Heegaard Floer homology, knots, and three-manifolds

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The central themes of this project are Low Dimensional Topology and Knot Theory. One of the main techniques involved is Heegaard Floer homology, developed by Peter Ozsvath and the principal investigator. The investigator will study knots, three-manifolds and smooth 4-manifolds by using recent advances and developing new techniques. In a different direction the PI investigates the relationship between different homological invariants for knots and links. The PI also studies foundational problems in Heegaard Floer homology and knot Floer homology.The project studies various three-dimensional spaces (three-manifolds), and also embedded loops inside three-manifolds. Heegaard Floer homology, developed by the principal investigator and Peter Ozsvath, plays a central role in the project. This theory has close ties to gauge theoretical invariants that originated from interactions between Mathematics and Mathematical Physics. Some of the recent results in Knot and Heegaard Floer homologies are also related to different mathematical fields, such as combinatorial constructions and algebraic techniques. A better understanding of the central problems in the project are expected to lead to further discoveries and new bridges between different mathematical fields. The project is also aimed to enhance graduate student research and introduce graduate students to the areas of Low Dimensional Topology and Heegaard Floer invariants.