The nonabelian Hodge correspondence gives a diffeomorphism between the

moduli of flat connections and the moduli of higgs bundles on a smooth

Riemann surface. The two moduli, however, are completely different as

algebraic varieties. Thus natural structures on one side of the

correspondence, such as Hitchin’s integrable system, become rather

mysterious on the other side.

I will discuss joint work with Zsuzsanna Dancso and Vivek Shende, which

defines a ‘microlocal model’ of the correspondence in a neighborhood of a

singular fiber of the Hitchin system. In particular, we prove a conjecture

of Hausel and Proudfoot relating the weight and perverse filtrations on

either side of this microlocal correspondence.