Department of Mathematics, MIT
In 2006 at MSRI, nine tropical geometers and combinatorialists met and announced the list of ten key open problems in (algebraic and combinatorial side of) tropical geometry. Axiomatization of tropical oriented matroids was one of them. After the work of Develin and Sturmfels, tropical oriented matroids were conjectured to be in bijection with subdivisions of product of simplices as well as with tropical pseudohyperplane arrangements. Ardila and Develin defined tropical oriented matroid, and showed one direction that tropical oriented matroids encode subdivision of product of simplices. Recently, in joint work with Oh, we proved that every triangulation of product of simplices encodes a tropical oriented matroid.
In this talk, I will give a survey on this topic, and discuss this well known conjecture. I will also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.