Abstract: To study the asymptotic behavior of orbits of a dynamical system, one can look at orbit closures or invariant measures. When the underlying system has a homogeneous structure, usually coming from a Lie group, with appropriate assumptions a wide range of rigidity theorems show that ergodic invariant measures and orbit closures have to be well-behaved and can often be classified. I will describe joint work with Brown, Eskin, and Rodriguez Hertz, which establishes rigidity results for quite general smooth dynamical systems having some hyperbolicity. I will also explain some of the necessary assumptions as well as the homogeneous structures that emerge.

Bio: Simion Filip received his PhD from the University of Chicago in 2016 under the supervision of Alex Eskin. After spending several years as a Clay Fellow at Harvard and the Institute for Advanced Study, he has been on the faculty of the University of Chicago since 2019. His research is concerned with the interactions between dynamical systems and geometry, particularly complex and algebraic geometry.