L6

A trace on a group is a positive-definite conjugation-invariant function on it. In the past couple of decades, the study of traces has led to exciting connections to the rigidity, stability, and dynamics of groups. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will see the different behaviours of these spaces for free groups vs. higher-rank lattices. This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.

Let X be a n by n unitary matrix, drawn at random according to the Haar measure on U_n, and let m be a natural number. What can be said about the distribution of X^m and its eigenvalues? 

The density of the distribution \tau_m of X^m can be written as a linear combination of irreducible characters of U_n, where the coefficients are the Fourier coefficients of \tau_m. In their seminal work, Diaconis and Shahshahani have shown that for any fixed m, the sequence (tr(X),tr(X^2),...,tr(X^m)) converges, as n goes to infinity, to m independent complex normal random variables (suitably normalized). This can be seen as a statement about the low-dimensional Fourier coefficients of \tau_m. 

In this talk, I will focus on high-dimensional spectral information about \tau_m. For example: 

(a) Can one give sharp estimates on the rate of decay of its Fourier coefficients?

(b) For which values of p, is the density of \tau_m  L^p-integrable? 

Using works of Rains about the distribution of X^m, we will see how Item (a) is equivalent to a branching problem in the representation theory of certain compact homogeneous spaces, and how (b) is equivalent to a geometric problem about the singularities of certain varieties called (Weyl) hyperplane arrangements.

Based on joint works with Julia Gordon and Yotam Hendel and with Nir Avni and Michael Larsen.

How to uniformly, or at least almost uniformly, choose an element from a finite group G? When G is too large to enumerate all its elements, direct (pseudo)random selection is impossible. However, if we have an explicit set of generators of G (e.g., as in the Rubik's cube group), several methods are available. This talk will focus on one such method based on the well-known product replacement algorithm. I will discuss how recent results on property (T) by Kaluba, Kielak, Nowak and Ozawa partially explain the surprisingly good performance of this algorithm.

to follow

Profinite rigidity explores the extent to which non-isomorphic groups can be distinguished by their finite quotients. Many interesting examples of this phenomenon arise in the context of group extensions—short exact sequences of groups with a fixed kernel and quotient. This talk will outline two main mechanisms that govern profinite rigidity in this setting and provide concrete examples of families of extensions that cannot be distinguished by their finite quotients.

The talk is based on my DPhil thesis.

In this talk, I will introduce fusion categories as categorical versions of finite rings. We will discuss some examples which may already be familiar, like the category of representations of a finite group and the category of vector spaces graded over a finite group. Then, we will define Tambara-Yamagami categories, which are a certain type of fusion categories which have one simple object which is non-invertible. I will provide the classification results of Tambara and Yamagami on these categories and give some small examples. Time permitting, I will discuss current work in progress on how to generalize Tambara-Yamagami fusion categories to locally compact groups. 

This talk will not assume familiarity with category theory further than the definition of a category and a functor.

In 1911, Otto Toeplitz posed the intriguing "Square Peg Problem," asking whether every Jordan curve admits an inscribed square. Despite over a century of study, the problem remains unsolved in its full generality. However, significant progress has been made over the years. In this talk, we explore recent advancements by Andrew Lobb and Joshua Greene, who approach the problem through the lens of Lagrangian Floer homology. Specifically, we outline a proof of their result: every smooth Jordan curve inscribes every rectangle up to similarity.

In operator algebra theory central sequences have long played a significant role in addressing problems in and around amenability, having been used both as a mechanism for producing various examples beyond the amenable horizon and as a point of leverage for teasing out the finer structure of amenable operator algebras themselves. One of the key themes on the von Neumann algebra side has been the McDuff property for II_1 factors, which asks for the existence of noncommuting central sequences and is equivalent, by a theorem of McDuff, to tensorial absorption of the unique hyperfinite II_1 factor. We will show that, for a topologically free minimal action of a countable amenable group on the Cantor set, the von Neumann algebra of the associated dynamical alternating group is McDuff. This yields the first examples of simple finitely generated nonamenable groups for which the von Neumann algebra is McDuff. This is joint work with Spyros Petrakos.

Embedding the group algebra into a division ring has proven to be a powerful tool for detecting structural properties of the group, especially in relation to its homology. In this talk, we will show how division rings can be used to identify residual properties of groups, one-ended groups, and coherent groups. We will place special emphasis on the class of free-by-cyclic groups to provide a clear, explicit exposition.

This talk is rescheduled and will take place on 21 January 2025