Perverse filtrations and the P=W conjecture

Given a proper morphism between algebraic varieties, the rational cohomology groups of the source
are endowed with the perverse filtration. In this talk, I will introduce five families of Hitchin
systems which are Hilbert schemes of points on surfaces. I will discuss why the perverse filtrations
associated with these Hitchin maps are multiplicative. The same method works for Hilbert schemes of
points of elliptic K3 surfaces. I will also discuss the connection between the multiplicativity of
the perverse filtration and the P=W conjecture for reductive groups.