We use the geometry of the stellahedral toric variety to study matroids. We identify the
valuative group of matroids with the cohomology ring of the stellahedral toric variety and …

This article is based on the Clay Mathematics Senior Scholar Lecture that was delivered by
Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture was the tropical …

We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier
divisors and intersection products between these two (without passing to rational …

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry.
Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over …

We introduce certain torus-equivariant classes on permutohedral varieties which we call
“tautological classes of matroids” as a new geometric framework for studying matroids …

Tropicalizations form a bridge between algebraic and convex geometry. We generalize
basic results from tropical geometry which are well-known for special ground fields to …

In a recent paper, the first author proved the log-concavity of the coefficients of the
characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a …

We construct the moduli spaces of tropical curves and tropical principally polarized abelian
varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli …

We construct an intersection product on tropical cycles contained in the Bergman fan of a
matroid. To do this we first establish a connection between the operations of deletion and …

We introduce a binary operation over complex numbers that is a tropical analog of addition.
This operation, together with the ordinary multiplication of complex numbers, satisfies …