Rigidity, lattices and invariant measures beyond homogeneous dynamics

TL;DR

This paper explores recent advances in understanding rigidity phenomena in geometry and dynamics through invariant measures, focusing on non-homogeneous settings and their implications for conjectures and submanifold structures.

Contribution

It highlights the importance of studying invariant measures beyond homogeneous spaces, connecting results on Zimmer's conjecture and hyperbolic submanifolds to broader rigidity questions.

Findings

Progress on Zimmer's conjecture with Brown and Hurtado

Characterization of totally geodesic submanifolds in hyperbolic manifolds

Emphasis on invariant measures outside homogeneous dynamics

Abstract

This article discusses two recent works by the author, one with Brown and Hurtado on Zimmer's conjecture and one with Bader, Miller and Stover on totally geodesic submanifolds of real and complex hyperbolic manifolds. The main purpose of juxtaposing these two very disparate sets of results in one article is to emphasize a common aspect: that the study of invariant and partially invariant measures outside the homogeneous setting is important to questions about rigidity in geometry and dynamics. I will also discuss some open questions including some that seem particularly compelling in light of this juxtaposition.