|Dept.||University of Wisconsin - Madison|
Matroids are combinatorial generalizations of configuration of points in vector spaces, or equivalently, hyperplane arrangements. I will discuss two conjectures in matroid theory. The first is a “top-heavy” conjecture by Dowling and Wilson in the 70’s, and the second is some non-negativity conjecture about the Kazhdan-Lusztig polynomial of matroids introduced recently by Elias-Proudfoot-Wakefield. I will explain the proofs of the conjectures in the realizable case (the first conjecture by Huh and myself, and the second by E-P-W). The proof uses Hodge theory of the matroid analogous of the Schubert varieties. I will also talk about some work in progress of extending the proof to the non-realizable case, which is joint with Tom Braden, June Huh, Jacob Matherne and Nick Proudfoot.