Measured Group Theory and Dynamics

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Groups are algebraic objects that formally describe the concepts of symmetry. They play a central role in all branches of mathematics. This NSF award supports research and education that contain several directions to study Dynamical Systems in the presence of large groups. These include dynamical properties in situations with many explicit or implicit symmetries. Such systems tend to attain additional special properties. Conversely, some dynamical properties can often serve as indications of rich underlying symmetries. Educational impact of the project is through Ph.D. supervision, dissemination efforts, and high school outreach. This research project will encompass a number of topics at the intersection of dynamical systems, group theory, geometry and operator algebras. The common theme of the research directions lies in the synthesis and interactions between these areas. Many of the aspects of these interactions involve rigidity, a phenomenon of robustness of a structure under perturbations. Specific projects are related to special phenomena in dynamical systems that occur in the presence of large groups, such as explicit or implicit symmetries, rigidity of certain exotic geometries, and applications of dynamical methods to geometry and analysis. The three main topics under consideration are: measure equivalence rigidity, the Lyapunov spectrum for random walks and generalizations, and the length spectrum for Gromov hyperbolic groups. One focus of this research is the application of dynamical methods in the study of group theory and geometry, and especially rigidity phenomena. The other focus will be the application of group theory to shed light on problems in dynamical systems and geometry. 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.