To a Riemann surface with boundary we associate the character variety, which parameterizes irreducible representations of the fundamental group, which are basically solutions to a multiplicative matrix problem. On the other hand, the non abelian Hodge correspondence provides a more linear model of the character variety as the module space of stable Higgs bundles. Hausel and Thaddeus initiated study of the cohomology rings of these moduli spaces, and Hausel, Letellier and Rodriguez-Villegas conjectured an explicit generating function formula for their mixed Hodge polynomials, suggesting interesting properties, for instance the curious hard Lefschetz and a connection to the cohomology rings of the Nakajima quiver varieties. Motivated by this, De Cataldo, Hausel and Migliorini formulated the P=W conjecture. I will explain how recent progress in cohomological Hall algebra computations helps us to understand the cohomology rings. One application is the proof of the P=W conjecture. Another (work in progress) will hopefully give a description of the cohomology rings and a proof of the mixed Hodge polynomials formula.