CAREER: New Methods and Applications for Smooth Rigidity of Algebraic Actions

Michigan State University, East Lansing MI

Investigators

Abstract

The field of dynamical systems originated from differential equations and celestial mechanics. It studies the long run behavior of a system subject to laws of motion. The study of rigidity phenomenon is one of the central themes in dynamical systems, with applications to number theory, geometry, and mathematical physics. Roughly speaking, an action is rigid if various properties of the system are preserved under appropriate modifications. So far, systems with strong chaotic properties have been well understood. In many systems of interest however, only weak chaotic behaviors can be observed. The goal of this project is to develop new tools to study such systems and then apply these results to study other areas of mathematics, such as number theory and representation theory. The research plan is complemented by educational and outreach activities involving the training of undergraduates, graduate students, and postdoctoral associates, and fostering collaborations among female researchers in different areas. The proposed research plan consists of several coherent projects, ranging from dynamical systems, harmonic analysis for Lie groups, representation theory and number theory. The principal investigator plans to establish cocycle rigidity and to study (twisted) cohomological equations for a large class of algebraic partially hyperbolic and parabolic actions by using representation theory. The results and techniques from the study of parabolic actions will be applied to number theory for the study of effective equidistribution for certain unipotent flows and maps on some homogenous spaces of semidirect product groups. The principle investigator will also combine KAM approach and representation theory to general partially hyperbolic and parabolic algebraic actions to study local rigidity. A large class of new examples whose geometric properties are distinctly different from existing examples will be explored. 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.