Heegaard Floer homology and Low Dimensional Topology

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The main theme of this project is the application of Heegaard Floer homologyand gauge theory for problems in low-dimensional topology and knot theory. New surgery formulas in Heegaard Floer homology allow the PI to study applications for surgery problems in three-manifolds, and surgery characterization for knots.In a different direction Floer homology is used to investigate unknotting numbers for knots in the 3-sphere. The project includes foundational problems for knot Floer homology and the study of combinatorial techniques as well. In a different direction, gauge theoretical techniquesare used in the search for small exotic 4-manifolds. The project studies three and four dimensional spaces and some central problemsin knot theory and low dimensional Topology. One of the main tools is a relatively new theory called Heegaard Floer homology. This theory has close ties 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.Another integral part of the project is the study of unknotting numbers for knots.While this is a classical invariant in knot theory, the unknotting process is also relevant in the study of knotted DNA. A way to measure the complexity is to study how many times the strand has to be cut by an enzyme and recombine itself to be unknotted.