C1

Wreath-like products are a new class of groups, which are close relatives of the classical wreath products. Examples of wreath-like product groups arise from every non-elementary hyperbolic groups by taking suitable quotients. As a consequence, unlike classical wreath products, many wreath-like products have Kazhdan's property (T). 

I will present several rigidity results for von Neumann algebras of wreath-like product groups. We show that any group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group von Neumann algebra L(G) remembers the isomorphism class of G. This provides the first examples of W*-superrigid groups with property (T). For a wider class wreath-like products with property (T), we show that any isomorphism of their group von Neumann algebras arises from an isomorphism of the groups. As an application, we prove that any countable group can be realized as the outer automorphism group of L(G), for an icc property (T) group G. These results are joint with Ionut Chifan, Denis Osin and Bin Sun.  

Time permitting, I will mention an additional application of wreath-like products obtained in joint work with Ionut Chifan and Daniel Drimbe, and showing that any separable II_1 factor is contained in one with property (T). This provides an operator algebraic counterpart of the group theoretic fact that every countable group is contained in one with property (T).

The pioneering work of Murray and von Neumann shows that any countable discrete group G gives rise in a canonical way to a group von Neumann algebra, denoted L(G). A main theme in operator algebras is to classify group von Neumann algebras, and hence, to understand how much information does L(G) remember of the underlying group G. In the amenable case, the classification problem is completed by the work of Connes from 1970s asserting that for all infinite conjugacy classes amenable groups, their von Neumann algebras are isomorphic.

In sharp contrast, in the non-amenable case, Popa's deformation rigidity/theory (2001) has led to the discovery of several instances when various properties of the group G are remembered by L(G). The goal of this talk is to survey some recent progress in this direction.

The notion of amenable actions by discrete groups on C*-algebras has been introduced by Claire Amantharaman-Delaroche more than thirty years ago, and has become a well understood theory with many applications. So it is somewhat surprising that an established theory of amenable actions by general locally compact groups has been missed until 2020. We now present a theory which extends the discrete case and unifies several notions of approximation properties of actions which have been discussed in the literature. We also present far reaching results towards the weak containment problem which asks wether an action $\alpha:G\to \Aut(A)$ is amenable if and only if the maximal and reduced crossed products coincide.

In this lecture we report on joint work with Alcides Buss and Rufus Willett.

An anomalous symmetry of an operator algebra A is a mapping from a group G into the automorphism group of A which is multiplicative up to inner automorphisms. To any anomalous symmetry, there is an associated cohomology invariant in H^3(G,T). In the case that A is the Hyperfinite II_1 factor R and G is amenable, the associated cohomology invariant is shown to be a complete invariant for anomalous actions on R by the work of Connes, Jones, and Ocneanu.

In this talk, I will introduce anomalous actions from the basics discussing examples and the history of their study in the literature. I will then discuss two obstructions to possible cohomology invariants of anomalous actions on simple C*-algebras which arise from considering K-theoretic invariants of the algebras. One of the obstructions will be of algebraic flavour and the other will be of topological flavour. Finally, I will discuss the classification question for certain classes of anomalous actions.

abstract-Oxford.pdf

Since Segal's formulation of axioms for 2d CFTs in the 80s, it has remained a major problem to construct examples of CFTs that satisfy the axioms.

I will report on ongoing joint work with James Tener in that direction.

A coarse structure is a way of talking about "large-scale" properties.  It is encoded in a family of relations that often, but not always, come from a metric.  A coarse structure naturally gives rise to Hilbert space operators that in turn generate a so-called uniform Roe algebra.

In work with Bruno Braga and Joe Eisner, we use ideas of Weaver to construct "quantum" coarse structures and uniform Roe algebras in which the underlying set is replaced with an arbitrary represented von Neumann algebra.  The general theory immediately applies to quantum metrics (suitably defined), but it is much richer.  We explain another source based on measure instead of metric, leading to the new, large, and easy-to-understand class of support expansion C*-algebras.

I will present the big picture: where uniform Roe algebras come from, how Weaver's framework facilitates our definitions.  I will focus on a few illustrative examples and will not presume familiarity with coarse structures or von Neumann algebras.

It is well known that if a group von Neumann algebra of a (nontrivial) discrete group is a factor, then it is a factor of type II_1. During the talk, I will answer the following question: which types appear as types of injective factors being group von Neumann algebras of discrete quantum groups (or looking from the dual perspective - von Neumann algebras of bounded functions on compact quantum groups)? An important object in our work is the subgroup of real numbers t for which the scaling automorphism tau_t is inner. This is joint work with Piotr Sołtan.

We are surrounded by systems that involve many elements and the interactions between them: the air we breathe, the galaxies we watch, herds of animals roaming the African planes and even us – trying to decide on whom to vote for.

As common as such systems are, their mathematical investigation is far from simple. Motivated by the realisation that in most cases we are not truly interested in the individual behaviour of each and every element of the system but in the average behaviour of the ensemble and its elements, a new approach emerged in the late 1950s - the so-called mean field limits approach. The idea behind this approach is fairly intuitive: most systems we encounter in real life have some underlying pattern – a correlation relation between its elements. Giving a mathematical interpretation to a given phenomenon and its emerging pattern with an appropriate master/Liouville equation, together with such correlation relation, and taking into account the large number of elements in the system results in a limit equation that describes the behaviour of an average limit element of the system. With such equation, one hopes, we could try and understand better the original ensemble.

In our talk we will give the background to the formation of the ideas governing the mean field limit approach and focus on one of the original models that motivated the birth of the field – Kac’s particle system. We intend to introduce Kac’s model and its associated (asymptotic) correlation relation, chaos, and explore attempts to infer information from it to its mean field limit – The Boltzmann-Kac equation.

 I will start this talk with a brief introduction and summary of the outcome of a joint work with James Gabe. An important special case of the main result is that for any countable discrete amenable group G, any two outer G-actions on stable Kirchberg algebras are cocycle conjugate precisely when they are equivariantly KK-equivalent. In the main body of the talk, I will outline the key arguments toward a special case of the 'uniqueness theorem', which is one of the fundamental ingredients in our theory: Suppose we have two G-actions on A and B such that B is a stable Kirchberg algebra and the action on B is outer and equivariantly O_2-absorbing. Then any two cocycle embeddings from A to B are approximately unitarily equivalent. If time permits, I will provide a (very rough) sketch of how this leads to the dynamical O_2-embedding theorem, which implies that such cocycle embeddings always exist in the first place.