CAREER: Rigidity in Mapping class groups and homeomorphism groups

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将使用来自动力系统、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，在国内机构指导研究生，以及研讨会组织。对称在数学中是一个普遍的概念。在微分拓扑的研究中，全对称群是流形的微分同构群。微分同构群有两面：一面是映射类群，即微分同构群的连通成分群；另一个是微分同构群的恒等分量，微分同构群是连通拓扑群。PI将使用几何群论来研究这些群通过研究这些群如何作用于某些复合体和动力学工具通过研究这些群如何作用于其他流形。该奖项反映了美国国家科学基金会的法定使命，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。

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.