I will give a light introduction to the theory of compact quantum groups from the operator algebraic perspective. I will highlight some aspects of their general theory (representation theory, Haar states, related operator algebras) as well as several concrete examples. Time permitting, I will also point out some recent applications of quantum groups in combinatorics and quantum information theory. Slides

We will briefly describe the construction of a crossed product $C^*$-Algebra from an action of a group $G$ on a $C^*$-algebra $A$. We will provide a few examples. Then we will look in more detail at the situation when $G= \mathbb{Z}$ and the situation when $G$ is a finite group. Samples of the many known theorems will be given. We will conclude with some current research on crossed products and large subalgebras. Slides

K-theory is a generalized homology in which one associates abelian groups to a C*-algebra. While K-theory provides some of the most important and useful invariants for C*-algebras, it is also notoriously technical. Simply defining the abelian groups can be cumbersome and involves a lot of machinery. Furthermore, K-theory is motivated by and has applications to a vast number of topics (e.g., topology, geometry, index theory, classification of C*-algebras), all of which can easily cause newcomers to get mired in details and lose track of the forest in the trees. In this talk we shall present a bird’s eye view of K-theory, including a descriptive walk-through of the definitions of the K-groups and how to use them. Throughout, we will discuss the origins and motivations for K-theory, explain how it relates to other subjects, examine some of its most useful properties, and present the major C*-algebraic results that is has provided. Slides

The algebra $L^\infty(\Omega, \mathbb{P})$ of essentially bounded random variables on some probability space is a (commutative) von Neumann algebra with a state given by the conditional expectation; it is tempting, then, to think of an arbitrary von Neumann algebra as an algebra of non-commutative random variables. This approach is free probability. Many results and techniques from probability theory have analogues in operator algebras, including the free central limit theorem and free information theory. These results are an analogue of their classical versions, though, not extensions, and it is necessary to introduce a new notion of independence suited to the non-commutative world. I will give an introduction to free probability theory, discuss the dictionary between it and "classical" probability theory, and spend some time proving the free central limit theorem, before closing with some brief remarks about how free probability can be used to establish structural properties of von Neumann algebras. Slides

We have various ways of describing the extent to which two countably infinite groups are "the same." Are they isomorphic? If not, are they commensurable? Measure equivalent? Quasi-isometric? Orbit equivalent? W*-equivalent? Von Neumann equivalent? In this expository talk, we will define these notions of equivalence, discuss the known relationships between them, and work out some examples.

Some of the most important results in C*-algebra theory depend on the theory of von Neumann algebras, aka W*-algebras. In this talk I'll describe the abstract connection between C*- and W*-algebras, which depends on the theory of duality in Banach spaces. Specifically, the double dual of every C*-algebra is a von Neumann algebra. I will describe this duality in detail, then give a tour of very useful applications. Slides

I will introduce groupoid C*-algebras using examples and a process that I informally call the C*-game. I will emphasize the special case of etale equivalence relations and C*-algebraic constructions coming from dynamical systems. The overall goal is to convince the audience that groupoids give not only a very general way of constructing C*-algebras but are also a useful tool in the study of C*-algebras in general. Slides

A subfactor is an inclusion of $\rm{II}_{1}$ factors $N\subseteq M$. The generalized symmetries of this inclusion are captured by an incredibly rich mathematical structure called the standard invariant, which has connections with a wide range of topics in mathematics and physics from low dimensional topology to quantum field theory. We will give a gentle introduction to these concepts, and explain some open problems in this area. Slides

Model theorists study mathematical structures through the lens of first-order logic. The model theoretic study of von Neumann algebras is quite new and has been expanding at a rapid pace, producing applications to purely operator algebraic problems as well as raising interesting model theoretic questions. In this talk I will give a survey of some of these results, the focus being on how this perspective sheds light on questions around ultraproducts of tracial von Neumann algebras. Slides

An operator system is a unital subspace of $B(H)$ that is closed under the involution operation $*$. This talk will introduce some basic results on operator systems and their connection to C*-algebras. Our end goal will be to describe some operator systems that seem tame but demonstrate some surprisingly complex behaviour. Slides

We review the Dykema-Radulescu notion of interpolated free group factors. We also discuss our recent work with Sorin Popa showing that all such factors are group von Neumann algebras. Slides

C*-algebras constructed from directed graphs form a class of operator algebras that can be analyzed by examining the combinatorics of the underlying graph. We will go over the basic notions of graph algebras, survey some major results in the field and see some connections between these algebras and groupoids algebras. Slides

Planar algebras are an algebraic (even categorical) way to look at the standard invariant of a subfactor. In this talk we'll explore why subfactors have diagrammatic operations, and constructing subfactors via their planar algebras. Slides

Let u and v be two unitary matrices that almost commute, meaning the commutator uv-vu is small. Must there be nearby unitary matrices that actually commute?
The question has several interesting precise versions, and the answer is ‘depends on the version’. For example, Voiculescu showed that if ‘small’ and ‘nearby’ are measured by the operator norm, and if the implicit constants are wanted independent of dimension, the answer is ‘no'.
I’ll start by explaining the close relation between Voiculescu’s (counter-)example and the Bott periodicity theorem from operator K-theory. I’ll then touch on connections to index theory and representation theory, among other things. Slides

I will start by discussing Voiculescu's asymptotic freeness theorem, a general result which gives a natural way in which freely independent noncommutative variables arise from random matrices. I will then use this as motivation to discuss the notion of 1-bounded entropy, an invariant for von Neumann algebras which arises out of an attempt to "count" how many matricial approximations a von Neumann algebras. I will discuss applications to structural results of free group factors. Time permitting, I may even discuss some of my results. Slides

Can K-theory (and related gadgets) tell you when two C*-algebras are isomorphic? In this expository talk I'll give a gentle introduction to the classification theory for nuclear C*-algebras, starting with the case of AF algebras, and present some recurring themes found in research being carried out in the field today. Slides

In the the mid thirties F. J. Murray and J. von Neumann found a natural way to associate a von Neumann algebra L(G) to every countable discrete group G. Classifying L(G) in terms of G emerged as a natural yet quite challenging problem as these algebras tend to have very limited “memory” of the underlying group. This is perhaps best illustrated by Connes’ famous result asserting that all icc amenable groups give rise to isomorphic von Neumann algebras; thus in this case, besides amenability, the algebra has no recollection of the usual group invariants like torsion, rank, or generators and relations. In the non-amenable case the situation is radically different; many examples where the von Neumann algebraic structure is sensitive to various algebraic group properties have been discovered via Popa’s deformation/rigidity theory.
In my talk I will focus on an extreme situation, namely when L(G) completely remembers the underlying group G; such groups G are called W*-superrigid. Currently there have been identified only two type of group theoretic constructions that lead to W*-superrigid groups: some classes of generalized wreath products groups with abelian base (Ioana-Popa -Vaes '10) and amalgamated free products (Chifan-Ioana '16). After briefly surveying these results I will introduce several new constructions of W*-superrigid groups which include direct product groups, semidirect products with non-amenable core, and tree groups (iterations of amalgams and HNN-extensions). In addition, I will present several applications of these results to the study of rigidity in the C*-setting. This is based on a very recent joint work with Alec Diaz-Arias and Daniel Drimbe. Slides

Since the work of von Neumann, the theory of operator algebras has been closely linked to the theory of groups. On the one hand, operator algebras constructed from groups provide an important source of examples and insight. On the other hand, many problems about groups are most naturally studied within an operator-algebraic framework. In this talk I will give an overview of some problems relating the structure of a group to the structure of a corresponding C*-algebra. I will discuss recent results and some possible future directions. Slides