Professor of Mathematics; Personnel Committee Chair

### About

I work on the interface of differential geometry, dynamical systems and ergodic theory and discrete subgroups of Lie groups. The main focus of my work lies with the investigationof rigidity properties of geometric and dynamical structures under either weak symmetryor extremal assumptions. The goal typically is to force such structures to be of classical,often algebraic nature. These ideas and results build on the celebrated works of Mostow,Prasad, Margulis, Ratner and Zimmer, and expand prior work of my own.

Currently, my main emphasis is on smooth actions of higher rank abelian and semisimpleLie groups and their lattices. At the heart of this lies the discovery that the higher rankrigidity results for discrete subgroups of semisimple Lie groups and in differential geometry have counterparts in dynamics, often with radically different proofs yet similar in spirit.

On the geometric side, I am fascinated by rigidity properties of systems that have extremal properties, for example higher spherical, Euclidean or hyperbolic ranks and suitable curvature bounds.

More generally, I work on dynamical and geometric properties and their relations, inparticular for Riemannian and Hilbert geometries, CAT(0) spaces and other more generalgeometric structures. On the dynamics front, I am most interested in systems with stringuniform hyperbolicity, and more generally group actions with strong dynamical features. Geodesic and frame flows enter as a bridge between geometry and dynamics.

I have written several survey articles on various aspects of these ideas.