Abstract

We will survey aspects of algebraically defined link homology theories and discuss several key open questions that come out of these structures.

About the Speaker

Mikhail Khovanov

Columbia University

Mikhail Khovanov is a recognized leader of the “categorification” program, which plays an important role in modern mathematics and physics.

Khovanov received his Ph.D. in 1997 under the supervision of Professor Igor Frenkel at Yale University. Shortly thereafter he came up with his famous idea of categorifying the Kauffman bracket, which is a version of the celebrated Jones polynomial of links in a 3-sphere. This was the first example of the categorification which interprets polynomial invariants as Poincare polynomials of new homology theories. The construction of Khovanovhomology was amazingly fruitful and very unexpected.

A further categorification of the HOMFLY-PT polynomial of links and a categorification of quantum groups are other major achievments of Khovanov which have now important implications in low-dimensional topology, algebraic and symplectic geometry, geometric representation theory and string theory.