Abstracts

(Baker): One of the questions people like to ask about a matroid M is over what fields M is representable. 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, most matroids (asymptotically 100%, in fact) 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 ternary matroids (matroids representable over the field of three elements). The proof of this classification theorem relies crucially on Tutte's celebrated Homotopy Theorem. Among other things, our classification provides a conceptual proof of a 1997 theorem of Lee and Scobee which says that a matroid is both ternary and orientable if and only if it is dyadic (i.e. representable by a matrix with rational entries whose maximal minors have determinant equal to ± 2^k for some natural number k).

(Shah): Many geometric invariants of a normal toric variety X can be described in terms of its associated polyhedral fan F. Fulton and Sturmfels described the Chow ring of a complete toric variety using Minkowski weights, which are a ring of integer-valued functions on F satisfying a balancing condition. These weights have appeared in many contexts since their introduction, including in tropical geometry where they are central to tropical intersection theory.

Here, I will discuss a K-theoretic analogue of Minkowski weights which we call Grothendieck weights. These weights describe the "operational" K-theory of a complete toric variety, and their balancing condition can be expressed in terms of Ehrhart theory. We will see some of their properties, and connections they have to other invariants of toric varieties including the K-theories of vector bundles and coherent sheaves.

(Mészáros):

Past Seminars

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