Classifying Bounded Ratios on Cluster Algebras with U-Variables

Explore the classification of bounded ratios in finite cluster algebras through this mathematical lecture that connects totally positive matrices, cluster coordinates, and U-variables. Learn how the classic problem of finding inequalities satisfied by polynomials of minors of totally positive matrices relates to identifying bounded ratios of minors over the set of totally positive matrices. Discover the correspondence between matrix minors as cluster coordinates and how this connection enables the classification of bounded ratios in any finite cluster algebra. Examine the surprising role of U-variables and "binary geometries" as key tools in this classification, which have been proposed for studying scattering amplitudes. Understand the collaborative research findings with M. Gekhtman and D. Soskin that reveal the deep connections between algebraic geometry, cluster algebras, and amplitudes theory. Gain insights into how these mathematical structures provide a framework for understanding bounded ratios and their applications in both pure mathematics and theoretical physics contexts.