Dynamics and the Classification of Geometries on Manifolds

流形上的动力学和几何分类

• 批准号: 2203493
• 负责人: William Goldman
• 金额: $ 35万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2027-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203493&HistoricalAwards=false
• 关键词: Dynamics; Classification; Geometries; Manifolds

This project will conduct research into the classification of locally homogeneous geometric structures on manifolds. This is a subject on the interface of geometry and topology and has been particularly important in the study of topology in 2 and 3 dimensions. A simple example is that there is no metrically accurate world atlas: the topology of the sphere (the earth) is not compatible with Euclidean geometry, although the topology of a torus (an inner tube) is. The project is closely related to other fields of mathematics, including group theory, algebraic geometry, and dynamical systems. The research has an experimental component and involves visualization techniques. The project will include software development for these questions and graduate students will receive training through involvement in the research.The research includes four main threads: (1) the classification of affine and projective structures with solvable holonomy; (2) relations between the certain geometries and symplectic/contact geometry; (3) dynamical systems arising from mapping class group actions on deformation spaces and character varieties; and (4) explicit constructions of topological automorphisms and the computation of finer dynamical invariants. The latter involves natural Poisson/symplectic geometry on the deformation spaces. The project also includes mentoring and training of students as well as outreach activities via the Experimental Geometry Lab at the PI's home institution.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将研究流形的局部几何结构的分类。这是几何和拓扑界面上的主题，在2和3维度的拓扑研究中尤为重要。一个简单的例子是，没有指标准确的世界地图：球体的拓扑（地球）与欧几里得的几何形状不兼容，尽管圆环的拓扑（内管）是。该项目与其他数学领域密切相关，包括群体理论，代数几何和动态系统。该研究具有实验成分，并涉及可视化技术。该项目将包括针对这些问题的软件开发，研究生将通过参与研究来接受培训。研究包括四个主要主题：（1）具有可解决的自律的仿射和投射结构的分类； （2）某些几何形状与符号/接触几何形状之间的关系； （3）由映射组对变形空间和角色品种的群组动作产生的动态系统； （4）拓扑自动形态的显式结构和更精细的动力不变的计算。后者涉及变形空间上的自然泊松/符号几何形状。 该项目还包括通过PI家庭机构的实验几何实验室进行的指导和培训以及宣传活动。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准来通过评估来支持的。