Title: On the classification of hyperbolicity and stability preservers

Abstract: A linear operator $T$ on $\mathbb{C}[z]$ is called
hyperbolicity-preserving or an HPO for short if $T(P)$ is hyperbolic
whenever $P\in \mathbb{C}[z]$ is hyperbolic, i.e., it has all real
zeros. One of the main challenges in the theory of univariate complex
polynomials is to describe the monoid $\mathcal{A}_{HP}$ of all HPOs.
This reputably difficult problem goes back to P\'olya-Schur's well-known
characterization of multiplier sequences of the first kind, that is,
HPOs which are diagonal in the standard monomial basis of
$\mathbb{C}[z]$. P\'olya-Schur's 1914 result generated a vast literature
on this subject and related topics at the interface between analysis,
operator theory and algebra but so far only partial results under rather
restrictive conditions have been obtained. In this talk I will report on
the progress towards solutions to both this problem and its analog for
(Hurwitz) stable polynomials as well as their multivariate versions made
in an ongoing series of papers jointly with Petter Br\"and\'en and Boris
Shapiro.