Given a matroid $M$, we can think of $M$ as a polytope via its base polytope (the convex hull of its bases). Therefore, extension complexity is also a measure of how complicated a matroid is. In this talk, I will give a crash course on extension complexity, with particular emphasis on matroid polytopes.
We consider classes of matroids that have more excluded minors than
members, dubbed "fractal classes". Specifically, the ratio of the number of
excluded minors of size $n$ to the number of members of size
$n$ goes to zero. It turns out that there are some surprisingly
straightforward examples, and the above-mentioned ratio can in fact
achieve any value whatsoever