CAREER: Dynamical Rigidity Related to Group Actions and Arithmetics

The proposed project belongs to the field of dynamical systems, which is the study of long-term trajectories of movements that take place over time in a geometric space and are governed by a fixed set of mathematical rules. The theory of dynamical systems has its roots in celestial mechanics and has become an important tool in many other scientific fields. The proposed research deals with the question of whether a given dynamical system has rigidity properties, namely whether different trajectories must display a lot of common patterns. The principal investigator will conduct research on the topic, collaborate with other researchers, work with graduate and undergraduate students, and disseminate ideas developed in the project. The principal investigator will also organize seminars and conferences. In particular, the principal investigator will organize a series of research workshops for undergraduate students. This research will try to determine properties of certain classes of dynamical systems of group theoretical or number theoretical origins. One of such classes are actions on smooth manifolds by higher-rank commutative groups or by lattices in higher-rank semisimple Lie groups. These actions are often expected to have rigidity, a phenomenon characterized by the scarcity of invariant structures under the action. Another proposed line of research is the study of some special cases of the Mobius disjointness conjecture. These include certain distal topological systems and other systems whose trajectories have strict patterns in sufficiently long term. The approaches in the proposed research will use ideas from ergodic theory, smooth dynamics, representation theory and number 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.