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Abstract:
Geometric representation theory has revealed a deep connection between
geometry and quantum groups suggesting that quantum groups are shadows
of richer algebraic structures called categorified quantum groups.
Crane and Frenkel conjectured that these structures could be
understood combinatorially and applied to low-dimensional topology. In
this talk I'll explain joint work with Mikhail Khovanov on a
categorification of quantum groups using a simple diagrammatic
calculus that requires no previous knowledge of quantum groups. If
time permits I'll also survey the applications of this theory
including a new grading on blocks of the affine Hecke algebra and
Webster's recent work categorifying Reshetikhin-Turaev invariants of
tangles.
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