CAREER: Mapping class groups, diffeomorphism groups, and moduli spaces

拓扑学是研究空间及其基本性质的学科。与这个项目相关的有两个基本原则。首先，一个给定的空间可以通过理解它的对称性来进行卓有成效的探索。其次，如果你想研究一些空间集合(举个简单的例子，想想平面中的多边形)，那么把这个空间集合变成一个具有启发性的单一空间(通常称为模空间)会很有帮助。在教育部分，国际和平研究所将组织拓扑学讲习班，目的是向研究生介绍活跃的研究领域，并为他们提供为这些领域做出贡献的工具。PI将安排暑期定向阅读计划，帮助本科生为研究和扩大参与做准备。最后，PI将继续通过数学圈向高中生伸出援手。这项资助将通过在校园举办的活动帮助增加教师的参与。这个项目关注关于流形和模空间的集体行动。关于群作用，重点是确定一组同胚类的同态何时可以实现为一组同胚群。这个问题被称为尼尔森实现，可以追溯到20世纪初尼尔森的工作，并与动力学、叶化理论以及流形和纤维丛的几何有关。在这个方案中感兴趣的模空间与低维拓扑和非正曲率有关。一个中心目标是计算感兴趣的模空间的新的同调和同伦不变量。具体研究项目如下。(1)解决三维和四维流形的Nielsen实现问题。(2)利用与Borel猜想相关的刚性结果研究具有奇异光滑结构的非球面流形上的有限群作用。(3)通过将问题归结为热带几何和组合学，证明了扭转环面在模空间的有限覆盖的同调中的非平凡性。(4)利用曲线复形构造新的曲面丛特征类。(5)在Farrell-Jones工作的基础上，计算高维双曲流形的嵌入空间的同伦群。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。

Topology is the study of spaces and their fundamental properties. Related to this project, there are two foundational principles. First, a given space can be fruitfully probed by understanding its symmetries. Second, if one wants to study some collection of spaces (for a simple example, think of polygons in the plane), then it can be helpful to turn this collection of spaces into a single space (often called a moduli space) whose properties are illuminating. For the educational component, the PI will organize topology workshops with the aim of introducing graduate students to active areas of research and giving them tools to contribute to these areas. The PI will orchestrate summer directed reading programs that will help prepare undergraduate students for research and broaden participation. Finally, the PI will continue his outreach to high school students through math circles. Support from this grant will help increase teacher involvement through events hosted on campus.This project is concerned with group actions on manifolds and moduli spaces. Regarding group actions, the focus is on determining when a group of isotopy classes of homeomorphisms can be realized as a group of homeomorphisms. This problem, known as Nielsen realization, dates back to the work of Nielsen in the early 1900s and has connections to dynamics, foliation theory, and the geometry of manifolds and fiber bundles. The moduli spaces of interest in this proposal relate to low-dimensional topology and nonpositive curvature. A central goal is to compute new homological and homotopical invariants of the moduli spaces of interest. The specific research project are as follows. (1) Solve Nielsen realization problems for 3- and 4-dimensional manifolds. (2) Study finite groups actions on aspherical manifolds with exotic smooth structures using rigidity results related to the Borel conjecture. (3) Show non-triviality of twist tori in the homology of finite covers of moduli space by reducing the problem to tropical geometry and combinatorics. (4) Construct new characteristic classes of surface bundles using the curve complex. (5) Compute homotopy groups of embedding spaces for high dimensional hyperbolic manifolds, building on work of Farrell-Jones.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.