Cyril Houdayer: Noncommutative ergodic theory of higher rank lattices

I will survey recent results regarding the dynamics of positive definite functions and character rigidity of irreducible lattices in higher rank semisimple algebraic groups. These results have several applications to ergodic theory, topological dynamics, unitary representation theory and operator algebras. In the case of lattices in higher rank simple algebraic groups, I will explain the key operator algebraic novelty, which is a noncommutative Nevo–Zimmer theorem for actions on von Neumann algebras. I will also present a noncommutative analogue of Margulis’ factor theorem and discuss its relevance regarding Connes’ rigidity conjecture for group von Neumann algebras of higher rank lattices.