LEAPS-MPS: Topological Symmetries of Non-Compact Riemann Surfaces

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。该项目涉及拓扑学领域的研究，拓扑学是数学的一个分支，专注于理解空间的全局大尺度结构，而几何则专注于局部精细结构。从提出关于宇宙形状的问题到理解复杂网络的大规模特性，拓扑学的抽象性质使其成为贯穿科学的有用工具。这个项目的重点是理解黎曼曲面的拓扑对称性，黎曼曲面是二维物体，包括复平面、二维球体和看起来像甜甜圈表面的物体。黎曼曲面几乎出现在数学的每一个分支中，并且自然地出现在科学中，特别是通过弦理论和微分方程的解。此外，该项目还有几个外联部分，旨在支持学生从事数学科学方面的职业。PI将创建一个组织，致力于建立校友网络，以促进社区关系并创造实习机会。此外，PI将举办几场职业讨论会，邀请在不同领域工作的前学生参加，并将为本科生提供研究经验。研究的重点是理解黎曼曲面（拓扑）映射类群的代数结构。在有限面积情况下，映射类群的结构被很好地理解，该理论与几何群论和teichmller理论有着深刻的联系。这个项目调查的是无限面积的情况，在那里人们所知相对较少。主要目的是刻画无限面积黎曼曲面的映射类群的可数指标法向子群。这项研究将在低维拓扑和拓扑群论的最新发展之间建立新的联系。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。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project concerns research within the field of topology, a branch of mathematics with a focus on understanding the global large-scale structure of spaces, in contrast to geometry’s focus on local fine structure. The abstract nature of topology has made it a useful tool throughout the sciences, from asking questions regarding the shape of the universe to understanding the large-scale properties of complex networks. This project focuses on understanding topological symmetries of Riemann surfaces, which are two-dimensional objects, including the complex plane, the two-dimensional sphere, and objects that look like the surface of a doughnut. Riemann surfaces appear in almost every branch of mathematics and naturally arise in science, especially via string theory and via the solutions of differential equations. In addition, the project has several outreach components aimed at supporting students in pursuing a career in the mathematical sciences. The PI will create an organization dedicated to building a network of alumni to foster relationships in the community and create internship opportunities. Additionally, the PI will host several career panels featuring former students working in a diverse range of fields and will provide research experiences for undergraduates. The research is focused on understanding the algebraic structure of the (topological) mapping class group of a Riemann surface. In the finite-area case, the structure of mapping class groups is well understood, and the theory has deep connections to geometric group theory and Teichmüller theory. This project investigates the infinite-area case, where relatively little is known. The main goal is to characterize the countable-index normal subgroups of mapping class groups of infinite-area Riemann surfaces. The investigation will forge new connections between low-dimensional topology and recent developments in topological group theory.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.