CA : Analytic group theory

The rich theory of infinite groups and dynamical/algebraic objects associated to them intertwines techniques and ideas from various areas of mathematics such as representation theory of groups, harmonic analysis, dynamical systems and operator algebras. A striking example of this phenomenon is the resolution to the longstanding problem in 2014-15 to determine when the C*-algebra associated to a such a group is simple. This course consists of an introduction to some of these concepts and their interactions.

It will roughly consist of three parts:
1. Unitary representations of groups: locally compact groups, unitary representations, amenability, Property (T)
2. Dynamical systems on groups: proximality, minimality, stationary measures, amenable actions, Furstenberg and Poisson boundaries
3. Operator algebras: Banach algebras, C*-algebras, von Neumann algebras, Gelfand Theory, C*-algebras of amenable groups, Crossed products.