Topology & Dynamics of Moduli Spaces of Geometric Structures

DMS-0103889威廉·M·戈德曼这个项目继续了PI对流形上的局部齐次几何结构、它们的模空间及其对称性的研究。完整对应将几何结构的模同基本群的表示理论联系起来。主要的几何结构包括共形平坦的Lorentz结构和Riemannian结构，实射影结构和仿射结构，以及实双曲结构和复双曲结构。将研究基本群的作用、各种流(包括和推广测地线流、周期流和框架流)以及模空间的自同构的动力学问题。映射类群的动力学与模空间的全局拓扑之间的密切相互作用尤为显著。模空间通常具有辛结构，这导致了重要而神秘的代数结构，例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都扮演着越来越重要的角色。洛伦兹流形是作为相对论中的宇宙学模型出现的。模空间已经成为量子物理中的中心，映射类群作用和TeichMuller空间在二维量子引力中已经变得重要。这项研究涉及几何/拓扑学、动力系统、分析和代数之间的界面。尽管它们的性质复杂，但所研究的例子是非常明确的。因此，人们可以从实验上对它们进行有益的研究。我们目前的计算机技术适合于这类项目。在理查德·施瓦茨博士和马里兰大学数学系的支持下，实验几何实验室已经运营了一年多。本实验室包括研究生、研究生，尤其是本科生进行的软件项目。

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.