Rigidity of abelian actions

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There are three main topics in the proposal. The first is the study of the cohomology of abelian higher-rank parabolic actions; in particular we study the cohomological equation and cocycle rigidity for actions completely absent of hyperbolicity. The second considers the smooth action rigidity for the Cartan action on homogeneous space generated by quaternion groups by geometric method; this complements the project of studying smooth rigidity for higher-rank algebraic actions. Finally, we study the general smooth action rigidity for Cartan action and parabolic action; here we apply the famous Kolmogorov-Arnold-Moser (KAM) method to homogenous function space. Dynamical systems appeared first because of Newton's discovery that the motions of mechanical objects can be described by the solutions of ordinary differential equations. More generally, many other natural and social phenomena,such as radioactive decay, chemical reaction, population growth, or dynamics of prices on the market, maybe be modeled with various dynamical systems. So the subject of dynamical systems hasfound numerous applications not only in other branches of mathematics, such asnumber theory, geometry and representation theory, but also in physics, biology, economics and other sciences.The proposed research focuses on smooth rigidity results of algebraic actions, that is, the stabilityproperties related to change of parameters. The PI proposes to develop new methods and techniques from representation theory andharmonic analysis in the study of dynamical systems. The PI believes that the new tool is powerful in solving problems and inspiring new directions in dynamical systems. The PI intends to work with some graduate students and teach graduate-level course during the period of this award.