Conference Agenda

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Session Overview
Session
MS154, part 4: New developments in matroid theory
Time:
Friday, 12/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F-106
30 seats, 54m^2

Presentations
3:00pm - 5:00pm

New 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)

 

Gain matroids and their applications

Viktoria Kasznitzky
Eötvös Loránd University

Let (G=(V,E)) be an undirected graph and let (k) and (ell) be two integers. (G) is said to be emph{((k,ell))-sparse} if (|E'|leq k|V'|-ell) holds for every subgraph (G'=(V',E')) of (G). The edge sets of the ((k,ell))-sparse subgraphs form a matroid on the edge set of a graph (H) called the emph{((k,ell)) count matroid}.

Gain matroids are a generalisation of count matroids. Let (Gamma) be a group. Now assign an element of (Gamma) (a emph{gain}) and a reference direction to every edge in (E). The emph{gain} of a (not necessarily directed) closed walk is defined as the group element obtained by the multiplication of the gains of its edges where the inverse gain should be used for the edges used in the reverse direction. The gain group corresponding to the edge set of a connected subgraph is the set of gains of closed walks starting at one of its vertices, (v). (After some observations (v) can be dropped from the definition and it can be extended to arbitrary subgraphs.)

 

Matroid threshold hypergraphs

José Alejandro Samper
Miami

In this talk we introduce the notion of a matroid threshold hypergraph: a collection of bases of a matroid obtained by capping the total weight of the bases under given a function of the ground set. Focusing on the uniform matroid yields the classical theory of threshold hypergraphs. In this talk we will motivate the definition, explain a few their interesting properties and speculate about the uses of the theory.

 

Whitney Numbers for Cones

Galen Dorpalen-Barry
Minnesota

An arrangement of hyperplanes dissects space into connected components called chambers. A nonempty intersection of halfspaces from the arrangement will be called a cone. The number of chambers of the arrangement lying within the cone is counted by a theorem of Zaslavsky, as a sum of certain nonnegative integers that we will call the cone's "Whitney numbers of the 1st kind". For cones inside the reflection arrangement of type A (the braid arrangement), cones correspond to posets, chambers in the cone correspond to linear extensions of the poset, and these Whitney numbers refine the number of linear extensions. We present some basic facts about these Whitney numbers, and interpret them for two families of posets.