Starting from the celebrated result of Parke and Taylor, and more drastically in the past two decades, it has become more and more evident how textbook tools of Quantum Field Theory provide an unsatisfactory description of scattering amplitudes, and that new physical ideas are required to replace them. One clue of what these could be came from the discovery of the Amplituhedron, which allows to define and compute scattering amplitudes in (planar) N=4 super Yang-Mills without any reference to a Lagrangian or Feynman diagrams, but rather casting the computation in terms of a purely geometrical problem. This has motivated the search for similar structures, nowadays called Positive Geometries, which could underlie amplitudes in other field theories, possibly closer to the ones describing our world. The goal of my talk is to describe the Positive Geometry which describes a simple colored scalar theory. As an application, I will present a novel unified Schwinger presentation for the amplitudes of this theory, valid at any order of the 1/N expansion, and which allows to practically compute amplitudes at very large multiplicity.
