Abstract: We study how a homogeneous structure M can be randomly expanded to a larger language in an Aut(M)-invariant manner. We show that under certain conditions, such an expansion is not just Aut(M)-invariant but fully Aut((N, =))-invariant, which allows us to classify such expansions. The problem of classifying Aut(M)-invariant Keisler measures may be seen as a special case of this problem, and the resulting classifications of Aut(M)-invariant Keisler measures yield many natural examples of very tame simple theories with non-forking formulas that are universally measure zero. This is joint work with Colin Jahel and Paolo Marimon, based on https://arxiv.org/abs/2408.08370
Abstract: Ramsey’s theorem is an often overlooked result in mathematical logic which resolves a special case of the Entscheidungsproblem. That being said, some somewhat more influential “theorems on combinations" appear in the first section of Ramsey’s paper, and from these “theorems on combinations” a beautiful branch of pure mathematics (Ramsey theory) was formed (largely by Erdős, at least according to Graham and Nešetřil). Now that I (hopefully) have your attention, this will broadly be a survey talk, in which I will discuss some results that appear in joint works with Meir (https://arxiv.org/abs/2212.08027 and https://arxiv.org/abs/2307.14468), and Meir and Touchard (https://arxiv.org/abs/2307.14468). The goal of the talk is to illustrate some rather interesting connections between model theory and (structural) Ramsey theory.
Abstract: Tannaka and Krein independently established around 1940 a duality theory in abstract harmonic analysis: any compact group can be fully recovered from the data contained in its unitary representations. These results can be seen as an analogue of the Pontryagin--van-Kampen duality for locally compact abelian groups.
After an introduction on abstract harmonic analysis, where I will go over the mentioned duality theories, I will explain how Tannaka's and Krein's results can be extended to Roelcke-precompact non-archimedean Polish groups (think Aut(M) for an ω-categorical first order structure M). Along the way, we obtain two realizations of the Hilbert compactification of such a group.
Abstract: In response to a question of Yongle Jiang, we show that a measurable map between Polish groups that is multiplicative on a large set must be identical to an actual homomorphism on a large set. We also discuss an automatic continuity result for Baire measure cocycles, which is analogous with a recent result of Meyerovich and Solan for Haar measurable cocycles.
Abstract: Given two Fraisse-like classes with generic limits, we ask whether we can merge the two classes into one class with a generic limit. We then study the properties of these merges and their generics, as well as their connections to structural Ramsey theory and the Hrushovski property (EPPA).
Abstract: A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: `How “sparse" must a given graph be, if I know that it has no “dense” subgraphs?’. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk.
So far so good, but this is a logic seminar and the title says the words “Model Theory"… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz’s problem in “geometric” contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz’s problem, when we restrict our attention to semilinear/semibounded o-minimal structures, Presburger arithmetic, and various kinds of Hrushovski constructions.
The new results that will appear in the talk were obtained jointly with Pantelis Eleftheriou.
Abstract: We introduce the notion of a weak A2 space (or just wA2-space). wA2 spaces satisfy a part of Todorčević's axioms for topological Ramsey spaces, and is also a generalisation of countable vector spaces. It turns out that the abstract Kastanas game (introduced in 2023 by Cano and Di Prisco) on wA2-spaces serves as an intersection of Ramsey theory of topological Ramsey spaces, and the study of strategically Ramsey subsets of countable vector spaces. In this talk, I will discuss some properties of Kastanas Ramsey subsets of wA2-spaces, and use them to give quick proofs of classical results of Ramsey subsets of topological Ramsey spaces, and of strategically Ramsey subsets of countable vector spaces.
Abstract: In joint work with Christian Rosendal, we investigate a notion of asymptotically spherical topological groups, which says that spheres of large radius with respect to any maximal length function are still spherical with respect to any other maximal length function. This is a strengthening of a related condition introduced by Sebastian Hurtado, which we call bounded eccentricity. Our main result is a partial characterization of which groups are asymptotically spherical, and we also give an example of a discrete, bounded eccentric group who fails to be asymptotically spherical. In this first of two talks (the second of which will be given by Christian Rosendal next week), we will motivate and define the aforementioned notions, present the main results, and begin to discuss the proofs.
Abstract: This is the continuation of Jenna Zomback's talk on our joint work on asymptotically spherical groups.
The fundamental observation of geometric group theory states that a large class of topological groups ,such as f.g. discrete, compactly generated locally compact or monogenic Polish groups, carry an inherent quasimetric structure. This means that the various metrics defining this structure are biLipschitz for large distances. However, for the subclass of asymptotically spherical groups defined in the first talk, the different metrics are asymptotically dilations of each other.
We will present a sufficient condition for a topological group to be asymptotically spherical and also present a partial converse that indicates that this is close to being necessary.
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