Polya-Schur-Lax problems: hyperbolicity and stability preservers
May 28 to June 1, 2007
American Institute of Mathematics,
Palo Alto, California
and Victor Vinnikov
This workshop will be devoted to bringing together researchers working
on the following topics and their interplay:
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.
- Polya-Schur problems: classification of linear preservers of
polynomials and entire functions in one or several variables with
prescribed zero sets.
- Lax-type problems: determinantal representations of multivariate
Gårding-hyperbolic polynomials and related objects.
- Properties and applications of stable polynomials and polynomials
with the half-plane property.
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 and a problem list.
another group photo