ICTS:30025

Video URL

Total positivity, directed networks and integrable cluster dynamics - III

Total positivity, directed networks and integrable cluster dynamics - III

(2024). Total positivity, directed networks and integrable cluster dynamics - III. SciVideos. https://youtube.com/live/DfUSTPjKYw4

Total positivity, directed networks and integrable cluster dynamics - III. SciVideos, Oct. 23, 2024, https://youtube.com/live/DfUSTPjKYw4 

          @misc{ scivideos_ICTS:30025,
            doi = {},
            url = {https://youtube.com/live/DfUSTPjKYw4},
            author = {},
            keywords = {},
            language = {en},
            title = {Total positivity, directed networks and integrable cluster dynamics - III},
            publisher = {},
            year = {2024},
            month = {oct},
            note = {ICTS:30025 see, \url{https://scivideos.org/icts-tifr/30025}}
          }
Michael Gekhtman
Talk numberICTS:30025
Source RepositoryICTS-TIFR
Source EventDiscrete integrable systems: difference equations, cluster algebras and probabi…
Collection

Abstract

Totally positive (TP) matrices are matrices in which each minor is positive. First introduced in 1930's by I. Schoenberg and F. Gantmakher and M. Krein, these matrices proved to be important  in many areas of pure and applied mathematics. The notion of total positivity was generalized by G. Lusztig in the context of reductive Lie groups and inspired the discovery  of cluster algebras by S. Fomin and A. Zelevinsky.

In this mini-course, I will first review some basic features of TP matrices, including their spectral properties and discuss some of their classical applications. Then I will focus on weighted networks parametrization of TP matrices due to A. Berenstein, S. Fomin and A. Zelevinsky. I will show how elementary transformations of planar networks lead to criteria of total positivity and important examples of mutations in the theory of cluster algebras. Finally, I will explain how particular sequences of mutations can be used to construct exactly solvable nonlinear dynamical sy...