Georgia Institute of Technology College of Sciences

The moduli space of Higgs bundles lies at the crossroads of different areas of mathematics. Its cohomology plays a central role in Ngo's proof of the fundamental lemma of the Langlands program, and it is the subject of recent results such as topological mirror symmetry and the P=W conjecture. Even though these developments seem unrelated, they all ultimately rely on a (partial) understanding of the Decomposition Theorem for the associated Hitchin fibration. In this talk, I will report on a complete and uniform description of the Decomposition Theorem in the logarithmic case, fully generalizing Ngo's results beyond the elliptic locus. This is joint work in progress with Mark de Cataldo, Roberto Fringuelli, and Mirko Mauri.