Conference Agenda

The interactions between Matroid Theory, Algebra, Geometry, and Topology have long been deep and fruitful. Pertinent examples of such interactions include breakthrough results such as the g-Theorem of Billera, Lee and Stanley (1979); the proof that complements of finite complex reflection arrangements are aspherical by Bessis (2014); and, very recently, the proof of Rota's log-concavity conjecture by Adiprasito, Huh, and Katz (2015).

The proposed mini-symposia will focus on the new exciting development in Matroid Theory such as the role played by Bergman fans in tropical geometry, several results on matroids over a commutative ring and over an hyperfield, and the new improvement in valuated matroids and about toric arrangements. We plan to bring together researchers with diverse expertise, mostly from Europe but also from US and Japan. We are going to include a number of postdocs and junior mathematicians.

Recently, the notions of G-Tutte polynomials and G-plexifications were introduced to build a general framework for studying hyperplane, toric, q-reduced arrangements and their "Tutte-like" polynomials (Tutte, arithmetic Tutte, characteristic (quasi-)polynomials) en masse. Like the above-mentioned (quasi-)polynomials, the G-Tutte polynomials possess Deletion-Contraction and convolution formulas, but unlike them, the G-Tutte polynomials may have negative coefficients. We are currently interested in under what conditions their coefficients all are positive? In this talk, we will propose some ideas and partial answers. This talk is based on two recent joint works with Ye Liu and Masahiko Yoshinaga.

G-Tutte polynomial is a generalization of arithmetic Tutte polynomial. I will discuss some results on enumerative aspects of G-Tutte polynomial.

Arrangements of hyperplanes have long offered a geometric point of view on matroids - at times leading to structural advances even in the nonrealizable case. The theory of arrangements recently broadened its scope beyond the case of hyperplanes to include arrangements in the torus, in products of elliptic curves and, more generally, in Abelian Lie groups. This development spurred the search for suitable enrichments of matroid theory.

In this context, I will introduce the foundations of a theory of group actions on (semi)matroids, focussing mainly on applications to the structure of intersection posets of arrangements. I will also outline how this framework relates to (arithmetic) Tutte polynomials, arithmetic matroids and G-Tutte polynomials. A further ramification of this setup will be illustrated in A. D’Alì’s talk on generalized Stanley-Reisner rings.

The material I will present is partly drawn from joint works with Alessio D’Alì, Giacomo d’Antonio, Noriane Girard, Giovanni Paolini and Sonja Riedel.

The Stanley-Reisner correspondence, which assigns a commutative ring to each finite simplicial complex, is a useful and well-studied bridge between commutative algebra and combinatorics, yielding particularly nice results for the independence complex of a matroid. In 1987 Sergey Yuzvinsky proposed a construction that allows to see the Stanley-Reisner ring of any finite simplicial complex as the ring of global sections of a sheaf of rings on a poset. Motivated by applications in the theory of Abelian arrangements, E. Delucchi and I extend Yuzvinsky's construction to the case of (possibly infinite) finite-length simplicial posets. We show that this generalization behaves well with respect to quotients of simplicial complexes and posets by group actions such as those introduced in E. Delucchi's talk.