Mon, August 10, 3pm ET (8pm BST, 7am Tue NZST)
Yelena Yuditsky, Ben-Gurion University
Typical structure of hereditary graph families
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Abstract:
A family of graphs $\cal F$ is hereditary if it is closed under isomorphism and taking induced subgraphs. For example, for a given graph $H$, a hereditary family is the family of all $H$-free graphs, that is graphs without an induced copy of $H$.

Alon, Balogh, Bollobás and Morris showed that for every hereditary family $\cal F$ there exist $\epsilon >0$ and $l\in \mathbb{N}$ such that the number of graphs in $\cal F$ on $n$ vertices is $2^{(1-1/l)n^2/2+o(n^{2-\epsilon})}$. They showed this bound by deriving various structural properties of almost all graphs in $\cal F$. We study and obtain additional structural properties of almost all graphs in some restricted hereditary families. As an application of our results, we prove the existence of an infinite family of counterexamples for the recent Reed-Scott conjecture about the structure of almost all $H$-free graphs.

This is a joint work with Segey Norin.

Mon, August 3, 3pm ET (8pm BST, 7am Tue NZST)
Jim Geelen, University of Waterloo
Towards the excluded-minors for GF(5)-representability
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Abstract:
I will discuss a strategy towards finding the excluded-minors for GF(5)-representability of matroids. This includes some joint work with Rutger Campbell and Geoff Whittle.

Mon, July 27, 3pm ET (8pm BST, 7am Tue NZST)
Zach Walsh, University of Waterloo
Quadratically Dense Matroids
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Abstract:
The extremal function of a class of matroids is the function whose value at an integer $n$ is the maximum number of elements of a simple matroid in the class of rank at most $n$. We present a result concerning the role of group-labeled graphs in minor-closed classes of matroids, and then use it to determine the extremal function, for all but finitely many $n$, for the class of complex-representable matroids which exclude a given rank-2 uniform matroid as a minor. This talk will focus on our original motivation, and on the connection between group-labeled graphs and representable matroids.

Mon, July 20, 3pm ET (8pm BST, 7am Tue NZST)
Pascal Gollin, Institute for Basic Science
Obstructions for bounded branch-depth in matroids
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Abstract:
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the cycle matroid of a large fan graph as a minor. In this talk, I present a proof that matroids of sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width.

This is joint work with Kevin Hendrey, Dillon Mayhew and Sang-il Oum.