Lagrangian correspondence between Hitchin and de Rham moduli spaces.

In 2008 Simpson conjectures the existence of a holomorphic Lagrangian foliation in the de Rham moduli space of holomorphic $G$-connections for a complex reductive group $G$. I will present an algebraic geometry description of the Lagrangian correspondence of conformal limits, based on the work of Simpson for $SL_2(C)$-Higgs bundles defined on a smooth connected projective curve $C$ of genus at least 2. This talk is based on joint work with Jennifer Brown and Motohico Mulase.