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

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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.