# Polya-Schur-Lax problems: hyperbolicity and stability preservers

May 28 to June 1, 2007

at the

American Institute of Mathematics, San Jose, California

organized by

Julius Borcea, Petter Branden, George Csordas, and Victor Vinnikov

## Original Announcement

This workshop will be devoted to bringing together researchers working on the following topics and their interplay:
1. Polya-Schur problems: classification of linear preservers of polynomials and entire functions in one or several variables with prescribed zero sets.
2. Lax-type problems: determinantal representations of multivariate Gårding-hyperbolic polynomials and related objects.
3. Properties and applications of stable polynomials and polynomials with the half-plane property.
The first topic goes back to Laguerre and Polya-Schur and is intimately connected with the other two, as shown by the recent solutions to the Polya-Schur problem for univariate hyperbolic (i.e., real-rooted) polynomials and the 1958 Lax conjecture for Garding-hyperbolic polynomials, respectively. One of the goals of this workshop is to extend the Polya-Schur characterization to other fundamental classes of univariate and multivariate polynomials and entire functions. In the process we hope to shed new light on a number of related problems, such as describing Fourier transforms with all real zeros.

The recently established Lax conjecture has already proved to be very fruitful and is expected to have many more far-reaching consequences as we are yet to fully explore it. In particular, appropriate analogs of the Lax conjecture for multivariate (real) stable polynomials - i.e., polynomials which are non-vanishing whenever the imaginary parts of its variables are all positive - would be most useful. Indeed, in just a few years these polynomials have become an important tool in several apparently unrelated areas (matroid theory, quantum statistical mechanics, the theory of Hermitian matrices, negative dependence/probability theory) and seem to provide an appropriate framework for studying a number of difficult problems in these areas.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

On the stability of Taylor sections of a function $\sum_k=0^\inftyz^k/a^k^2, a > 1$
by  Olga M. Katkova and Anna M. Vishnyakova
A sufficient condition for a polynomial to be stable
by  Olga M. Katkova and Anna M. Vishnyakova
Negative dependence and the geometry of polynomials
by  Julius Borcea, Petter Brändén, and Thomas M. Liggett
Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products
by  Julius Borcea and Petter Brändén
Polya-Schur master theorems for circular domains and their boundaries
by  Julius Borcea and Petter Brändén
Multivariate Polya-Schur classification problems in the Weyl algebra
by  Julius Borcea and Petter Brändén
Additive invariants on quantum channels and applications to regularized minimum entropy
by  Shmuel Friedland
Remarks on BMV conjecture
by  Shmuel Friedland
Matrices Totally Positive Relative to a Tree
by  Charles R. Johnson, Roberto S. Costas-Santos, and Boris Tadchiev