3:00pm - 5:00pmNew developments in matroid theory
Chair(s): Alex FInk (Queen Mary), Ivan Martino (Northeastern University, United States of America), Luca Moci (Bologna)
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.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Cohomology rings of projective models of toric arrangements
Giovanni Gaiffi
PIsa
I will describe, by providing generators, relations and examples, the cohomology rings of projective models of toric arrangements (joint work with Corrado De Concini).
Arithmetic matroids, posets and cohomology of toric arrangements
Roberto Pagaria
Pisa
Matroids are cryptomorphic to geometric lattices and from an oriented matroid can be build an Orlik-Solomon algebra. This algebra is the cohomology algebra of the complement of an arrangement of hyperplanes or of pseudospheres, hence the Tutte polynomial specializes to the Poincaré polynomial of the complement.
Recent works introduced arithmetic matroids and studied their relations with the cohomology algebra of toric arrangements. We will discuss the relation between arithmetic matroids and posets of layers of toric arrangements. This study leads to a construction - from the poset of layers - of a "toric Orlik-Solomon algebra" isomorphic to the cohomology algebra of the complement of the toric arrangement. Indeed, the Poincaré polynomial of the complement is a specialization of the arithmetic Tutte polynomial.
Categories of matroids, Hopf algebras, and Hall algebras
Jaiung Jun
SUNY Binghamton
In their recent paper, Baker and Bowler introduced the notion of matroids over partial hyperstructures which unifies various generalizations (including oriented, valuated, and phase matroids). One can generalize the notion of minors and direct sums (of matroids) to the case of matroids over partial hyperstructures. In particular, this allows one to generalize the matroid-minor Hopf algebra to this setup. We then investigate the category of (ordinary) matroids, showing that the matroid-minor Hopf algebra is dual to the Hall algebra associated to the category of matroids. This is joint work with Chris Eppolito and Matt Szczesny.