Stability, hyperbolicity, and zero localization of functions

This workshop, sponsored by AIM and the NSF, 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. For surveys of these and further developments, see http://www.math.hawaii.edu/~tom/mathfiles/czdssurvey.pdf and http://www.ams.org/journals/bull/2011-48-01/S0273-0979-2010-01321-5/home.html

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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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