Heegaard diagrams and holomorphic disks

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In recent work with Zoltan Szabo, the principal investigator constructed invariants for three- and four-dimensional spaces, called Heegaard Floer homology. In further joint work with Robert Lipshitz and Dylan Thurston, these invariants are extended down, to define bordered Floer homology. Bordered Floer homology gives invariants for two-dimensional surfaces and three-dimensional spaces with two-dimensional boundary. These bordered invariants shed light on Heegaard Floer homology in its original setting. In particular, they can be used to give conceptual calculations of certain Heegaard Floer invariants for closed manifolds. The project aims to understands the invariants better, extending the "bordered theory" to a broader context.The introduction of equations with origins in mathematical physics has lead to great advances in our understanding of the topological properties of three and four-dimensional spaces over the past twenty-five years. Further progress in this area is facilitated by an alternative, more geometric understanding of the data derived from these equations, known as "Heegaard Floer homology", developed by the investigator in collaboration with Zoltan Szabo. This three and four-dimensional story can be extended to cover two-dimensional objects, as well, in a a new theory, "bordered Floer homology", developed by the investigator in collaboration with Robert Lipshitz and Dylan Thurston. The proposal aims to further develop both of these tools and apply them to study topological questions.