Dynamics of Moduli Spaces and Surface Groups

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This project concerns the classification of geometric structures onmanifolds, the geometry of their moduli spaces, and dynamical systemson moduli spaces arising from automorphisms of surfaces. For example,complete affine 3-manifolds behave similarly to hyperbolic Riemannsurfaces. For a large class of examples, the deformation space hasrecently been shown to be a bundle of convex cones over thedeformation space of hyperbolic surfaces. One aspect of the project isthe classification and dynamical study of such manifolds, and theirconformal and constant curvature generalizations. Another aspect ofthis project deals with dynamical systems arising from theclassification of geometric structures on surfaces. The action of themapping class group on the character variety is a dynamical systemassociated to the fundamental group of the surface and the geometryassociated to a Lie group. These moduli spaces support invariantsymplectic structures, as well as finer structures depending on aRiemann surface with the given fundamental group. How do thesestructures change as the Riemann surface varies? How do the geometryof these structures influence the dynamics of their topologicalsymmetries of the surface? How does dynamical complexity of themapping class group action accompany topological complexity of themoduli space?These questions lie at the interface between geometry and topology,unified through the notion of symmetry. While geometry involvesmetric quantitative relationships, topology concerns the loosequalitative organization of ``points'' in abstract ``spaces.''Geometry relates to topology through their respective symmetries,which can be studied algebraically through the techniques of grouptheory. These questions are natural for experimental investigation.The Experimental Geometry Lab at the University of Maryland providesinterested students opportunities for mathematical experimentation ina stimulating environment. Its goal is the dissemination andpromotion of mathematics, related to the projects described above.The Lab will continue to be a center for Undergraduate ResearchExperiences, outreach presentations, and interactive projects at all levels.