Geometric Structures on Surfaces

PI建议继续他在低维流形上的几何结构、离散群的变形和曲面群表示的模的研究。其中一个问题涉及完成适当仿射作用的分类问题，这是几何学中的一个基本问题，其历史可以追溯到20世纪初欧几里得晶体群的分类。在过去的25年里，三维的分类已经归结为二维双曲几何的问题。这个普遍问题的另一个方面源于这样的观察，即几何结构的变形空间本身是在基本群的表示空间上局部建模的。这自然导致了关于映射类组的动作的动态问题。表面群表示的模空间支持不变的泊松结构，以及依赖于共形结构的更精细的结构。当Riemann曲面在Teichmueller空间上变化时，这些依赖关系对于理解表面群表示的动力学是至关重要的。拓扑学关注的是点模式的松散定性组织，而几何学关注的是距离、角度和面积等定量测量。这两种观点通过对称性进行比较，从而导致涉及群的代数计算。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。上述许多项目都有一个实验部分，国际和平协会将继续与学生(从高中生到博士后)合作，开发用户友好的软件，以协助他们的调查和传播。

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.