Topology & Dynamics of Moduli Spaces of Geometric Structures

DMS-0103889William M.戈德曼这个项目继续PI的调查局部齐性几何结构的流形，他们的模空间和他们的对称性。完整对应将几何结构的模与基本群的表示论联系起来。 本研究的主要几何结构包括共形平坦的洛伦兹和黎曼结构，真实的射影和仿射结构，以及真实的和复双曲结构。动力学的问题都涉及基本组的行动，各种流（包括和generalizingthe测地线，horocycle和框架流）和模空间的自同构将被研究。映射类群的动力学和模空间的全局拓扑之间的密切相互作用尤其值得注意。 模空间通常具有辛结构，这些辛结构导致了一些重要而神秘的代数构造，如基于光滑曲面上的曲线的李代数，这些对象在数学和物理中发挥着越来越重要的作用。 洛伦兹流形是相对论中的宇宙学模型。模空间已经成为量子物理学的中心，映射类群作用和Teichmuller空间在二维量子引力中变得突出。 这项研究在于几何/拓扑，动力系统，分析和代数之间的接口。尽管其复杂的性质，研究的例子是非常明确的。 因此，人们可以通过实验对它们进行有益的研究。我们目前的计算机技术适用于这类项目。在Richard Schwartz博士和马里兰州大学数学系的支持下，实验几何实验室已经运作了一年多。这个实验室包括了由研究生、研究生，特别是本科生完成的软件项目。

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.