Curves, Surfaces, and 3-Manifolds: Geometry, Topology, and Dynamics in the Mapping Class Group and Beyond

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Of central importance in the study of geometry and topology is an algebraic object called the mapping class group, which roughly corresponds to the set of symmetries of the space under consideration. In this project the spaces of interest are one-, two- and three-dimensional. These are called curves, surfaces, and three-manifolds. A common thread in this research program is leveraging combinatorial constructions of curves on surfaces to answer broader geometric and algebraic questions. At the heart of the project is the interplay between several fields of mathematics, namely- topology, combinatorics, and algebra, that arises in the study of surfaces through their mapping class groups. The project includes directions that are suitable for both graduate and undergraduate research. The PI will continue conference organization as well as her engagement with community-wide initiatives to address issues of equity and justice in mathematics and community-building initiatives for members of underrepresented groups. The mapping class group of a surface captures its topological symmetries. This group appears across mathematics from algebraic geometry (e.g., the moduli space of a Riemann surface) to low-dimensional topology (e.g., Heegaard splittings of 3-manifolds) to dynamics (e.g., the topological entropy of braids). Thus, the study of surfaces can provide insights into a wide range of mathematical problems while making use of the structure and rigidity found in dimensions one and two. The PI and her collaborators will investigate the geometry and topology of surfaces, using a three-pronged approach: 1) the algebra and dynamics of the mapping class group of surfaces of both finite and infinite type; 2) the length spectra of non-positively curved metrics on surfaces; and, 3) the coarse-geometry of various graphs naturally associated to 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.