Combinatorics of Amplituhedra – Scattering Amplitudes and Triangulations

In this talk I will discuss about Amplituhedra – generalizations of polytopes inside the Grassmannian – introduced by physicists as one of the new geometric constructions encoding physical observables in Quantum Field Theories. In particular, I will explain how the problem of finding triangulations of Amplituhedra is related to computing scattering amplitudes of planar N=4 super Yang-Mills theory. We will discuss how the combinatorics of triangulations interplays with T-duality from String Theory, in connection with a dual object – the Momentum Amplituhedron. A generalization of T-duality led us to discover a striking duality between triangulations of Amplituhedra of m=2 type and the ones of a seemingly unrelated object – the Hypersimplex. The latter is a polytope with beautiful combinatorics with connections to matroid theory and tropical geometry. Finally, we comment on how the search for triangulations of Amplituhedra beyond the physical m=4 type opens up a new surprising zoo of complexity which points towards a generalization of the way “amplitudes” can factorise.