Geometric Structures on Surfaces

PI建议继续他在低维流形上的几何结构，离散群的变形和表面群表示的模量方面的研究。一个问题涉及完成适当的仿射行动的分类问题，这是几何学中的一个基本问题，其历史可以追溯到世纪初的欧几里得晶体群的分类。在过去的25年里，三维的分类已经被简化为二维双曲几何的问题。这个一般问题的另一面源于观察到几何结构的变形空间本身是局部地在基本群的表示空间上建模的。这自然会导致动态问题的映射类组的行动。表面群表示的模空间支持不变的泊松结构，以及取决于共形结构的更精细的结构。当Riemann曲面在Teichmueller空间上变化时，这些关系对于理解曲面群表示的动力学是至关重要的。拓扑学关注点的模式的松散的定性组织，几何学关注定量测量，如距离，角度和面积。这两种观点是比较通过对称性，导致代数计算涉及群体。实验几何实验室一直是PI及其学生和合作者（各级）开发这些计算和可视化工具的中心。上述许多项目都有实验性的组成部分，公共信息研究所将继续与学生（从高中到博士后）合作，开发方便用户的软件，以协助其调查和传播。

The PI proposes to continue his research in geometric structures on low-dimensional manifolds, deformations of discrete groups, and moduli of surface group representations. One problem involves completing the classification problem for proper affine actions, a basic problem in geometry whose history stretches back to the classification of Euclidean crystallographic groups in the early 20th century. In the last 25 years, the classification in 3 dimensions has been reduced to a problem in 2-dimensional hyperbolic geometry. Another side of this general problem stems from the observation that deformation spaces of geometric structures are themselves locally modeled on representation spaces of fundamental groups. This naturally leads to dynamical questions on actions of mapping class groups. Moduli spaces of surface group representations support invariant Poisson structures, as well as finer structures depending on a conformal structure. The dependence of these as the Riemann surface varies over Teichmueller space is crucial to understanding dynamics of surface group representations.This research focuses on the relation between topology and geometry. While topology concerns the loose qualitative organization of patterns of points, geometry concerns quantitative measurements such as distance, angle and area. These two viewpoints are compared through symmetries, leading to algebraic calculations involving groups. The Experimental Geometry Lab has been a center for the PI and his students and collaborators (at all levels) to develop tools for these calculations and visualization. Many of the above projects have an experimental component and the PI will continue to work with students (from high school to postdocs) to develop user-friendly software to assist in their investigation and dissemination.