i will illustrate a recent result obtained in collaboration with l. spolaor and b. velichkov concerning the regularity of the free boundaries in the two phase bernoulli problems. the new main point is the analysis of the free boundary close to branch points, where we show that it is given by the union of two $c^1$ graphs. this complete the analysis started by alt caffarelli friedman in the '80s.

the p=w conjecture by de cataldo, hausel, and migliorini suggests a surprising connection between the topology of hitchin systems and hodge theory of character varieties. in this talk, we will focus on interactions between compact and noncompact hyper-kaehler geometries. such connections, together with symmetries coming from the moduli of compact hyper-kaehler manifolds, lead to new progress on the p=w conjecture. i will discuss the results we obtained and the difficulties we met. based on joint work with mark de cataldo and davesh maulik.