Lorenzo Vecchi:Hilbert series of matroids

Abstract: During the last decade, matroid theory has been revolutionized by the introduction of algebro-geometric techniques: long-standing combinatorial conjectures were solved by first studying the cohomology of a variety associated to realizable matroids, and then by constructing an abstract combinatorial analogue in the non-realizable case. Nowadays, the Chow ring of a matroid and all the Kazhdan-Lusztig invariants can be rightfully considered an essential tool for anyone interested in matroids. All the new polynomials that arose in this context are conjectured to have nice combinatorial properties, for example being real-rooted. The plan of the talk is to introduce these invariants and then give some very recent developments in tackling these open problems.

This is based on a joint work with Luis Ferroni and George Nasr, and an ongoing project with Luis Ferroni, Jacob Matherne and Matthew Stevens.