Noncommutative Ergodic Theory of Higher Rank Lattices

Explore recent advancements in noncommutative ergodic theory of higher rank lattices in this 46-minute lecture by Cyril Houdayer. Delve into the dynamics of positive definite functions and character rigidity of irreducible lattices in higher rank semisimple algebraic groups. Discover applications to ergodic theory, topological dynamics, unitary representation theory, and operator algebras. Examine the noncommutative Nevo–Zimmer theorem for actions on von Neumann algebras, a key operator algebraic innovation for lattices in higher rank simple algebraic groups. Investigate a noncommutative analogue of Margulis' factor theorem and its implications for Connes' rigidity conjecture. Follow the comprehensive syllabus covering topics such as simple algebraic groups, examples of higher rank lattices, dynamics of AP(A), character classification, boundary structures on von Neumann algebras, and more.

Intro
Simple algebraic groups
Examples of higher rank lattices
Motivation
Dynamics of AP(A)
Examples of characters
Existence and classification of characters
Dichotomy for topological dynamics
Strategy of proof
Structure theory of G/P
Boundary structures on von Neumann algebras
Examples of boundary structures
The noncommutative Nevo-Zimmer theorem
Connes' rigidity conjecture for higher rank lattices
The noncommutative Margulis factor theorem