Matroid theorists are of course very interested in questions concerning representability of matroids over fields. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid $M$, which governs the representations of $M$ over all partial fields. Unfortunately, almost all matroids are not representable over any partial field, and in this case, the universal partial field gives no information.
Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid. The foundation of $M$ is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. Pastures form a natural class of field-like objects within Lorscheid’s theory of ordered blueprints, and they have desirable categorical properties (e.g., existence of products and coproducts) that make them a natural context in which to study algebraic invariants of matroids. The foundation of a matroid $M$ represents the functor taking a pasture $F$ to the set of rescaling equivalence classes of $F$-representations of $M$; in particular, $M$ is representable over a pasture $F$ if and only if there is a homomorphism from the foundation of $M$ to $F$.
As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for matroids having no $U(2,5)$ or $U(3,5)$ minors. The proof of this classification theorem relies crucially on Tutte’s Homotopy Theorem and the theory of cross-ratios for matroids. Among other things, our classification provides a short conceptual proof of the 1997 theorem of Lee and Scobee which says that a matroid is both ternary and orientable if and only if it is dyadic.
This is part 1 of a series of two talks. The second talk will be given the following week by Oliver Lorscheid.