Heegaard Floer homology and its applications to low-dimensional topology

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This project deals with invariants in low-dimensional topology which come from gauge theory and symplectic geometry, especially Heegaard Floer homology. The focus will be the applications of Heegaard Floer homology to low-dimensional topology, and the connection between Heegaard Floer homology and other aspects of low-dimensional topology. One problem we plan to study is unbalanced sutured manifold decompositions for Heegaard Floer homology. This is related to characterizing incompressible surfaces using Heegaard Floer homology. Another problem we will study is the botany problem in Heegaard Floer homology, namely, to what extent one may determine a manifold using its Heegaard Floer homology. These problems are related to questions about Dehn surgeries and Khovanov homology. We will also address the applications of Floer homology to 4-dimensional topology, for example, the topology of knot surgeries on the K3 surface.In the microscopic world, macromolecules are often visualized as knots and links in the three-dimensional space. The knot invariants studied in this project thus provide important tools in analyzing the structures of macromolecules: Some questions our methods can study are, how firmly the macromolecules are interlocked, how to detect their chirality, and how to change their topological structure. These features are extremely significant in nanotechnology and pharmacology.