Petter Brändén: Tropicalization of hyperbolic polynomials

Hyperbolic polynomials and their relatives appear in various areas of mathematics. The study of the tropical image of hyperbolic polynomials was initialized by David Speyer, who used it to give an alternative proof of Horn's conjecture on eigenvalues of sums of Hermitian matrices. We look further into the intricate structure of the tropicalization of hyperbolic polynomials and describe discrete concavity properties satisfied by it.