Compactifications of Hitchin's moduli space

The moduli space of rank 2 Higgs bundles has several natural compactifications, depending upon its
algebraic, differential geometric, or topological realizations. In this talk, I will compare the compactification coming from the algebraic geometry of the C-star action with the one coming from the analytic "limiting configurations" of solutions to the Hitchin equations.
I will explain how the nonabelian Hodge correspondence extends as a map between these compactifications. Somewhat surprisingly, the extension is not continuous, but fails in a controlled
way. This is joint work with Siqi He, Rafe Mazzeo, and Xuesen Na.