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
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.
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.
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.
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. DinnerWe 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 Here are some pictures from the dinner:
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