CAREER: Rigidity in Mapping class groups and homeomorphism groups

University Of Maryland, College Park, College Park MD

Investigators

Abstract

In geometry and topology, one of the most fundamental objects is to study various geometric groups and their features. This project will investigate the rigidity problems concerning mapping class groups and homeomorphism groups of manifolds. The PI will use methods from dynamical systems, geometric group theory, low dimensional topology, and differential geometry. The educational activities include high school outreach, undergraduates research through REU projects, mentoring graduate students in the home institution, and workshops organizations. Symmetry is a pervasive concept in mathematics. In the study of differential topology, the full symmetry group is the diffeomorphism group of a manifold. There are two sides of a diffeomorphism group: one is the mapping class group, the group of connected components of a diffeomorphism group; the other is the identity component of a diffeomorphism group, which is a connected topological group. The PI will study these groups using both geometric group theory through the study of how those groups act on certain complexes and dynamical tools through the study of how those groups act on other manifolds. 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.