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.