Abstract:  Hitchin’s integrable systems lie in the crossroads of geometry, representation theory, and mathematical physics. I will discuss two central conjectures raised in the last two decades which greatly influenced the development for the algebraic geometry of Hitchin moduli spaces. The first is the P=W conjecture, which concerns the interaction of the topology of the Hitchin system and the non-abelian Hodge correspondence. The second is the topological mirror symmetry conjecture which connects the Langlands duality of groups and the mirror symmetry for Hitchin systems. I will explain that both conjectures can be proved in a uniform way, via vanishing cycles techniques and support theorem. Based on joint work with Davesh Maulik.