Monday, July 12, 3pm ET (8pm BST, 7am Tue NZST)
Eimear Byrne, University College Dublin
Some recent results on q-matroids

Abstract:
The q-analogue of a (poly)matroid has been a topic of recent interest among the coding theory community, due to their connections with rank-metric codes (see e.g. [GJLR20], [JP18], [Shi19]). When defining a q-(poly)matroid, we associate a rank function with the lattice of subspaces of a finite dimensional space E. As one might expect, there are several cryptomorphic descriptions of a q-matroid, in terms of independent spaces, circuits, flats, etc. We will go through some of these, highlighting the difference to the classical case. We will also outline some applications of q-(poly)matroids to the construction of the q-analogue of a t-design.

Some of the results of this talk are based on joint work with Michela Ceria, Sorina Ionica, Relinde Jurrius and Elif Sacikara.

[JP18] Relinde Jurrius and Ruud Pellikaan. Defining the q-analogue of a matroid. Electronic Journal of Combinatorics, 25(3):P3.2, 2018.

[GJLR20] Elisa Gorla, Relinde Jurrius, Hiram H L´opez, and Alberto Ravagnani. Rank-metric codes and q-polymatroids. Journal of Algebraic Combinatorics, 52:1–19, 2020.

[Shi19] Keisuke Shiromoto. Codes with the rank metric and matroids. Designs, Codes and Cryptography, 87(8):1765–1776, 2019.

Monday, July 5, 3pm ET (8pm BST, 7am Tue NZST)
Shayla Redlin, University of Waterloo
Extensions of cliques

Abstract:

In this talk, we will investigate the extensions of the cycle matroid of a complete graph. I will show that the number of these extensions is surprisingly large.

This is joint work with Peter Nelson and Jorn van der Pol.

Monday, June 28, 3pm ET (8pm BST, 7am Tue NZST)
Carla Groenland, Utrecht University
Universal Graphs and Labelling Schemes

Abstract:
An induced universal graph for a graph class contains all graphs in the class as an induced subgraph. We construct induced universal graphs for all hereditary graph classes, and derive reachability labelling schemes for digraphs and comparability labelling schemes for posets from this. All these results are asymptotically optimal. This talk aims to give some intuition about these concepts and our techniques (which includes Szemerédi’s regularity lemma). We will also discuss a new venue of research: what if we strengthen induced to isometric? Several interesting questions are left open.

This is based on joint work with Marthe Bonamy, Louis Esperet, Cyril Gavoille and Alex Scott.

Monday, June 21, 3pm ET (8pm BST, 7am Tue NZST)
Sebastian Wiederrecht, LIRMM
Even Circuits in Oriented Matroids

Abstract:

This work generalises the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of the even dicycle problem. Then we show that the problem of detecting an even directed circuit in a regular matroid is polynomially equivalent to the recognition of non-even oriented matroids.
Our main result is a precise characterisation of the class of non-even oriented bond matroids in terms of forbidden minors, which complements an existing characterisation of non-even oriented graphic matroids by Seymour and Thomassen and reveals an extended class of obstructions.
This is joint work with Karl Heuer and Raphael Steiner.