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Department of Mathematics pres.

Student Combinatorics Seminar

Matroids II: Representability and the Curse of the Forbidden Minors

The two most important examples motivating the theory of matroids are edge sets of graphs and finite sets of vectors. However, as Alana showed us last week, the first example is actually a special case of the second. Does this mean that matroid theory is actually just linear algebra in disguise?

As this talk will explore, the answer is an emphatic no. Some matroids can't be represented by sets of vectors at all, while others can be, but only over particular ground fields. In this talk, we'll look at examples of these phenomena, and consider the tricky problem of characterizing when matroids are representable over certain fields. As time permits, we'll discuss an interesting fact: not only is matroid theory not linear algebra in disguise, but in a precise sense it never could have been, as shown in Vamos's delightfully titled paper "The Missing Axiom of Matroid Theory Is Lost Forever".

While it picks up where last week's talk left off, this talk should be accessible to anyone who knows what a matroid is. Speaker(s): Will Dana (UM)

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