Dynamics of Moduli Spaces and Surface Groups

这个项目涉及流形上几何结构的分类，流形上模空间的几何，以及由曲面的自同构产生的模空间上的动力系统。例如，完全仿射3-流形的行为类似于双曲黎曼曲面。对于一大类例子，变形空间最近被证明是双曲曲面变形空间上的一丛凸锥。该项目的一个方面是对这类流形的分类和动力学研究，以及它们的共形和常曲率推广。这个项目的另一个方面涉及到由表面几何结构分类产生的动力系统。映射类群在特征簇上的作用是一个与曲面的基本群相关的动力系统和与李群相关的几何系统。这些模空间支持不变的辛结构，以及依赖于具有给定基本群的Riemann曲面的更精细结构。当黎曼面变化时，这些结构是如何变化的？这些结构的几何形状如何影响其表面的拓扑对称性的动力学？映射类群作用的动力学复杂性如何伴随着模空间的拓扑复杂性？这些问题存在于几何和拓扑的交界处，通过对称性的概念统一起来。几何学涉及度量数量关系，而拓扑学则涉及抽象的‘空间’中‘点’的松散均衡组织。几何学通过它们各自的对称性与拓扑学有关，可以通过群论技术对其进行代数研究。这些问题对于实验研究来说是很自然的。马里兰大学的实验几何实验室为感兴趣的学生提供在模拟环境中进行数学实验的机会。它的目标是传播和推广与上述项目相关的数学。该实验室将继续成为本科生研究实验、外展报告和各级互动项目的中心。

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.