Petter Branden: Combinatorics of hyperbolic polynomials

Abstract

Hyperbolic polynomials have a rich combinatorial structure which is far from understood. We will talk about combinatorial properties of hyperbolic polynomials such as

(1) the tropicalization of the space of hyperbolic polynomials, which contains the hive cone as well as the tropical Grassmannian,

(2) hyperbolic polynomials in permanent theory as in Gurvits amazing proof of the Van den Waerden conjecture, and the recent proof of the monotone permanent conjecture by Haglund, Visontai, Wagner and the speaker, and

(3) their role in (poly-) matroid theory, and the consequences to semidefinite optimization.