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, and 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 lead to great advances in our understanding of the topological properties of three and four-dimensional spaces in the past twenty years. 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 study of the investigator, and the present proposal deals with further developments of these methods.
