C3

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.

The reduced twisted C*-algebra A of an étale groupoid G has a canonical abelian subalgebra D: functions on G's unit space. When G has no non-trivial abelian subgroupoids (i.e., G is principal), then D is in fact maximal abelian. Remarkable work by Kumjian shows that the tuple (A,D) allows us to reconstruct the underlying groupoid G and its twist uniquely; this uses that D is not only masa but even what is called a C*-diagonal. In this talk, I show that twisted C*-algebras of non-principal groupoids can also have such C*-diagonal subalgebras, arising from non-trivial abelian subgroupoids, and I will discuss the reconstructed principal twisted groupoid of Kumjian for such pairs of algebras.

The classification program of C*-algebras aims to classify simple, separable, nuclear C*-algebras by their K-theory and traces, inspired by analogous results obtained for von Neumann algebras. A landmark result in this project was obtained in 2015, building upon the work of numerous researchers over the past 20 years. More recently, Carrión, Gabe, Schafhauser, Tikuisis, and White developed a new, more abstract approach to classification, which connects more explicitly to the von Neumann algebraic classification results. In their paper, they carry out this approach in the stably finite setting, while for the purely infinite case, they refer to the original result obtained by Kirchberg and Phillips. In this talk, I provide an overview of how the new approach can be adapted to classify purely infinite C*-algebras, recovering the Kirchberg-Phillips classification by K-theory and obtaining Kirchberg's absorption theorems as corollaries of classification rather than (pivotal) ingredients. This is joint work with Jamie Gabe.

The (Local) Lifting Property ((L)LP) is introduced by Kirchberg and deals with lifting completely positive maps. We will discuss various examples, characterizations, and closure properties of the (L)LP and, if time permits, connections with some other lifting properties of C*-algebras.  Joint work with Dominic Enders.

I will give a light introduction to the concept of a quantum expander, which is an analogue of an expander graph that arises in quantum information theory.  Most examples of quantum expanders that appear in the quantum information literature are obtained by random matrix techniques.  I will explain another, more algebraic approach to constructing quantum expanders, which is based on using actions and representations of discrete quantum groups with Kazhdan's property (T).  This is joint work with Eric Culf (U Waterloo) and Matthijs Vernooij (TU Delft).