Heegaard Diagrams and Holomorphic Disks

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This National Science Foundation award supports research to develop new tools to study three- and four-dimensional spaces as well as knotted curves, bringing together techniques from many mathematical disciplines. As these objects closely relate to our physical world and the space-time, this line of research is partially inspired by modern physics. As such, it lies at a fertile intellectual crossroads, bringing new perspectives to neighboring subjects, and providing novel methods for attacking old problems. The PI and his collaborators pioneered new invariants, known as "Heegaard-Floer homology" and "knot Floer homology," and developed a technique known as "bordered Floer homology" for effectively utilizing simple component pieces of a space. Research funded by this award deals with further developing these bordered techniques for three-dimensional spaces and for knotted curves, to get both a better conceptual understanding of these invariants, and for giving effective computational techniques for studying them.In collaboration with Zoltan Szabo, the PI constructed an invariant for three- and four-dimensional spaces known as the "Heegaard-Floer homology." Heegaard-Floer homology brings together tools from various mathematical disciplines, including symplectic geometry, analysis, and homological algebra, to study problems in knot theory and low-dimensional topology, in a way that was partially inspired by modern physics. A variant of this construction, called "knot Floer homology," is used to study knots in three-dimensional manifolds. In collaboration with Robert Lipshitz and Dylan Thurston, the PI defined "bordered Floer homology," a technique for reconstructing one variant of Heegaard-Floer homology from a three-manifold that is decomposed into simple component pieces. In the research funded by this award, the PI aims to study bordered Floer homology as a tool for studying various versions of Heegaard-Floer homology and knot Floer homology. Part of the project will start by extending the bordered theory to include the full (unspecialized) Heegaard-Floer homology for three-manifolds with torus boundary. In a different direction, bordered Floer homology is extended to a tool for studying and computing knot Floer homology.