C3

TBC

We show that any finite coloring of an amenable group contains 'many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables.  This gives the first combinatorial proof and extensions of Bergelson and McCutcheon's non-commutative Schur theorem.  Our main new tool is the introduction of what we call `quasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.

We give a gentle introduction to the theory of self-similar sets and self-similar measures. Connections of this topic to Diophantine approximation on Lie groups as well as to additive combinatorics will be exposed. In particular, we will discuss recent progress on Bernoulli convolutions. If time permits, we mention recent joint work with Samuel Kittle on absolutely continuous self-similar measures. 
 

Heegner points are a powerful tool for understanding the structure of the group of rational points on elliptic curves. In this talk, I will describe these points and the ideas surrounding their generalisation to other situations.

We propose a natural version of Connes' Rigidity Conjecture (1982) that involves property (T) groups with infinite centre. Using methods at the rich intersection between von Neumann algebras and geometric group theory, we identify several instances where this conjecture holds. This is joint work with Ionut Chifan, Denis Osin, and Hui Tan.

Rota’s Alternierende Verfahren theorem in classical probability theory, which examines the convergence of iterates of measure preserving Markov operators, relies on a dilation technique. In the noncommutative setting of von Neumann algebras, this idea leads to the notion of absolute dilation.  

In this talk, we explore when a Fourier multiplier on a group von Neumann algebra is absolutely dilatable. We discuss conditions that guarantee absolute dilatability and present an explicit counterexample—a Fourier multiplier that does not satisfy this property. This talk is based on a joint work with Christian Le Merdy.

A countable group G is said to be W*-superrigid if G can be entirely recovered from its ambient group von Neumann algebra L(G). I will present a series of joint works with Milan Donvil in which we establish new degrees of W*-superrigidity: isomorphisms may be replaced by virtual isomorphisms expressed by finite index bimodules, the group von Neumann algebra may be twisted by a 2-cocycle, the group G might have infinite center, or we may enlarge the category of discrete groups to the broader class of discrete quantum groups.