Planar curve singularities, knot invariants and representation theory

The natural topological object attached to the planar curve singularity is its link, which is the intersection of a small three-sphere around the singularity and the curve. It turns out that the HOMFLY-PT invariant (and its categorification) of the link has a geometric interpretation in terms of topology of the compactified Jacobian of the curve and in terms of Hilbert scheme of points on the curve. When the curve in the question has $\mathbb{C}^*$ symmetry, the HOMFLY-PT polynomial has an interpretation as a character of the degenerate Double Affine Hecke Algebra. The talk is based on the joint results with Eugene Gorsky, Jake Rasmussen, Lev Rozansky, Vivek Shende and Zhiwei Yun.