For a smooth curve, the Hilbert schemes of points are just symmetric powers of the curve, and their cohomology is easily computed in terms of H1 of the curve. In joint work with Davesh Maulik, we generalize this formula to curves with planar singularities (as conjectured by Migliorini and proved independently by Migliorini and Shende). In the singular case, the compactified Jacobian will play an important role in the formula, and our proof uses Ngô's technique from his proof of the fundamental lemma.