Combinatorics of the amplituhedron

The tree amplituhedron $A(n,k,m)$ is the image in the Grassmannian $Gr(k,k+m)$ of the totally nonnegative part of $Gr(k,n)$, under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in $N = 4$ supersymmetric Yang-Mills theory. In this talk I'll start with a gentle introduction to the amplituhedron, then give an equivalent "orthogonal" description of it. I'll then describe what the amplituhedron looks like in various special cases. For example, one can use the theory of sign variation and matroids to show that the amplituhedron $A(n,k,1)$ can be identified with the complex of bounded faces of a cyclic hyperplane arrangement (and hence is homeomorphic to a closed ball). This is joint work with Steven Karp.