Classification of hyperbolicity and stability preservers Julius Borcea Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden julius@math.su.se A linear operator T on C[z] is called hyperbolicity-preserving or an HPO for short if T(P) is hyperbolic whenever P 2 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 AHP of all HPOs. This outstanding open 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 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 a complete solution of 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. The concepts of hyperbolicity and stability have natural multivariate extensions: a polynomial f 2 C[z1, . . . , zn] is stable if f(z1, . . . , zn) 6= 0 whenever =(zj) > 0, 1 j n. A stable polynomial with real coefficients is called real stable. Hence a univariate real stable polynomial is hyperbolic in the above sense. We generalize the notion of multiplier sequences to multivariate polynomials and give a complete characterization of higher-dimensional multiplier sequences. We then classify all operators in the Weyl algebra An of differential operators that preserve stability and show that real stability preservers in n variables are generated by real stable polynomials in 2n variables via the symbol map. One of the key ingredients in the proofs is a natural duality theorem for the Fischer-Fock spa