Group Actions and Rigidity

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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 (the use of wavefront sets to study fine analytic properties of solutions to equations) to theoretical computer science ((super)expanders, Kazhdan's property (T), Lafforgue's strong property (T) and coarse embedding problems). 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 breakthroughs on conjectures of Zimmer's. Another major focus is on the structure of hyperbolic manifolds where the PI recently made another major breakthrough on a question of McMullen and Reid. Other topics include studying families of expanders and superexpanders arising from dynamical constructions, both to understand their coarse geometry and to see if they have novel applications to computer science, dynamics or operator algebras.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.