CAREER: Heegaard Floer homology and low-dimensional topology

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Topology is the study of the shape of different spaces. One- and two-dimensional spaces are well-understood, as are dimensions five and above; roughly, in the former, there are not enough dimensions for interesting phenomena, and in the latter, there are so many dimensions that anything interesting has enough room to become uninteresting. Low-dimensional topology focuses on three- and four-dimensions, where many unique phenomena occur. One central question is whether a knotted loop in three-dimensions becomes unknotted when one allows a fourth dimension. One can also ask what happens upon cutting a knot out of space, and then filling in the resulting void in a different way. Knot theory has applications to the very small (e.g., the behavior of knotted strands of DNA) as well as the extremely large (e.g., the shape of the universe). In tandem with the research component, the PI plans to further her mentoring and outreach efforts, for example, by supervising undergraduate and graduate research, and by leading local math events for middle and high school students. She will also organize a workshop for undergraduates to present their research and learn more about careers in mathematics. Heegaard Floer homology, developed by Ozsvath and Szabo, is a powerful tool for understanding low-dimensional topology. The PI plans to use several recent developments to provide answers to long-standing questions in the field. For example, the recently defined involutive Heegaard Floer homology of Hendricks and Manolescu may have applications to understanding divisibility in the concordance group. In a different direction, the PI plans to study which manifolds arise from surgery on an n-component link, using the link surgery formula of Manolescu and Ozsvath. She also proposes to study homology cobordism, which is closely related to knot concordance, with the hope of providing obstructions to being homology cobordant to surgery on a knot.