Title: Cohomology ring of the compactified Jacobian of the plane curve $x^m=y^n$

Abstract: Joint work with Zhiwei Yun. In elementary terms the compactified Jacobian $JC_{m,n}$ is the moduli space of subspaces $L\subset \CC[[t]]$ of codimension $(m-1)(n-1)$ that are preserved by multiplication on $t^m$ and $t^n$. Together with Zhiwei Yun we described an action of the spherical rational Cherednik algebra $eH_{m/n}(S_n)e$ on $H^*(JC_{m,n})$ and the ring structure of the cohomology. I will also discuss perverse filtration on cohomology $H^*(JC_{m,n})$ and connections with $q,t$-Catalan numbers.