Monday, July 26, 3pm ET (8pm BST, 7am Tue NZST)
Matthew Sullivan, University of Waterloo
Rotation Systems and Simple Drawings of $K_n$

Abstract:

A complete rotation system on $n$ vertices is a collection consisting of cyclic permutations of the elements $[n]\backslash \{i\}$, one cyclic permutation for each $i$. If $D$ is a simple drawing of a labelled graph, then a rotation at vertex $v$ is the cyclic ordering of the edges at $v$. In particular, the collection of all vertex rotations of a simple drawing of $K_n$ is a rotation system.

If a complete rotation system can be represented by a simple drawing of $K_n$, then we call such a rotation system realizable. In 2011, Jan Kynčl published a proof using homotopy implying that if all 6 vertex rotation systems of an $n$ vertex rotation system $R_n$ are realizable, then $R_n$ is realizable.

In this talk, we will briefly review a full characterization of realizable rotation systems, present a structural characterization of edges and faces in simple drawings of $K_n$ and see a combinatorial proof of a weak characterization of realizable rotation systems.

Monday, July 19, 3pm ET (8pm BST, 7am Tue NZST)
Natasha Morrison, University of Victoria
Uncommon systems of equations

Abstract:

A system of linear equations $L$ over $\mathbb{F}_q$ is common if the number of monochromatic solutions to $L$ in any two-colouring of $\mathbb{F}_q^n$ is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of $\mathbb{F}_q^n$. Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs, the systematic study of common systems of linear equations was recently initiated by Saad and Wolf. Building on earlier work of Cameron, Cilleruelo and Serra, as well as Saad and Wolf, common linear equations have been fully characterised by Fox, Pham and Zhao.

In this talk I will discuss some recent progress towards a characterisation of common systems of two or more equations. In particular we prove that any system containing an arithmetic progression of length four is uncommon, confirming a conjecture of Saad and Wolf. This follows from a more general result which allows us to deduce the uncommonness of a general system from certain properties of one- or two-equation subsystems.

This is joint work with Nina Kamčev and Anita Liebenau.

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.