Nick Brettell

On excluded-minor characterisations of representable matroids

Matroids are abstract structures that encapsulate a notion of dependence appearing in graph theory, linear algebra, and discrete geometry. A matroid is said to be representable over a field F if it has a representation as a set of vectors in a vector space over F, where the notion of dependence is linear dependence. Characterising when a matroid is representable over a certain field, or set of fields, is one of the oldest problems in matroid theory, dating back to a 1935 paper of Hassler Whitney. In this talk, I will survey some of the main results in this area, and discuss recent progress towards finding characterisations of particular classes of matroids that are representable over all fields in some set. Prior familiarity with matroids will not be assumed.