Abstract:
For a non-amenable group G, there may be many (group) C*-algebras that lie naturally between the universal and the reduced C*-algebra of G. One way to construct such algebras is from L^p-integrability properties of matrix coefficients of unitary representations of G. After an introduction to the relatively young topic of exotic group C*-algebras, I will analyze this construction in the setting of discrete groups, Lie groups, and (non-discrete) locally compact groups acting on trees. For the latter two classes, we rely on methods from harmonic analysis and on spherical representations of the underlying groups.

Abstract:
We will review the classical theory of Furstenberg boundaries for topological groups, explaining some of the main arguments and examples. One of the goals of the presentation will be to show how this tool can be used to establish an Iwasawa decomposition for certain locally compact groups arising as Gelfand pairs.

Abstract:
A conjecture of George Elliott dating back to the early 1990's asks if separable, simple, nuclear C*-algebras are determined up to isomorphism by their K-theoretic and tracial data. Restricting to purely infinite algebras, this is the famous Kirchberg-Phillips Theorem. The stably finite setting proved to be much more subtle and has been a driving force in research in C*-algebras over the last 30 years. A series of breakthroughs were made in 2015 through the classification results of Elliott, Gong, Lin, and Niu and the quasidiagonality theorem of Tikuisis, White, and Winter. Today, the classification conjecture is now a theorem under two additional regularity assumptions: Z-stability and the UCT. In my recent joint work with José Carrión, Jamie Gabe, Aaron Tikuisis, and Stuart White a much shorter and more conceptual proof of the classification theorem in the stably finite setting was provided. I hope to give an overview of the classification problem for C*-algebras and discuss some of the new techniques that led to the new proof.

Abstract:
A self-similar group (G,X) consists of a group G acting faithfully on a homogeneous rooted tree such that the action satisfies a self-similarity condition. In this series of talks I will outline Nekrashevych's proof that that the Cuntz-Pimsner algebra of a self-similar group action is Spanier-Whitehead dual to the groupoid C*-algebra of an associated Smale space. Nekrashevych's result ties together ideas from operator algebras, dynamical systems and geometric group theory in a truly beautiful way.

Abstract:
Nuclear dimension, defined about 10 years ago as well as its various forerunners, such as decomposition rank or completely positive rank are all dimension concepts which regard certain finite dimensional approximations as non-commutative versions of partitions of unity, leading to a covering dimension for non-commutative C*-algebras. Nuclear dimension has played an important role in the classification programme in recent years and has been studied intensively. We will give a survey of its theory, starting from the definition, trying to unify certain aspects, outlining its role in classification theory and possibly mentioning some new variants. If time permits we will include related applications to AF-embeddability.