The Combinatorics of the Amplituhedron – Tiles, Tilings, and Cluster Algebras [1]

The amplituhedron is the image of the positive Grassmannian—the region of the Grassmannian where all Plucker  ̈

 

coordinates are nonnegative—under a totally positive linear map. It is a far-reaching generalization of cyclic polyto-pes and hyperplane arrangements, and the positive

Grassmannian itself. The “volume” of the amplituhedron encodes probabilities of particle interactions in the quantum field

theory N=4 super Yang-Mills, and calculating this volumeinvolves decomposing (or tiling) the amplituhedron into

smaller pieces (or tiles) and summing their volumes. This talk will delve into the rich combinatorics of these tiles and

tilings, presenting recent results on some of the central conjectures in this area, including the magic number, BCFW tiling, and cluster adjacency conjectures.