The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes

Scattering amplitudes are both a wonderful playground to discover novel ideas in Quantum Field Theory and simultaneously of immense phenomenological importance to make precision predictions for e.g.~particle collider observables and more recently also for gravitational wave signals. In this review chapter, we give an overview of some of the exciting recent progress on reformulating QFT in terms of mathematical, geometric quantities, such as polytopes, associahedra, Grassmanians, and the amplituhedron. In this novel approach, standard notions of locality and unitarity are derived concepts rather than fundamental ingredients in the construction which might give us a handle on a number of open questions in QFT that have evaded an answer for decades. We first give a basic summary of positive geometry, before discussing the associahedron -- one of the simplest physically relevant geometric examples -- and its relation to tree-level scattering amplitudes in bi-adjoint $\phi^3$ theory. Our second example is the amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory.