  We describe all linear operators on spaces of multivariate polynomials
preserving the property of being non-vanishing in open circular domains. This
completes the multivariate generalization of the classification program
initiated by P\'olya-Schur for univariate real polynomials and provides a
natural framework for dealing in a uniform way with Lee-Yang type problems in
statistical mechanics, combinatorics, and geometric function theory.
  This is an announcement with some of the main results in arXiv:0809.0401 and
arXiv:0809.3087.
