Non-abelian Hodge theory relates topological and algebro-geometric objects associated to a compact Riemann surface. More precisely, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field. This yields a transcendental matching between two very different moduli spaces: the character variety (parametrizing representations of the fundamental group) and the Hitchin moduli space (parametrizing Higgs bundles). In 2010, de Cataldo, Hausel, and Migliorini proposed the P=W conjecture, which links precisely the topology of the Hitchin integrable system and the Hodge theory of the character variety. I will introduce the conjecture, review its recent proofs, and discuss how the geometry hidden behind the P=W phenomenon is connected to other branches of mathematics.
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