Heegaard Diagrams and Holomorphic Disks

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The proposal deals with studying ``Heegaard Floer homology,'' the new invariants for three-and four-dimensional spaces constructed by the investigator in collaboration with Zoltan Szabo. These invariants give new insight into earlier invariants for spaces constructed using ``gauge theoretic'' techniques: the Donaldson and Seiberg-Witten invariants. In addition, they can be used to address older topological questions, including specifically questions about Dehn surgery on classical knots. This project aims to understand these invariants better, in veiw of recent computational advances in the theory, and to give further applications to knot theory and the topology of three- and four-dimensional manifolds.The introduction of equations with origins in mathematical physics has has lead to great advances in our understanding of the topoligical properties of three and four-dimensional spaces over the past twenty 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'', a new theory developed by the investigator in collaboration with Zoltan Szabo. Heegaard Floer homology has already shed new light on the structure of three- and four-dimensional spaces. These new methods have been the object of study of the investigator, and the present proposal deals further developments using these techniques.