C1

In 1989, Mazur defined the deformation functor associated to a residual Galois representation, which played an important role in the proof by Wiles of the modularity theorem. This was used as a basis over which many mathematicians constructed variations both to further specify it or to expand the contexts where it can be applied. These variations proved to be powerful tools to obtain many strong theorems, in particular of modular nature. In this talk I will give an overview of the deformation theory of Galois representations and describe two variants of Mazur's functor that allow one to properly deform reducible residual representations (which is one of the shortcomings of Mazur's original functor). Namely, I will present the theory of determinant-laws initiated by Bellaïche-Chenevier on the one hand, and an idea developed by Calegari-Emerton on the other.
If time permits, I will also describe results that seem to indicate a possible comparison between the two seemingly unrelated constructions.

I will survey Cartan respectively diagonal subalgebras of nuclear C*-algebras. This setup corresponds to a presentation of the ambient C*-algebra as an amenable groupoid C*-algebra, which in turn means that there is an underlying structure akin to an amenable topological dynamical system.

The existence of such subalgebras is tightly connected to the UCT problem; the classification of Cartan pairs is largely uncharted territory. I will present new constructions of diagonals of the Jiang-Su algebra Z and of the Cuntz algebra O_2, and will then focus on distinguishing Cantor Cartan subalgebras of O_2.

A complex irreducible admissible representation of a reductive p-adic group is typically infinite-dimensional. To quantify the "size" of such representations, we introduce the concept of canonical dimension. To do so we have to discuss the Moy-Prasad filtrations. These are natural filtrations of the parahoric subgroups. Next, we relate the canonical dimension to the Harish-Chandra local character expansion, which expresses the distribution character of an irreducible representation in terms of nilpotent orbital integrals. Using this, we consider the wavefront set of a representation. This is an invariant the naturally arises from the local character expansion. We conclude by explaining why the canonical dimension might be considered a weaker but more computable alternative to the wavefront set.

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.