In the study of physical and mathematical objects, a key role is often played by the symmetries of the object, particularly when the object has many symmetries. This project 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 multiple areas of mathematics. As a result, this research has connections with many areas from differential equations to theoretical computer science to descriptive set theory to number theory. The PI will also continue to organize many conferences, summer schools and other events in order to increase participation in mathematics. 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 paradigmatic model of symmetry. A major focus is the study of group actions on manifolds where the PI recently made significant advances on conjectures of Zimmer's. Another major focus is on the structure of negatively curved manifolds where the PI made several breakthroughs and discovered new connections. An additional topic is studying how hidden symmetries 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