This semester’s topic in our joint research seminar is Hodge theory of matroids. The meetings take place on Zoom on a bi-weekly basis during lecture time, Thursdays 15-18.

Talk 11: June Huh (Princeton University): Kazhdan-Lusztig and singular Hodge theory for matroids

Abstract: There is a remarkable parallel between the theory of Coxeter groups (think of the symmetric group or the dihedral group) and matroids (think of your favorite graph or vector configuration). After giving an overview of the similarity, I will report proofs of two combinatorial conjectures, the nonnegativity conjecture for Kazhdan-Lusztig coefficients and the top-heavy conjecture for the number of flats of matroids. The key step is to formulate and prove an analogue of the decomposition theorem for a resolution of singularities in a combinatorial setup, which is closely related to the validity of the hard Lefschetz theorem and the Hodge-Riemann relations for what would be an intersection Chow cohomology in realizable cases. The talk should be enjoyable without specialized knowledge. Joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang.