In this talk, I will present a noncommutative analogue of Margulis’ factor theorem for higher rank lattices. More precisely, I will give a complete description of all intermediate von Neumann subalgebras sitting between the von Neumann algebra of the lattice and the von Neumann algebra of the action of the lattice on the Furstenberg-Poisson boundary. As an application, we infer that the rank of the semisimple Lie group is an invariant of the pair of von Neumann algebras. I will explain the relevance of this result regarding Connes’ rigidity conjecture.

The classification by K-theory and traces of the category of simple, separable, nuclear, Z-stable C*-algebras satisfying the UCT is an extraordinary feat of mathematics. What's more, it provides powerful machinery for the analysis of the internal structure of these regular C*-algebras. In this talk, I will explain one such application of classification: In the subclass of classifiable C*-algebras consisting of those for which the simplex of tracial states is nonempty, with extremal boundary that is compact and has the structure of a connected topological manifold, automorphisms can be shown to be generically tracially chaotic. Using similar ideas, I will also show how certain stably projectionless C*-algebras can be described as crossed products.

Equivariant Jiang-Su stability is an important regularity property for group actions on C*-algebras.  In this talk, I will explain this property and how it arises naturally in the context of the classification of C*-algebras and their actions. Depending on the time, I will then explain a bit more about the nature of equivariant Jiang- Su stability and the kind of techniques that are used to study it, including a recent result of Gábor Szabó and myself establishing an equivalence with equivariant property Gamma under certain conditions.