Mathematics Colloquia and Seminars

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In the late 90's, Khovanov introduced a homology theory for knots categorifying the Jones polynomial. Like the Jones polynomial, Khovanov's construction admits an elementary/combinatorial definition. Khovanov and Rozansky later introduced a generalization of Khovanov homology, categorifying the sl_n quantum knot invariant, and subsequent work of many authors introduced various other sl_n knot homologies. In contrast to Khovanov's original work, none of these formulations of sl_n knot homology are elementary in nature. In this talk, we'll discuss recent work of the speaker (joint with H. Queffelec) which provides an elementary formulation of Khovanov-Rozansky sl_n link homology. Surprisingly, this construction is achieved by relating the theory to categorified quantum sl_m (not a typo!) via categorical skew Howe duality. Time permitting, we'll discuss various consequences of this construction, both representation-theoretic and topological, e.g. our work pairs with that of several authors to show that all of the differently-defined sl_n knot homologies are actually isomorphic. No background in higher representation theory or knot homology will be assumed. Joint with the geometry and topology seminar.