Categories and Knot Invariants

Due to the pioneering work of Khovanov, there has been a lot of recent interest in certain knot homology theories. I will explain a program to construct these knot homology theories using braid group actions on derived categories of coherent sheaves. In particular, I will consider derived categories of coherent sheaves on certain moduli spaces of vector bundles on curves. I will explain connections with spherical twists, Mukai flops, and Chuang-Rouquier's sl_2 categorification.