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，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富有成效的探索。其次，如果想要研究一些空间集合（举个简单的例子，考虑平面上的多边形），那么将这些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）可能会有所帮助。在教育方面，PI将组织拓扑研讨会，目的是向研究生介绍活跃的研究领域，并为他们提供在这些领域做出贡献的工具。该项目将组织夏季指导阅读项目，帮助本科生为研究做好准备，扩大参与范围。最后，PI将通过数学圈继续与高中生进行接触。这笔赠款的支持将有助于通过在校园举办的活动增加教师的参与。这个项目关注的是流形和模空间上的群作用。在群作用方面，重点是确定一组同位素类的同胚何时可以实现为一组同胚。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、叶理理论、流形和纤维束的几何形状有关。本文研究的模空间涉及低维拓扑和非正曲率。一个中心目标是计算感兴趣的模空间的新的同调和同调不变量。具体的研究项目如下：(1)解决三维和四维流形的Nielsen实现问题。(2)利用与Borel猜想相关的刚度结果研究了具有奇异光滑结构的非球面流形上的有限群作用。(3)通过将问题简化到热带几何和组合学，证明了模空间有限复盖同调中扭转环面的非平凡性。(4)利用曲线复形构造曲面束的新特征类。(5)在Farrell-Jones的基础上，计算高维双曲流形嵌入空间的同伦群。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。

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.