Linear operators preserving stability and the Lee-Yang program
Meeting Room 3, CMS
We characterize all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing when the variables are in (products of) prescribed open circular domains, which solves the higher dimensional counterpart of a long-standing classification problem going back to P\'olya-Schur. This also leads to a self-contained theory of multivariate stable polynomials and a natural framework for dealing in a uniform manner with Lee-Yang type problems in e.g. combinatorics, statistical mechanics, complex analysis, which we illustrate with several examples. This is joint work with Petter Brändén.