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.
A representation of the category of finite sets is a slightly unusual algebraic structure, consisting of a vector space for each finite set and a linear transformation between vector spaces for each map of sets. (It is a functor from finite sets to vector spaces). I will talk about how these representations arise in the homology of moduli spaces of curves, and how they can be used to study the asymptotic behavior of sequences of homology groups.
The tropical Grassmannian Trop G(k,n), introduced by Speyer and Sturmfels, parametrizes tropical linear spaces in tropical projective space. For k=2, it can be identified with the space of phylogenetic trees. Beyond applications to mathematical biology, it has seen striking new connections in physics to generalized scattering amplitudes via the CEGM framework.
Despite this, constructing a combinatorial model for the positive tropical Grassmannian at higher k has remained an open problem. I will describe such a model built from the planar basis, a distinguished basis of the space of tropical Plücker vectors whose elements are rays of the positive tropical Grassmannian, together with a duality between tropical u-variables and noncrossing tableaux, which provides an explicit inverse to the Speyer–Williams parameterization. For k=3, the model connects to SL(3) representation theory via a cross-ratio formula that computes tropical invariants directly from non-elliptic webs, and to CAT(0) geometry via diskoids in affine buildings.
Based on joint work with Thomas Lam.
