Zachary Greenberg - Classifying bounded ratios on cluster algebras with U-variables скачать в хорошем качестве

Zachary Greenberg - Classifying bounded ratios on cluster algebras with U-variables

This lecture was part of the Thematic Programme on "Amplitudes and Algebraic Geometry" held at the ESI February 19 - March 27, 2026. A classic problem is to find inequalities which are satisfied by polynomials of minors of any totally positive matrix. When these inequalities only involve multiplication, they are equivalent to finding ratios of minors which are bounded over the set of totally positive matrices. Furthermore, the set of matrix minors are cluster coordinates. With M. Gekhtman and D. Soskin, we used this correspondence to classify the bounded ratios in any finite cluster algebra. Surprisingly, the key tool are the U-variables and "binary geometries" which have been proposed as a tool for studying scattering amplitudes.