The Positive Grassmannian, the Amplituhedron, and Cluster Algebras

Explore the fascinating world of the positive Grassmannian and amplituhedron in this 47-minute lecture by Lauren K. Williams. Delve into the combinatorial structure of the positive Grassmannian and its connections to statistical physics, integrable systems, and scattering amplitudes. Discover how the amplituhedron emerges as the image of the positive Grassmannian under a positive linear map. Gain insights into the structure of these mathematical objects through the lens of oriented matroids, tropical geometry, and cluster algebras. Follow the comprehensive syllabus covering topics such as matroid stratification, cells of the positive Grassmannian, tiles of the amplituhedron, and the moment map for the Grassmannian. Enhance your understanding with visual aids from the accompanying presentation slides.

Intro
Overview of the talk
The Grassmannian and the matroid stratification
What is the positive Grassmannian?
Cells of the positive Grassmannian
What is the amplituhedron?
How can we understand the amplituhedron?
Tiles of the amplituhedron
Coordinates for the amplituhedron
Tiles and cluster algebras
Tilings of the amplituhedron
The moment map for the Grassmannian
Recap on hypersimplex and m = 2 amplituhedron
Example of theorem (when k = 1 and n = 4)
Summary