Stability, hyperbolicity, and zero localization of functions

December 5 to December 9, 2011

at the

American Institute of Mathematics, Palo Alto, California

organized by

Petter Branden, George Csordas, Olga Holtz, and Mikhail Tyaglov

Original Announcement

This workshop will be devoted to the emerging theory of stability and hyperbolicity of functions. These notions are well known in the univariate setting, where stability means that all zeros lie in the left-half plane and hyperbolicity means that all zeros are real. The multivariate generalizations go back to 1950s and are being actively explored now. Among recent applications of multivariate stability and hyperbolicity are the proof of Johnson's conjectures on mixed determinants, new proofs of van der Waerden and Schrijver-Valiant type conjectures, and the resolution of several conjectures on negative dependence in discrete probability theory.

Hyperbolic and stable multivariate polynomials arise surprisingly often in matrix-theoretic and combinatorial applications. For example, three major conjectures -- Lieb's "permanent-on-top" (POT) conjecture in matrix theory/combinatorics, the Bessis-Moussa-Villani (BMV) conjecture in quantum statistical mechanics, and Mason's conjecture in combinatorics can all be reformulated by means of real stable polynomials.

Another important class of examples is provided by analytic number theory, where the Riemann Zeta function and other L-functions are conjectured to have nontrivial zeros on the line Re z =1/2. This can be restated as the hyperbolicity of the Riemann Xi function (or other associated functions).

The aim of the workshop is both to contribute to the general theory of stability and hyperbolicity as well as to find ways of applying this theory to concrete problems of algebra, analysis, combinatorics, and mathematical physics.

Some of the main open problems that the workshop will focus on are the following.

  1. The Bessis-Moussa-Villani (BMV) conjecture originally arose in the theory of quantum mechanical system. It states that tr (exp(A-tB)) is the Laplace transform of a positive measure for any two positive definite matrices A and B. It can be restated as a conjecture about the positivity of all coefficients of specific polynomials. We plan to investigate the zero distribution of these polynomials.

  2. The Mason conjecture states the ultra-log concavity of the f-vector of a matroid or, equivalently, of the sequence of coefficient of the so-called independent set polynomial of that matroid. We plan to consider the graphical case of Mason's conjecture using a variety of newly developed techniques.

  3. Lee-Yang problems of statistical mechanics and multivariate stability/hyperbolicity: Phase transitions in statistical mechanics can be determined and analyzed using the corresponding partition functions. We would like to understand a general theoretical framework for stability and hyperbolicity of partition functions in statistical mechanics (Ising, Potts, and other models).

  4. Pólya-Schur problems: classification of linear (or non-linear) preservers of polynomials and entire functions in one or several variables with prescribed zero sets.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:
Classification theorems for operators preserving zeros in a strip
The generalized Laguerre inequalities and functions in the Laguerre-Polya class
Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials
Zero-free polynomial approximation on a chain of Jordan domains