String-Math Seminar: A new look at the quantum knot invariants

Seminar | October 23 | 2-3 p.m. | 402 LeConte Hall

 Aaron Lauda, University of Southern California

 Department of Mathematics

The Reshetikhin-Turaev construction associated knot invariants to the data of a simple Lie algebra and a choice of irreducible representation. The Jones polynomial is the most famous example coming from the Lie algebra sl(2) and its two-dimensional representation. In this talk we will explain Cautis-Kamnitzer-Morrison’s novel new approach to studying RT invariants associated to the Lie algebra sl(n). Rather than delving into a morass of representation theory, we will show how two relatively simple Lie theoretic ingredients can be combined with a powerful duality (skew Howe) to give an elementary and diagrammatic construction of these invariants. We will explain how this new framework solved an important open problem in representation theory, proves the existence of an (a,q)-super polynomial conjectured by physicists (joint with Garoufalidis and Lê), and leads to a new elementary approach to Khovanov homology and its sl(n) analogs (joint with Queffelec and Rose).