Chern-Simons invariants of knots have many applications in mathematics and in physics. Khovanov showed  in ‘99 that the simplest such invariant, the Jones polynomial, arizes as the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their meaning: what are they homologies of?

Mirror symmetry is another important strand in the interaction between mathematics and physics. Homological mirror symmetry, formulated by Kontsevich in ’94, naturally produces hosts of homological invariants. Sometimes, it can be made manifest, and then its striking mathematical power comes to fore. Typically though, it leads to invariants which have no particular interest outside of the problem at hand.

I showed recently that there is a vast new family of mirror pairs of manifolds, for which homological mirror symmetry can be made manifest. They do lead to interesting invariants: in particular, they solve the knot categorification problem.

I will describe a pair of recent applications of enumerative geometry: one is to classic problems coming from integrable lattice models, and the other to the geometric Langlands correspondence. This is based on joint works with Andrei Okounkov and Edward Frenkel.