The following subjects will be addressed: (1) state sum definitions of the colored Jones polynomial and its specializations by using the combinatorics of knot diagrams and H-triangulations; (2) a proof of the volume conjecture for the figure-eight knot; (3) Witten's approach to the volume conjecture through analytically continued Chern-Simons theory.
I will talk about an approach to the ring generated by the kappa classes via the moduli space of stable quotients. The main new result (with A. Pixton) is a proof of a conjecture by Faber and Zagier of an elegant set of relations. Whether these are all the relations is an interesting question. I will discuss the data on both sides.
The purpose of these lectures is to introduce Donaldson-Thomas theory, the enumerative theory of sheaves on Calabi-Yau threefold geometries, as well as some generalizations such as a motivic version and a "categorification". After introducing basic definitions, I'll explain the combinatorics of toric DT theory, known as the topological vertex. I will then explain Behrend's local approach, which leads to a motivic version of the theory. I will finally explain the approach via mixed Hodge modules and show some sample computations.