Heegaard Diagrams and Holomorphic Disks

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The proposal deals with studying Heegaard Floer homology, the newinvariants for three-and four-dimensional spaces constructed by theinvestigator in collaboration with Zoltan Szabo. These invariants givenew insight into earlier invariants for spaces constructed usinggauge theoretic techniques: the Donaldson and Seiberg-Witteninvariants. In addition, they can be used to address oldertopological questions, including specifically questions about Dehnsurgery on classical knots. This project aims to understand theseinvariants better, and give further applications to knot theory andthe topology of three- and four-dimensional manifolds.The introduction of equations with origins in mathematical physics has lead to great advances in our understanding of the topologicalproperties of three and four-dimensional spaces in the past twentyyears. Further progress in this area is facilitated by an alternative,more geometric understanding of the data derived from these equations.These alternative more geometric methods have been the object of studyof the investigator, and the present proposal deals with furtherdevelopments of these methods.