Dynamics of Moduli Spaces and Surface Groups

University Of Maryland, College Park, College Park MD

Investigators

Abstract

This project concerns the classification of geometric structures on manifolds, the geometry of their moduli spaces, and dynamical systems on moduli spaces arising from automorphisms of surfaces. For example, complete affine 3-manifolds behave similarly to hyperbolic Riemann surfaces. For a large class of examples, the deformation space has recently been shown to be a bundle of convex cones over the deformation space of hyperbolic surfaces. One aspect of the project is the classification and dynamical study of such manifolds, and their conformal and constant curvature generalizations. Another aspect of this project deals with dynamical systems arising from the classification of geometric structures on surfaces. The action of the mapping class group on the character variety is a dynamical system associated to the fundamental group of the surface and the geometry associated to a Lie group. These moduli spaces support invariant symplectic structures, as well as finer structures depending on a Riemann surface with the given fundamental group. How do these structures change as the Riemann surface varies? How do the geometry of these structures influence the dynamics of their topological symmetries of the surface? How does dynamical complexity of the mapping class group action accompany topological complexity of the moduli space? These questions lie at the interface between geometry and topology, unified through the notion of symmetry. While geometry involves metric quantitative relationships, topology concerns the loose qualitative organization of ``points'' in abstract ``spaces.'' Geometry relates to topology through their respective symmetries, which can be studied algebraically through the techniques of group theory. These questions are natural for experimental investigation. The Experimental Geometry Lab at the University of Maryland provides interested students opportunities for mathematical experimentation in a stimulating environment. Its goal is the dissemination and promotion of mathematics, related to the projects described above. The Lab will continue to be a center for Undergraduate Research Experiences, outreach presentations, and interactive projects at all levels.