TBA

In 2008, Toms constructed a counterexample to the Elliott conjecture: a pair of simple, separable, nuclear and unital C*-algebras which are indistinguishable by the Elliott invariant, but are not isomorphic. The key to distinguishing this pair of carefully crafted C*-algebras lies with a rather refined invariant called the Cuntz semigroup. Consequently, Toms’s counterexample highlighted the importance of the Cuntz semigroup to the classification of C*-algebras.

In this talk, we will discuss the Cuntz semigroup in the context of graph C*-algebras, a highly diverse class of mostly non-simple C*-algebras. In particular, we will accentuate how the highly organised structure of a unital graph C*-algebra is reflected in its Cuntz semigroup and if enough time permits, mention properties of unital graph C*-algebras that are revealed by these Cuntz semigroups.

Much recent research has gone into understanding the first order theory of II1 factors. Very recently, Peterson released a preprint which develops deformation rigidity in the ultrapower setting. His techniques give many explicit examples of non-isomorphic ultrapowers for natural families of II1 factors. In this talk, I will introduce some of Peterson's techniques and results, including an analogue of amenability in the ultrapower setting and the interplay between property T and malleable deformations.

One of the problems posed by Kadison in 1967 asks whether every separably acting von Neumann algebra is generated by a single element. The problem remains open in its full generality but significant progress has been made since. One can of course ask the same question in the C*-algebraic setting where, however, counterexamples are abundant even among commutative C*-algebras. I will give an overview of the history of the problem and then discuss some recent results on single generation of C*-algebras associated to graphs and C*-algebras with Cartan subalgebras.

We consider the question of whether almost-homomorphisms are close to honest homomorphisms. I’ll survey a few historical results, with different source/target collections of algebras, and also consider what to take as the definition of “almost-homomorphisms”. If we end up having time, I will sketch an elementary proof that almost-characters from commutative C*-algebras are close to honest characters.

Descriptive set theory provides a useful framework for studying the complexity of classification problems in operator algebras. In this talk I will discuss how C*-algebras can be encoded as points in a Borel space, and introduce several equivalent parametrizations, including a new one in terms of ideals of a universal C*-algebra. I will then discuss examples of natural classes of C*-algebras that form Borel sets, as well as a parametrization of *-homomorphisms and recent results on the Borelness of certain functors. Time permitting, I will introduce KK-theory and the Kasparov product, and explain a new result showing that the Kasparov product is Borel in a certain appropriate parametrized setting.