke Meißner. 2-ample theories and pseudo-buildings
Zilber showed the following for strongly minimal theories T: if T is 1-ample, or equivalently non-linear, then T interprets a pseudoplane. I will show a way in which this result can be generalised to theories which are 2-ample, or equivalently not CM-trivial. We will call the corresponding interpretable objects pseudo-buildings, where the notion of a building was originally introduced by Tits in order to study algebraic groups. This is joint work with Katrin Tent.
24 Oct - Floris Vermeulen. Uniform upper bounds for rational points on varieties
Given a variety over the rational numbers, a central objective in number theory is a good understanding of its set of rational points. For applications it is often useful to have upper bounds on counting rational points which are largely independent of the variety itself. For projective varieties, a general uniform upper bound was conjectured by Heath-Brown and Serre, and is now known in almost all cases by work of Browning, Heath-Brown and Salberger.
I will give a general introduction to this topic of counting rational points, and discuss some recent work on dimension growth for affine varieties. This is partially joint work with Raf Cluckers, Pierre Dèbes, Yotam Hendel and Kien Nguyen.
31 Oct - Tingxiang Zou. Elekes-Szabó -- beyond the group configuration theorem
In this talk, I will review the Elekes-Szabó theorem and its proof, with the emphasis on the use of the group configuration theorem. I will then talk about a recent work joint with Martin Bays on generalisations of the Elekes-Szabó theorem to asymmetric and higher dimensional cases, and mention how to go beyond the group configuration theorem to obtain these results. As an application, we obtain asymmetric and higher dimensional generalisations of the Elekes-Ronyai theorem.
4 Nov - Sam Hughes. On finite quotients of discrete groups. Monday, 14:15-15:15 Joint with the topology seminar
Room: M3, see also https://www.uni-muenster.de/Topologie/researchseminars/ostoposs2425.html
In this talk I will survey a number of recent results regarding (relative) profinite rigidity of certain groups (3-manifold groups, Coxeter groups, free-by-cyclic groups, Kaehler groups). Here profinite rigidity asks how much of information about a finitely generated residually finite group can be recovered from its finite quotients. From an algebraic geometry viewpoint this is essentially asking when the algebraic fundamental group determines an aspherical projective variety up to biholomorphism (assuming residual finiteness of the topological fundamental group). Much of the input will come from developments around the world of 3-manifold topology, building on the Virtual Fibring Theorem of Agol and Wise. With this in hand (and time permitting) I will discuss work of Wilton—Zalesskii, Wilkes, and Liu on rigidity amongst 3-manifold groups, work of myself and Kudlinska on rigidity amongst freeby-cyclic groups, and work of myself, Llosa Isenrich, Py, Spitler, Stover, and Vidussi on rigidity amongst Kaehler groups.
7 Nov - Alf Onshuus. Lie groups and o-minimality
It has been known for some time that any group definable in an o-minimal expansion of the real field can be endowed definably with the structure of a Lie group, and that any definable homomorphisms between definable groups is a Lie homomorphism (under the above mentioned Lie structure).
In this talk we explore the converse: We will give a brief characterization of when a Lie group has a Lie isomorphic group which is definable in an o-minimal expansion of the real field, and concentrate then on when Lie isomorphisms, hopefully focussing on two questions:
- That any Lie isomorphism between definable groups can be added to the language without losing o-minimality and
- Give a brief overview and some interesting open questions about "non definable Lie isomorphic definable groups".
14 Nov - Kai-Uwe Bux. The Boone–Higman conjecture for groups acting on locally finite trees
(joint with Xiaolei Wu and Claudio I. Llosa)
We develop a method for proving the Boone-Higman Conjecture for groups acting on locally finite trees. As a consequence, the Boone-Higman Conjecture holds for all Baumslag-Solitar groups and for all free(finite rank)-by-cyclic groups. This resolves two cases that have been raised explicitly by Belk, Bleak, Matucci and Zaremsky.
21 Nov - Vincent Bagayoko. Equations over valued groups
Solving equations over groups while satisfying first-order conditions is difficult, yet necessary when studying elementary classes of expansions of groups. For certain groups coming from pure group theory, o-minimality or commutative algebra, we may rely on valuations that behave well with respect to commutativity on such groups. I will explain how to use valuations in order to solve equations over groups.
28 Nov - Sylvy Anscombe. The core of a valued field: molten or solid?
Ax, Kochen, and Ershov found the complete axiomatization of the theory of the field Qp of p-adic numbers, for each prime p. A proof of the completeness of their axiomatization usually goes through two stages: first an analysis of the model theory of henselian valued fields of equal characteristic zero, then a smidgen of structure theory of CDVRs in mixed characteristic. The same pattern applies more generally to axiomatizations of the theories of finitely ramified valued fields. The role of the CDVRs at the core of such theories is played in the infinitely ramified case by a different kind of valued field: maximal and R-valued. Moreover, the theory of these core (valued) fields is an invariant of the theory of the original valued field. When their residue fields are perfect, these are tame and some known model theory may be applied, although there are still plenty of open questions, including a "composition AKE" problem. In the imperfect case, things are yet wilder. This is ongoing joint work with Jahnke and Ketelsen.
5 Dec - Dana Bartosova. Strength of Ramsey theorems
It is often the case that one Ramsey problem is solved by encoding a problem in one already solved and transferring the solution back. In that vein, one can consider the latter Ramsey theorem to be stronger that the former. In recent years, there have been attempts to find general notions for such a procedure. One of them, a semi-retraction, was introduced by Lynn Scow, whose primary motivation was a study of generalized indiscernible. Together with Scow, we found the most general conditions on countable structures under which a semi-retraction transfers the Ramsey property, and more generally Ramsey degrees, from one structure to another. We will compare this notion to the category theoretic notion of pre-adjunction studied by Dragan Mašulović.
12 Dec - Blaise Boissonneau. AI (disambiguation)
(joint with Franziska Jahnke, Anna de Mase, Pierre Touchard)
In 2011, Cluckers and Halupczok, building from work of Gurevitch and Schmitt, showed that the theory of ordered abelian groups (OAG) admit quantifier elimination down to their spines. Building in turn on their work, we discuss "structural" questions about OAG equipped with their spines: not all substructures of an OAG equipped with its spines are themselves OAG equipped with spines. However, convex subgroups are always well behaved in this regard. Continuing our study, we show that all OAG are AI, where AI stands of course for augmentable at infinity: for any OAG Γ, there exists an OAG Δ such that Γ is an elementary substructure of the direct sum Δ⊕Γ, lexicographically ordered. In other words, there is an elementary extension of Γ containing Γ itself as a convex subgroup.
9 Jan - Shujie Yang. Projective Fraïssé Limits of Trees
We will discuss the application of projective Fraïssé theory to finite rooted trees. By focusing on specific classes of epimorphisms between these structures, we construct projective Fraïssé limits that lead to interesting continua, including the Mohler-Nikiel universal dendroid, the Ważewski dendrite, and a new continuum that has yet to be fully characterized.
We also introduce the present partial result of the Ramsey property of classes, which, through the Kechris-Pestov-Todorcevic (KPT) correspondence, relates to topological dynamics properties and allow us to calculate the universal minimal flows of automorphism groups of continua.
This talk is based on joint work with W. Charatonik, A. Kwiatkowska, and R. Roe.16 Jan - Eduardo Silva. Poisson boundaries of Baumslag-Solitar groups
The Poisson boundary of a countable group G endowed with a probability measure μ is a probability space that encodes all bounded μ-harmonic functions on G. Alternatively, it captures the asymptotic directions of the μ-random walk on G. A natural problem is to identify an explicit model of the associated Poisson boundary, described in terms of the geometry of the group G.
I will discuss the identification problem of Poisson boundaries for random walks with finite entropy on Baumslag-Solitar groups. Our results are expressed in terms of both the action of a Baumslag-Solitar group on its Bass-Serre tree and the action by affine transformations on the rational numbers Q. These results generalize the work of Kaimanovich and Cuno-Sava-Huss to a setting where no conditions are imposed on the moments of the measures. This is joint work with Kunal Chawla.
23 Jan - Radhika Gupta
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