Double affine Hecke algebras and topological field theory

Double affine Hecke algebras (DAHA's) are certain combinatorially defined algebras with surprisingly deep connections throughout representation theory of algebraic groups.  In recent years work of several different groups points to a new construction of DAHA's in the framework of topological field theory and knot invariants:

1)  Gorsky, Oblomkov, Rasmussen and Shende have observed a conjectural identity between the Khovanov-Rozansky homology of torus knots and characters of certain DAHA representations.  Since then many authors have contributed in this direction.
2)  Cherednik has proposed a conjectural construction of a "DAHA-Jones polynomial", lifting the Jones polynomial to the DAHA.  Cherednik, Danilenko and Elliot have computed these for torus knots and their cablings.
3)  Aganagic and Shakirov have seen hints of the DAHA and its relation to quantum A-polynomials in their "refined Chern-Simons theory".
4)  Berest and Samuelson have seen hints of the DAHA in skein modules of certain knot complements

In this talk I'll give a light touch overview of this fascinating web of conjectures, and I will outline two further appearances of the DAHA, directly in the language of topological fiheld theory.  These are based on joint work with David Ben-Zvi, Adrien Brochier, Martina Balagovic, and Monica Vazirani.