Abstract: The P=W conjecture of de Cataldo, Hausel, and Migliorini relates the topology of Hitchin fibrations to the Hodge theory of character varieties via Simpson’s nonabelian Hodge theory. In recent joint work with Junliang Shen, we proved a compact analog of this conjecture, which relates the topology of Lagrangian fibrations to the Hodge theory of compact hyper-Kaehler manifolds. I will present circle of ideas as well as some applications of our result.