Anatomy of the amplituhedron

We initiate a comprehensive investigation of the geometry of the amplituhedron, a recently found geometric object whose volume calculates the integrand of scattering amplitudes in planar N = 4 $$ \mathcal{N}=4 $$ SYM theory. We do so by introducing and studying its stratification, focusing on four-point amplitudes. The new stratification exhibits interesting combinatorial properties and positivity is neatly captured by permutations. As explicit examples, we find all boundaries for the two and three loop amplitudes and related geometries. We recover the stratifications of some of these geometries from the singularities of the corresponding integrands, providing a non-trivial test of the amplituhedron/scattering amplitude correspondence. We finally introduce a deformation of the stratification with remarkably simple topological properties