Positive Geometries in the Real World

Our understanding of the fundamental laws of Nature is based on the study of scattering amplitudes, traditionally computed using Feynman diagrams.In the past three decades, the shortcomings of this representation have become increasingly clear. Scattering amplitudes enjoy a simplicity which is destroyed by Feynman diagrams, resulting in an overwhelming apparent complexity of computations.This has motivated the search for alternatives to Feynman diagrams, which culminated in the discovery that scattering amplitudes in two toy theories, maximally supersymmetric Yang-Mills theory, and the simplest theory describing colored scalars, can be computed as ""Volumes"" of positive geometries: regions in kinematic space carved out by inequalities.In the new representation it is the usual properties kept manifest by Feynman diagrams, Locality and Unitarity, which are now obscured while the simplicity of scattering amplitudes is restored.These developments are both conceptually intriguing and practically useful