Graphs and Matroids Seminar - Oliver Lorscheid

Title: Foundations of Matroids without Large Uniform Minors, Part 2

Abstract:

In this talk, we take a look under the hood of last week’s talk by Matt Baker: we inspect the foundation of a matroid.

The first desired properties follow readily from its definition: the foundation represents the rescaling classes of the matroid and shows a functorial behaviour with respect to minors and dualization. It requires however deep results by Tutte, Dress-Wenzel and Gelfand-Rybnikov-Stone to gain an understanding of the foundation in terms of generators and relations.

For small matroids, this allows us to determine the foundation explicitly. This, in turn, lets us derive the structure theorem for foundations of matroids without large uniform minors, which is the central result behind the applications from last week.

In a concluding part of the talk, we turn to potential future directions and open problems.