Hitchin fibrations are remarkable families of (degenerate) abelian varieties which appear in geometry, arithmetic, representation theory, etc. The study of the singularities of these maps has been the focus of much work, at least since Ngo's proof of the fundamental lemma. One way to encode some of these singularities is to look at the summands appearing in the decomposition theorem. In this talk, we focus on the subvarieties of the target of the Hitchin map that support such summands. I will report on work on done in the case of SL(n) and in joint work in progress with J. Heinloth and L. Migliorini in the case of GL(n). My goal is to walk the audience through the main concepts, results and techniques. Speaker(s): Mark de Cataldo (Stony Brook University)