"Vamos matroid" by David Eppstein - Own work. Licensed under CC0 via Commons - https://commons.wikimedia.org/wiki/File:Vamos_matroid.svg#/media/File:Vamos_matroid.svg

Recent Developments in Matroid Theory

Minisymposium, 7th European Congress of Mathematics, TU Berlin

9:00 - 11:00, Tuesday July 19, 2016


This minisymposium aims to bring together researchers from different branches of matroid theory to talk about recent developments in the field. It will also strengthen the interaction between matroid theorists and researchers in other fields of mathematics, by discussing novel connections with areas like commutative algebra, analysis, and set theory.

Speakers

Nathan Bowler (Universität Hamburg) - Infinite matroids
I'll give an overview of the theory of infinite matroids, explaining what they are, why we care about them and what is known about them. I'll also introduce some of the central open problems.

Petter Brändén (KTH) - Hyperbolic matroids
Hyperbolic matroids is a family of matroids that properly contains the family of matroids representable over the complex numbers. These matroids are associated to so called hyperbolic polynomials, which although they were introduced in PDE theory, recently have been studied in combinatorics, optimization, probability theory and theoretical computer science. We address the question of representability of hyperbolic matroids, and show that members of a large class of hyperbolic matroids which generalize the Vámos matroid fail to be representable over any (skew) field.
This is joint work with Nima Amini.

Alex Fink (Queen Mary) - Matroids over rings
Matroids arise in many algebra-flavoured combinatorial problems which feature lists of vectors over a field. But often one's data are elements in a module over some other ring, and there is more information to be extracted than the field-agnostic linear algebra that the matroid can see. Luca Moci and I have defined the notion of matroid over a ring to expose this extra information.
I will begin by reviewing matroids and introducing matroids over rings. I'll discuss two examples of situations where matroids leave some combinatorics uncaptured that the theory over rings does take in, one related to subtorus arrangements and the other to tropical geometry. Further topics will include their analogue of the Tutte polynomial, matroid polytopes, and their moduli space, as time permits.

Anna de Mier (UPC) - Extending transversal matroids
A matroid N is a single-element extension of a matroid M if N=M\x for some element x of N. If M and N belong to some class of matroids that admits a particular type of representation, one may ask if a representation of M can be extended to yield a representation of N. We look at this question for the class of transversal matroids (those whose independent sets are the partial transversals of a set system (A_1,...,A_r)). Clearly, adding a new element to some of the sets of a set system representing the transversal matroid M gives the representation of an extension of M, but since M could have many different representations, one may actually get the same extension several times. We present several results about the structure of the set of transversal extensions of a transversal matroid, showing in particular that all extensions can be obtained from minimal presentations only. We also discuss about the application of these results in enumerating transversal matroids.

Dinner

We will have a group dinner on Tuesday evening after the event, together with participants of the Toric, tropical, and combinatorial geometry - a panorama minisymposium. It will take place at the restaurant

SCHNITZELEI
Röntgenstraße 7
Tuesday July 19 at 19:15
http://schnitzelei.de/

Some of us will meet at 18:45 in front of the TU main building to walk to the restaurant.

Here are some pictures from the dinner:
   

Related events

"Toric, tropical, and combinatorial geometry - a panorama" minisymposium  (7ECM, July 20)
Symposium Diskrete Mathematik 2016  (Berlin, July 15-16)

Organizers

Matthias Lenz (Université de Fribourg)
Felipe Rincón (University of Oslo)