Polya-Schur-Lax problems: hyperbolicity and stability preservers

This workshop, sponsored by AIM and the NSF, 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.