Structures in Khovanov-Rozansky homology

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A knot is a closed loop in three-dimensional space, a link is a union of several such loops, possibly linked with each other. Besides the immediate mathematical applications, knot theory has implications in physics of quantum systems and in the study of chemical and biological properties of long knotted molecules. The central questions in knot theory are the classification problem (Can a knot be transformed to another knot without tearing its strands? Can it be untangled to look like an ordinary circle?) and the study of the geometric properties of knots or links. Both of these questions can be partially answered with the help of a link invariant: a collection of numbers that does not change under continuous stretching of a link. Two links are different if their invariants are different. The project is focused on uncovering and studying the patterns and symmetries of link invariants. The project will provide research training opportunities for students and post-docs.In more detail, the project is focused on Khovanov-Rozansky link homology which generalizes the celebrated HOMFLY polynomial. To a link such a theory associates a triply graded vector space which carries an action of many interesting operators. The project will build upon and unify a variety of existing operations (such as Rasmussen's differentials, tautological classes and Witt algebra action), to study the commutation relations between these and to define new ones. The action of such a large algebra is expected to unravel some patterns and symmetries in link homology. Another motivation comes from geometric models for link homology: these include sheaves on Hilbert schemes of points on the plane, braid varieties, Hilbert schemes on singular curves and affine Springer fibers. In many cases, geometric representation theory then predicts an action of large algebras in homology, and the investigator will translate these into explicit actions in link homology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.