Noncommutative ergodic theory of higher rank lattices

Introduction

Noncommutative ergodic theory of higher rank lattices is a current topic that has seen several exciting developments in the last five years. Among these recent advancements, in joint work with Rémi Boutonnet (2019), we proved a noncommutative analogue of Nevo-Zimmer's structure theorem for actions of higher rank lattices on von Neumann algebras.

Applications and Results

First of all, we derived several novel applications to:

• Ergodic theory
• Topological dynamics
• Unitary representation theory
• Operator algebras associated with higher rank lattices

Then we obtained a noncommutative analogue of Margulis' factor theorem that provides strong evidence towards Connes' rigidity conjecture for the group von Neumann algebra of higher rank lattices. These results revealed deep and unexpected interactions between the field of discrete subgroups of semisimple Lie groups and the field of operator algebras.

Research Proposal

In this research project, I plan to build upon these recent achievements to develop new directions in noncommutative ergodic theory of higher rank lattices. This research proposal is centered around two main interconnected themes.

Theme 1: Dynamics of Positive Definite Functions

Firstly, I plan to work on several problems and conjectures around the dynamics of the space of positive definite functions and character rigidity for higher rank lattices. These include:

1. The classification of characters for higher rank lattices of product type
2. More general lattices with dense projections in product groups
3. Stiffness results for stationary positive definite functions

Theme 2: Connes' Rigidity Conjecture

Secondly, drawing inspiration from Margulis' superrigidity theorem, I plan to tackle Connes' rigidity conjecture for higher rank lattices by developing a novel strategy combining techniques from boundary theory, C*-algebras, and von Neumann algebras. These methods will lead to new rigidity phenomena for operator algebras arising from irreducible lattices in higher rank semisimple connected Lie groups.