Matching fields and tropical hyperplane arrangements

An (n,d)-matching field is a collection of matchings such that there is a unique matching for each d-subset of [n]. They naturally arise as minimal matchings of a weighted complete bipartite graph, and can be thought of as a matroid-like structure for tropical geometry. They have relationships to multiple combinatorial objects, in particular arrangements of tropical hyperplanes. In this talk, we show how machinery from tropical geometry can be applied to matching fields via this relationship, and use it to prove two outstanding conjectures of Sturmfels and Zelevinsky on matching fields.