A diagrammatic categorification of quantum groups

Series: 
Job Candidate Talk
Wednesday, January 26, 2011 - 15:00
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Columbia University
The Jones polynomial is a link invariant that can be understood in terms of the representation theory of the quantum group associated to sl2. This description facilitated a vast generalization of the Jones polynomial to other quantum link and tangle invariants called Reshetikhin-Turaev invariants. These invariants, which arise from representations of quantum groups associated to simple Lie algebras, subsequently led to the definition of quantum 3-manifold invariants. In this talk we categorify quantum groups using a simple diagrammatic calculus that requires no previous knowledge of quantum groups. These diagrammatically categorified quantum groups not only lead to a representation theoretic explanation of Khovanov homology but also inspired Webster's recent work categorifying all Reshetikhin-Turaev invariants of tangles.