Gluing Young tableaux into a ball

Fix a partition and a maximum value, and build the following simplicial complex: one facet for each semistandard Young tableaux, with two glued together if their union is one of Buch's set-valued tableaux. Theorem: this is homeomorphic to a ball, with the set-valued tableaux as the interior faces.

I'll explain how the algebraic geometry of vexillary matrix Schubert varieties forced this discovery on us; why being a ball is pleasant, surprising, and important; and why Buch only discovered the interior faces. This work is joint with Ezra Miller and Alex Yong.