Rigidity in Dynamics and Geometry

In the study of mathematical objects, a key role is often played by the symmetries of the object—particularly when the object has many symmetries. This research investigates ways of characterizing, describing, and studying spaces with many symmetries in various dynamical, geometric, and topological settings. These questions often require learning, adapting, and applying ideas and techniques from many areas of mathematics. This work has connections with diverse areas of mathematics: from differential equations to theoretical computer science to descriptive set theory to number theory. Graduate student funding will be used to train a new generation of experts. The main thrust of the project is to exploit connections between a wide set of areas to further understand fundamental structures related to lattices in Lie groups. A major focus is the study of group actions on manifolds where the PI recently made major advances on conjectures of Zimmer's. Another major focus is on the structure of hyperbolic manifolds where the PI recently made a breakthrough on a question of McMullen and Reid. Other topics include studying how manifolds with many hidden symmetries (dense commensurators) relate to several classical questions in geometry, topology, and group theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.