Seminar
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| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
A braided tensor 2-category from link homology
An early highlight of quantum topology was the observation of Reshetikhin and Turaev that the Jones polynomial — and many other knot and link invariants — arise from, and may be expressed in terms of braided tensor categories of representations of certain quantum groups (although not yet using that language).
Not much later, Khovanov discovered his link homology which refines a `categorifies' the Jones polynomial, in that it assigns graded chain complexes to links from which the earlier link polynomials may be recovered. It was therefore widely expected that Khovanov homology and its variants are themselves expressible in terms of certain braided tensor 2-categories which `categorify' the familiar braided tensor categories. However, a major roadblock in realizing this dream is the problem of coherence: Link homology theories live in the world of homological algebra, and constructing a braided tensor structure (at chain level) in principle requires an infinite amount of higher and higher homological coherence data.
In this talk, I will sketch a proposed solution to this problem, joint with Leon Liu, Aaron Mazel-Gee, Catharina Stroppel and Paul Wedrich, and explain how we use the language of infinity-categories to build a braided tensor 2-category (more precisely, an E_2-monoidal (infinity,2)-category) which categorifies the Hecke braided tensor category underlying the HOMFLYPT link polynomial.