The proposal studies "Heegaard Floer homology," which is an invaraint for three-and four-dimensional spaces constructed by the investigator in collaboration with Zoltan Szabo. A variant of this construction, called "knot Floer homology", can be used to study knotted curves in three-dimensional space. A three-dimensional space can be built up out of smaller pieces which fit together along their boundaries. A further aspect of the proposal deals with how to reconstruct the Heegaard Floer homology of a three-dimensional space in terms of data associated to its component pieces. This reconstruction procedure, entitled "bordered Floer homology", is studied in collaboration with Robert Lipshitz and Dylan Thurston. The proposal aims to further strengthen the bordered theory, and explore its applications. A better understanding of these constructions will lead to further applications of these new constructions to knot theory and the topology of three- and four-dimensional spaces. 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 which was 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 proposal aims to study bordered Floer homology as a tool for studying various versions of Heegaard Floer homology and knot Floer homology. Part of the proposal 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 will be extended to a tool for studying and computing knot Floer homology. Applications of these structures include a study of new concordance homomorphisms from knot Floer homology.
