Scattering Amplitudes: their structure, their singularities and their geometry

Explicit computations of scattering amplitudes have revealed a wealth of interesting structures such as: iterated integrals, their symbols and their Hopf algebra, momentum twistor geometry and cluster algebras, positive geometry and more. In this talk I will review some of these results and I will also describe a program to compute the iterated integral form from a detailed study of singularities (Landau equations) and the monodromies around branching hypersurfaces. This relies on older results of Leray, Thom and Pham.