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# Mikhail Khovanov (Columbia University), Brauer Lectures

## March 30 @ 3:30 pm - 4:30 pm

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**Lecture 2 Title:** Categorifications of quantum link invariants

**Abstract:** The Jones polynomial has many generalizations, including the two-variable HOMFLYPT polynomial. We will review approaches to the categorification of the HOMFLYPT polynomial and its specializations. Several new ideas proved crucial, including evaluating foams (two-complexes embedded in 3-space) via systems of Frobenius algebras that come from cohomology rings of flag varieties and constructing functorial tangle invariants via matrix factorizations. Matrix factorizations are generalized complexes with the square of the differential equal to theĀ multiplication by a polynomial. The resulting structures have been related to many canonical objects in representation theory and algebraic geometry.