CAREER: Bordered Floer homology and applications

Dartmouth College, Hanover NH

Investigators

Abstract

Low-dimensional topology studies the shapes of spaces in dimensions one through four, and has applications ranging from physics and cosmology in which the shape of the universe is studies to biochemistry, which seeks to understand the behavior of knotted DNA. Closely related to the study of 3- and 4-dimensional spaces is the study of knots, which can be viewed as tied in space. This project will further develop and apply recent cut-and-paste tools in low-dimensional topology. Part of the project concerns the question of what kinds of geometric structures, specifically what kind of “contact structures”, a given 3-dimensional space can support. In addition to direct applications to mathematics, contact structures have found numerous applications in physics, including classical mechanics, thermodynamics, and control theory. In parallel to the research component, the PI will further their educational and outreach efforts. For example, the PI will supervise undergraduate and graduate research, and establish a high school enrichment program. In the early 2000s, Ozsvath and Szabo developed a package of powerful invariants for knots and 3- and 4-dimensional spaces, generally known as Heegaard Floer homology. Heegaard Floer homology has since taken a major place in low-dimensional topology, and has helped researchers obtain many new results and settle numerous old conjectures. Bordered Floer homology generalizes Heegaard Floer homology to manifolds with boundary, and provides nice techniques for computing the Heegaard Floer invariants of closed manifolds, by cutting a manifold into pieces (e.g. a knot into tangles), and studying the individual pieces and their gluing. This project seeks to develop further the bordered Heegaard Floer tools we currently have. The PI plans to continue to develop an invariant from bordered Floer homology for contact 3-manifolds with convex boundary, and use it to address open questions in contact topology; extend bordered Floer homology and tangle Floer homology to integral coefficients; understand and develop the connections between knot Floer homology and quantum algebra. 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.