The positive Grassmannian, the amplituhedron, and cluster algebras

Abstract

The positive Grassmannian ${\mathrm{Gr}}_{k,n}^{\ge 0}$ is the subset of the real Grassmannian where all Plücker coordinates are nonnegative. It has a beautiful combinatorial structure as well as connections to statistical physics, integrable systems, and scattering amplitudes. The amplituhedron ${\mathcal{A}}_{n,k,m}(Z)$ is the image of the positive Grassmannian ${\mathrm{Gr}}_{k,n}^{\ge 0}$ under a positive linear map ${\mathbb{R}}^n\to {\mathbb{R}}^{k+m}$. We will explain how ideas from oriented matroids, tropical geometry, and cluster algebras shed light on the structure of the positive Grassmannian and the amplituhedron.