Award: DMS 1611765, Principal Investigator: Alexander Furman
Study of symmetry is one of the central themes in mathematics: one investigates mathematical laws and various associated objects by analysing their symmetries (i.e., transformations that do not change them) and further analyses how the laws and objects change under various transformations. In modern language one studies groups of symmetries and groups of transformations. The overarching theme of the project is to study phenomena in dynamics and geometry that occur in the presence of large groups (i.e., situations with a lot of explicit or implicit symmetries). This research continues important work over the last few decades that connects topics in geometry, probability theory, dynamical systems, and number theory.
More specifically, the proposal focuses on three areas: (1) measured group theory - an ergodic-theoretic analogue of geometric group theory; (2) further study of higher-rank super-rigidity phenomena; and (3) study of Lyapunov exponents in the classical context of the multiplicative ergodic theorem in the presence of large groups. The first topic includes study of rigidity for rank-one lattices in analogy with previously established rigidity results for higher-rank lattices. Another promising direction is development of a measured analogue of Gromov-hyperbolic groups. The second theme includes study of rigidity phenomena for groups that resemble but are different from higher-rank lattices, but admit rich Weyl groups. The third topic develops a method of establishing simplicity of the Lyapunov spectrum using certain group actions - this investigation connects now classical results on random products of matrices, geodesic flows on negatively curved manifolds, and recent developments in Teichmuller dynamics.
