Stability, hyperbolicity, and zero localization of functions

This workshop, sponsored by AIM and NSF, was 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 [CM, Post, Fisk, Rah]. The multivariate generalizations go back to 1950s and are being actively explored now [BB2,WagB]. 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 [Gur2, Gur3, Wag2, BBS, BB1, BB2, BBL]. 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 The workshop participants included experts in real and complex analysis, orthogonal polynomials and special functions, combinatorics and probability, statistical mechanics, as well as matrix and operator theory. While some of the participants knew each other and some even collaborated previously, many also met for the first time during the workshop. Thus, a broader goal of the workshop was to promote interaction and future collaboration among researchers in distinct but interrelated fields. The following main open problems were initially selected as focal points of this workshop: