CAREER: Rigidity in Mapping class groups and homeomorphism groups

在几何学和拓扑学中，最基本的对象之一就是研究各种几何群及其特征。本课题将研究流形的映射类群和同胚群的刚性问题。PI将使用动力系统、几何群论、低维拓扑学和微分几何的方法。教育活动包括高中外展，通过REU项目进行本科生研究，在家庭机构指导研究生，以及组织研讨会。对称性是数学中一个普遍存在的概念。在微分拓扑学的研究中，全对称群是流形的微分同胚群。微分同胚群有两个方面：一个是映射类群，即微分同胚群的连通分支群；另一个是微分同胚群的单位分支，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体来研究这些群体，并通过研究这些群体如何作用于其他流形来使用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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.