Discrete Geometry and Combinatorics Seminar

Steven KarpUniversity of California at Berkeley
The $m=1$ amplituhedron and cyclic hyperplane arrangements

Monday, October 17, 2016 - 2:30pm
Malott 206

The totally nonnegative part of the Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspaces of $R^n$ whose Plucker coordinates are all nonnegative. The amplituhedron is the image in $Gr(k,k+m)$ of the totally nonnegative part of $Gr(k,n)$, through a $(k+m)\times n$ matrix with positive maximal minors. It was introduced in 2013 by Arkani-Hamed and Trnka in their study of scattering amplitudes in $N=4$ supersymmetric Yang-Mills theory. Taking an orthogonal point of view, we give a description of the amplituhedron in terms of sign variation. We then use this perspective to study the case $m=1$, giving a cell decomposition of the $m=1$ amplituhedron and showing that we can identify it with the complex of bounded faces of a cyclic hyperplane arrangement. It follows that the $m=1$ amplituhedron is homeomorphic to a ball. This is joint work with Lauren Williams.