Combinatorics Seminar: "Singular Hodge theory of matroids"

Abstract: A theorem of de Bruijn and Erdős says that a collection of n lines in a projective plane intersect in at least n points.  This is a special case of the more general “Top-Heavy Conjecture” of Dowling and Wilson (1974).

This conjecture was formulated for all matroids and was proven for hyperplane arrangements (realizable matroids) by Huh and Wang in 2017.  A key idea of their proof is to use the Hodge theory of a certain singular projective variety, called the Schubert variety of the arrangement.  For arbitrary matroids, no such variety exists; nonetheless, I will discuss a proof of the Top-Heavy Conjecture for all matroids, which proceeds by finding combinatorial stand-ins for the cohomology and intersection cohomology of these Schubert varieties and by studying their Hodge theory.  I will also discuss a closely related problem: the nonnegativity of the coefficients of Kazhdan-Lusztig polynomials of matroids (polynomials mirroring the classical Kazhdan-Lusztig polynomials for Coxeter groups).  This is joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.

Host: Martha Precup