Cohomological Hall algebras and P=W conjecture

Let S be a smooth surface, and Coh(S) the moduli stack of properly supported coherent sheaves on S. One can equip the Borel-Moore homology of Coh(S) with a convolution algebra structure; this is called the cohomological Hall algebra (CoHA) of S. While understanding CoHA algebraically is more or less hopeless for a general S, I will give an explicit presentation of its part, which corresponds to sheaves with zero-dimensional support. I will further explain how this presentation was used to prove the P=W conjecture of de Cataldo-Hausel-Migliorini about the perverse filtration on the cohomology of the Hitchin system of a smooth projective curve C.