CAREER: Rigidity in Mapping class groups and homeomorphism groups

职业：映射类群和同胚群中的刚性

• 批准号: 2339110
• 负责人: Lei Chen
• 金额: $ 49.12万
• 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: 2024
• 资助国家: 美国
• 起止时间: 2024-08-01 至 2029-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2339110&HistoricalAwards=false
• 关键词: CAREER; Rigidity; Mapping; class; groups

In geometry and topology, one of the most fundamental objects is to study various geometric groups and their features. This project will investigate the rigidity problems concerning mapping class groups and homeomorphism groups of manifolds. The PI will use methods from dynamical systems, geometric group theory, low dimensional topology, and differential geometry. The educational activities include high school outreach, undergraduates research through REU projects, mentoring graduate students in the home institution, and workshops organizations. Symmetry is a pervasive concept in mathematics. In the study of differential topology, the full symmetry group is the diffeomorphism group of a manifold. There are two sides of a diffeomorphism group: one is the mapping class group, the group of connected components of a diffeomorphism group; the other is the identity component of a diffeomorphism group, which is a connected topological group. The PI will study these groups using both geometric group theory through the study of how those groups act on certain complexes and dynamical tools through the study of how those groups act on other manifolds.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.

在几何学和拓扑学中，最基本的目标之一就是研究各种几何群及其特征。本计画将研究流形上映射类群与同胚群的刚性问题。PI将使用动力系统，几何群论，低维拓扑和微分几何的方法。教育活动包括高中外展，通过REU项目的本科生研究，指导研究生在家庭机构，和研讨会组织。对称性是数学中的一个普遍概念。在微分拓扑学的研究中，全对称群是流形的同构群。复同态群有两个方面：一个是映射类群，复同态群的连通分支群;另一个是复同态群的单位分支，它是一个连通拓扑群。PI将通过研究这些群体如何作用于某些复合物，使用几何群论和动力学工具来研究这些群体。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。