Rigidity Phenomena of Higher Rank Actions

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This project belongs to the field of dynamical systems, which aims to understand long-term behaviors 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 a special class of dynamical systems, where the rules governing the movement form a sufficiently large group so that a point is allowed to move in one of several possible ways in each step. In such cases, there is often a small collection of possible structures that remain unchanged over time. The goal of this project is to determine such structures. The PI will conduct research on the topic, collaborate with other researchers, work with graduate and undergraduate students, and disseminate ideas developed in the project. This research will study dynamical systems of group actions. The principal objects are actions on tori, nilmanifolds and other homogeneous spaces 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. The main objective is to classify such group actions under suitable assumptions, as well as invariant structures under these actions. The method of the research will combine tools from ergodic theory, smooth dynamics and representation theory.