Tropical Geometry Forwards and Backwards

Abstract

Tropical geometry is a relatively recent network of ideas at the intersection of geometry and combinatorics. In its purest form, tropical geometry is a combinatorial analogue of algebraic geometry and can be studied logically independently from it. The basic objects in tropical geometry are things like graphs or simplicial complexes. They are decorated with some additional data that allows them to coarsely mimic objects from algebraic geometry. The properties of these objects are constrained by combinatorial analogues of the basic theorems in geometry. For example, the tropical analogue of a Riemann surface is a graph, and there are purely combinatorial analogues of the Riemann-Roch and Riemann-Hurwitz theorems for graphs. The surprising turn is often that theorem statements in tropical geometry are the same as or similar to parallel ones in algebraic geometry, even though all the words mean something completely different.