Surfacehedra

Scattering amplitudes are one the most fundamental observables in physics and in recent years it has been appreciated how they are intimately connected with various branches of discrete mathematics, such as Combinatorics and Tropical Geometry. The recurrent theme of these connections is that the singularity properties of scattering amplitudes, such as the patterns of poles and residues that they are allowed to have, are mirrored by the boundary structure of certain geometrical objects defined by some notion of positivity.

In this talk I will review a recent and particularly simple instance of this general phenomenon which sees on the physical side of the correspondence a colored cubic scalar field theory, and on the mathematical side a class of convex polytopes that describe the combinatorics of the crossing of curves on Riemann surfaces. The emphasis will be put on describing these new polytopes, dubbed Surfacehedra, which generalize the classical Associahedron to any surface.