Amplitudes, tropical geometry and cluster algebras

Recently there are hints that there should be an underlying geometrical framework for describing scattering amplitudes in N=4 super Yang-Mills theory. Cluster algebras encode many interesting physical statements about the nature of singularities of these amplitudes and their relation to each other. We find the notion of tropical geometry allows one to obtain finite sets of singularities even when the cluster algebras become infinite and moreover leads to a natural definition of algebraic singularities of exactly the type seen in recent explicit calculations.