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

职业：映射类群、微分同胚群和模空间

• 批准号: 2236705
• 负责人: Bena Tshishiku
• 金额: $ 54.96万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2028-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2236705&HistoricalAwards=false
• 关键词: CAREER; Mapping; class; groups; diffeomorphism

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.

拓扑是对空间及其基本特性的研究。与该项目有关，有两个基本原则。首先，可以通过理解其对称性来良好地探测给定的空间。其次，如果一个人想研究一些空间集合（例如一个简单的例子，请考虑到平面中的多边形），那么将这个空间集合成一个属性的单个空间（通常称为模量空间）会很有帮助。对于教育组成部分，PI将组织拓扑研讨会，目的是将研究生介绍给活跃的研究领域，并为他们提供为这些领域做出贡献的工具。 PI将策划夏季定向的阅读计划，以帮助本科生为研究和扩大参与做准备。最后，PI将通过数学圈继续向高中生的宣讲。这笔赠款的支持将通过在校园内举办的活动来帮助增加教师的参与。该项目涉及对流形和模量空间的小组行动。关于小组行动，重点是确定何时可以将一组同态同态类别作为一组同构形态学。这个被称为尼尔森实现的问题可以追溯到尼尔森在1900年代初的工作，并且与动力学，叶面理论和歧管和纤维束的几何形状有联系。该提案中感兴趣的模量空间与低维拓扑和非阳性曲率有关。一个核心目标是计算感兴趣的模量空间的新的同源和同型不变。特定的研究项目如下。 （1）解决了尼尔森（Nielsen）的3维流形的实现问题。 （2）研究有限组对具有外在平滑结构的非球面歧管的作用，使用与鲍尔猜想有关的刚性结果。 （3）通过将问题减少到热带几何形状和组合物质，在模量空间有限覆盖的同源性中表现出扭曲托里的非平凡性。 （4）使用曲线复合物构建表面束的新特征类别。 （5）在Farrell-Jones的工作基础上计算高维双曲线歧管的嵌入空间的同拷贝组。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点评估来支持的，并具有更广泛的影响。