**Mon, September 7,** 3pm ET (8pm BST, 7am Tue NZST)**Matt Baker**, Georgia Tech**Foundations of Matroids without Large Uniform Minors, Part 1**

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**Abstract:**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.

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