Heegaard Diagrams and Holomorphic Disks

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This project in topology studies "Heegaard Floer homology," which is an invariant for three-and four-dimensional spaces. A variant of this construction, called "knot Floer homology," provides new methods for studying knotted curves in three-dimensional space and answering classical questions about such objects. Both Heegaard Floer homology and the associated knot invariant come in two versions: a simplified one, and a richer one. The simplified version provides interesting three-dimensional information, but the richer version also sheds light on four-dimensional aspects of the theory. 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. Heegaard Floer homology continues to provide novel methods for attacking old problems. The award provides support for graduate students to be engaged in related research.In collaboration with Robert Lipshitz and Dylan Thurston, the PI defined "bordered Floer homology," a technique for reconstructing the simplified version of Heegaard Floer homology from a three-dimensional space that is decomposed into simple component pieces. In collaboration with Szabo, the PI used similar methods to describe the simplified version of knot Floer homology. This project deals with further developing bordered techniques for three-dimensional spaces and also for knots, extending them to the richer versions. This project aims to both give a better conceptual understanding of these invariants, and to give effective computational techniques for calculating them. The project 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. Having established most of the algebraic foundations of this theory, the PI will now turn to the analytical aspects, with a view towards constructing modules for three-manifolds with torus boundary, and a pairing theorem for computing the (unspecialized) Heegaard Floer homology of a three-manifold decomposed along a torus, in terms of the modules assocaited to the pieces. In a different direction, bordered Floer homology is extended to a tool for studying and computing (unspecialized) knot Floer homology.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.