Matroids and the geometry of toric vector bundles

I'll give an overview of some recent work on the geometry of projectivized toric vector bundles.  A toric vector bundle is a vector bundle over a toric variety equipped with an action by the defining torus of the base.  As a source of examples, toric vector bundles and their projectivizations provide a rich class of spaces that still manage to admit a combinatorial characterization.  I'll begin with a recent classification result which shows that a toric vector bundle can be captured by an arrangement of points on the Bergman fan of a certain matroid.   Then I'll describe how to extract geometric information of the projectivization of the toric vector bundle when this data is nice. In particular, I'll discuss the Cox ring, the canonical class, the nef cone, and Fujita's freeness conjectures, focusing on the case when the matroid is uniform.  Then I'll describe how these properties interact with natural operations on toric vector bundles.  This involves the geometry of the closely related class of toric flag bundles and leads to some combinatorial questions about multilinear operations on matroids. This is joint work with Kiumars Kaveh, Courtney George, Austin Alderete, and Ayush Tibrewal.