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

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。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，侧重于理解空间的全球大尺度结构，而几何学则侧重于局部精细结构。拓扑的抽象性质使其成为整个科学领域的有用工具，从提出有关宇宙形状的问题到理解复杂网络的大规模属性。 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，自然也出现在科学中，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职业。 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将举办几个职业小组，其中包括在不同领域工作的前学生，并将为本科生提供研究经验。研究的重点是理解黎曼曲面的（拓扑）映射类群的代数结构。 在有限区域的情况下，映射类群的结构是很好理解的，并且该理论与几何群论和Teichmüller理论有很深的联系。 这个项目调查的无限区域的情况下，相对知之甚少。 主要目的是刻画无限面积黎曼曲面的映射类群的可数指数正规子群。 该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NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.