Abstract: Both the Higgs bundle moduli space and the moduli space of flat connections have a natural stratification induced by a C* action. In both of these stratifications, each stratum is a holomorphic fibration over a connected component of complex variations of Hodge structure. While the nonabelian Hodge correspondence provides a homeomorphism between Higgs bundles and flat connections, this homeomorphism does not preserve the respective strata. The closed stratum on the Higgs bundle side is the image of the Hitchin section and the closed stratum in the space of flat connections is the space of opers. In this talk, we will show how many of the relationships between opers and the Hitchin section extend to general strata. In particular, we will show that the conformal limit identifies certain holomorphic Lagrangian subspaces of the stratifications.