Geometric Structures on Surfaces

表面上的几何结构

• 批准号: 0706781
• 负责人: William Goldman
• 金额: $ 89.13万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706781&HistoricalAwards=false
• 关键词: Geometric; Structures; Surfaces

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.

PI建议继续在低维流形，离散组的变形和表面组表示模量的几何结构上进行研究。一个问题涉及完成适当仿射动作的分类问题，这是几何学中的基本问题，其历史可以追溯到20世纪初期欧几里得晶体学组的分类。在过去的25年中，在二维双曲几何形状中，三个维度的分类已缩小为问题。这个一般问题的另一面是源于观察到的几何结构的变形空间本身是在基本组的表示空间上进行的。这自然会导致有关映射课程组的动作的动态问题。表面组表示的模量空间支持不变的泊松结构，以及根据共形结构的更精细的结构。这些研究的依赖性在Teichmueller空间上对于理解表面组表示的动力学至关重要。这项研究重点是拓扑结构与几何形状之间的关系。尽管拓扑涉及点的宽松定性组织，但几何形状涉及定量测量，例如距离，角度和区域。通过对称性比较这两个观点，从而导致涉及组的代数计算。实验几何实验室一直是PI及其学生和合作者（在各个层面）开发这些计算和可视化工具的中心。以上许多项目都有一个实验组件，PI将继续与学生（从高中到博士后）合作，以开发用户友好的软件，以协助他们的调查和传播。