CAREER: The Algebra, Geometry, and Topology of Infinite Surfaces

University Of Utah, Salt Lake City UT

Investigators

Abstract

This mathematics research project focuses on questions in geometry and topology, both of which are concerned with the study of the shapes of objects. The project studies the properties of surfaces, which fall into two categories: finite-type and infinite-type. The theory of finite-type surfaces has been historically more developed than that of infinite-type surfaces, partly because there is a simple classification of all finite-type surfaces. The primary goal of the research project is to significantly deepen understanding of infinite-type surfaces, which are ubiquitous in topology, geometry, and dynamics. The first part is aimed at characterizing their geometric symmetries (isometry groups). The second and third parts concern their topological symmetries (mapping class groups), with the long-term goal of completely classifying the different types of topological symmetries. The educational component of this project consists of three parts. The first part is a research training and professional development graduate student workshop for students in algebra, geometry, topology, and number theory. This workshop is aimed at early-career graduate students and intended to serve as a bridge between successful programs like the EDGE summer program and research-focused workshops for advanced graduate students. One of the goals is to support the participants in their transition between coursework and research-based mathematics. The second part of the educational component is the expansion of an existing high school outreach program in Salt Lake City. The third part is a speaker series featuring prominent individuals in STEM. The research focus of this project is on infinite-type surfaces and their groups of symmetries. Infinite-type surfaces arise naturally in many contexts, such as in the study of group actions on the plane, and are intimately related to the study of quasiconformal maps. The first part of this project, aimed at characterizing isometry groups of infinite-type surfaces, is inspired by Felix Klein's suggestion from 1872 that groups of geometric symmetries be used to better understand the geometry of Riemann surfaces. The results regarding isometry groups are used in-turn to produce algebraic invariants of the mapping class group. This group can be thought of as the group of topological symmetries of the surface and is the focus of the second and third parts of the project. In particular, the second part is aimed at producing algebraic invariants of the mapping class groups of infinite-type surfaces via subgroup constructions, and the third part focuses on using the actions of mapping class groups on hyperbolic graphs to produce a Nielsen-Thurston type classification for these surfaces. 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.