Dynamics and the Classification of Geometries on Manifolds

University Of Maryland, College Park, College Park MD

Investigators

Abstract

This project will conduct research into the classification of locally homogeneous geometric structures on manifolds. This is a subject on the interface of geometry and topology and has been particularly important in the study of topology in 2 and 3 dimensions. A simple example is that there is no metrically accurate world atlas: the topology of the sphere (the earth) is not compatible with Euclidean geometry, although the topology of a torus (an inner tube) is. The project is closely related to other fields of mathematics, including group theory, algebraic geometry, and dynamical systems. The research has an experimental component and involves visualization techniques. The project will include software development for these questions and graduate students will receive training through involvement in the research. The research includes four main threads: (1) the classification of affine and projective structures with solvable holonomy; (2) relations between the certain geometries and symplectic/contact geometry; (3) dynamical systems arising from mapping class group actions on deformation spaces and character varieties; and (4) explicit constructions of topological automorphisms and the computation of finer dynamical invariants. The latter involves natural Poisson/symplectic geometry on the deformation spaces. The project also includes mentoring and training of students as well as outreach activities via the Experimental Geometry Lab at the PI's home institution. 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.