CAREER: Connections between algebraic and geometric invariants in low-dimensional topology

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The principal investigator will probe the connection between two classes of "homology-type" invariants of objects in low-dimensional topology: Khovanov homology, whose (algebraic) construction originates in the higher representation theory of quantum groups and Heegaard Floer homology, whose (geometric/analytic) construction arises from ideas in symplectic geometry and gauge theory. This project builds on, and seeks to re-explain from first principles, foundational work of Ozsvath-Szabo, who constructed a deformation of the (reduced) Khovanov homology of a link, thereby connecting it to the Heegaard Floer homology of its double-branched cover. By understanding the relationship between "open" versions of the two theories, the P.I. will obtain applications to questions about braids and tangles.The broad aim of the present project is to improve our understanding of the topology of 3- and 4-dimensional spaces, i.e., the properties of these spaces that remain unchanged under stretching and contracting (but not under tearing and gluing). Topological ideas underpin the development of efficient computer chips and information networks. The shapes of molecules and proteins determine their electrical properties and biological functions. Basing quantum computing algorithms on large-scale features of a quantum system minimizes their susceptibility to random error. Moreover, knot theory, the study of loops imbedded in 3-dimensional space, has become increasingly important in our understanding of how DNA behaves in cells.