Three-Dimensional Manifolds, Heegaard Floer Homology and Knot Theory

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Some of the central objects studied in the project are knotted circles in the three-dimensional space. These mathematical objects are used to study more complicated three- and four-dimensional spaces called manifolds that form the foundations of an area of mathematics known as low dimensional topology. Building on the principal investigator's prior work with Peter Ozsvath, this project will use modern gauge-theoretical and topological methods to study various open problems and conjectures related to these spaces, investigate recent methods and developments, and develop new invariants for knots and links. The project also aims to enhance graduate student research by introducing them to different areas of low dimensional topology and teaching recent methods in knot and link Floer homology and Heegaard Floer theory. Heegaard Floer homology and knot Floer homology constructions use methods from Topology and Symplectic Geometry through the study of Heegaard diagrams and holomorphic disks in symmetric products of a Heegaard surface. There have been some recent breakthroughs in knot Floer homology, and the principal investigator will apply these new algebraic and symplectic techniques to study problems in Low Dimensional Topology. Specifically, the PI plans to generalize bordered Floer homology methods to links, with potential applications to the Thurston norm and other topological questions; develop tools to compute the master complex for knot Floer homology using the Pong algebra; study the double point knot Floer homology developed by Lipschitz; and use the HF mixed invariant to study exotic smooth structures on 4-manifolds. Other projects include investigating the Berge conjecture, cosmetic surgery conjecture, and mutations.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.