Topology & Dynamics of Moduli Spaces of Geometric Structures

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DMS-0103889William M. GoldmanThis project continues the PI's investigation of locally homogeneous geometric structures on manifolds, their moduli spaces and their symmetries. The holonomy correspondence relates moduli of geometric structures to the representation theory of the fundamental group. The primary geometric structures of this investigation include conformally flat Lorentz and Riemannian structures, real projective and affine structures, and real and complex hyperbolic structures. Dynamical questions concerning both the action of the fundamental group, various flows (including and generalizingthe geodesic, horocycle and frame flows) and the automorphisms of the moduli space will be studied. The close interaction between the dynamics of the mapping class group and the global topology of the moduli space is especially notable. The moduli spaces often possess symplectic structures which lead to important and mysterious algebraic constructions, such as the Lie algebra based on curves in a smooth surface.These objects play an increasingly important role both in mathematics and physics. Lorentzian manifolds arise as cosmological models in relativity. Moduli spaces have become central in quantum physics, and mapping class group actions and Teichmuller space have become prominent in 2-dimensional quantum gravity. This study lies in the interface between geometry/topology, dynamical systems, analysis and algebra. Despite their complicated nature, the examples studied areextremely explicit. Thus one can profitably investigate them experimentally. Our current computer technology is appropriate for such projects. With Dr. Richard Schwartz and the support of the University of Maryland Mathematics Department, the Experimental Geometry Lab has been operating for over a year. This lab has included software projects performed by postgraduates, graduate students, and especially undergraduates.