Low Dimensional Topology and Heegaard Floer homology

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The main theme of this project is the study of knots, three dimensional manifolds, and smooth 4-manifolds. One of the central tools is Heegaard Floer homology that was developed by Peter Ozsvath and the Principal Investigator. The PI studies new applications of these techniques for various surgery problems for three dimensional manifolds, slice knots, and exotic structures on smooth 4-manifolds.The proposal also deals with topological and computational aspects of Heegaard Floer homology. This involves the study of special multiply pointed Heegaard diagrams to build up a topological version of the theory. In a different direction new advances in Heegaard Floer homology are used to study three-manifolds with simple Floer homologies.The project studies three and four dimensional spaces and central problems in Knot Theory and Low Dimensional Topology. Heegaard Floer homology provides various new tools, such as topological invariants and surgery formulas, for low-dimensional mathematical objects. This theory is closely related to gauge theoretical invariants that originated from interactions between Mathematics and Mathematical Physics. It is expected that advances in Heegaard Floer homology will lead to a better understanding of these gauge theoretical invariants as well.