Seminar (online) 07.06.2023: D. Talalaev (MSU, YarSU, ITEP)
Title: Polynomial graph invariants, Sturm criterion, hyperbolic stability and total positivity
Abstract:
I will talk about various manifestations of the phenomenon of total positivity in such seemingly remote areas as polynomial invariants of graphs, problems of localization of the roots of a polynomial, the geometry of flag varieties. Nevertheless, in all these problems, the condition we are interested in is expressed in terms of total positivity, or n-positivity, of some auxiliary matrix: Toeplitz matrix, Hankel matrix, or Sturm matrix, depending on the problem.
In particular, I will talk on logarithmic concavity of the conditional flow and chromatic polynomial and the criterion of hyperbolic stability of a polynomial.
The report is based on several joint works with B. Bychkov, D. Golitsyn, A. Kazakov, A. Kutuzova, E. Lerner and S. Mukhamedzhanova.