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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。这个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，而几何学则专注于局部精细结构。拓扑学的抽象性质使其成为贯穿整个科学的有用工具，从询问关于宇宙形状的问题到理解复杂网络的大规模属性。这个项目专注于了解黎曼曲面的拓扑对称性，黎曼曲面是二维对象，包括复杂平面、二维球体和看起来像甜甜圈表面的对象。黎曼曲面几乎出现在数学的每一个分支中，并在科学中自然而然地出现，特别是通过弦理论和微分方程式的解。此外，该项目有几个外展部分，旨在支持学生从事数学科学事业。PI将创建一个致力于建立校友网络的组织，以促进社区关系并创造实习机会。此外，PI将主办几个职业小组，其中包括以前的学生在不同领域工作的情况，并将为本科生提供研究经验。主要研究黎曼曲面的(拓扑)映射类群的代数结构。在有限区域情形下，映射类群的结构被很好地理解，并且该理论与几何群论和Teichmüler理论有很深的联系。这个项目调查的是无限大区域的情况，而人们对此知之甚少。主要目的是刻画无限区域黎曼曲面映射类群的可数指标正规子群。这项研究将在低维拓扑学和拓扑群理论的最新发展之间建立新的联系。这一奖项反映了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。

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.