Bergman fans, amalgams, and tropical correspondences

Tropical geometry extends the combinatorial dictionary of toric geometry to subvarieties that are not themselves torus-invariant. A fundamental example is the tropicalization of a linear subspace, which produces a Bergman fan depending only on the associated matroid. These fans give rise to the Chow rings of matroids, objects whose intersection theory has played a central role in recent breakthroughs connecting matroid theory and algebraic geometry.

In this talk, I will discuss two further instances of this dictionary. First, I will explain how the tropical analogue of the categorical fibre product naturally leads to the matroidal notion of a proper amalgam. Second, I will introduce tropical correspondences between Bergman fans and explain how certain graph correspondences refine the matroidal notion of weak maps. Along the way, I will emphasize the interpretation of matroid Chow rings in terms of piecewise-linear functions on Bergman fans.