Some applications of polynomials with the half-plane property

A multivariate polynomial has the half-plane property if it is non-zero whenever all variables are in the open right half-plane. These polynomials and their cousins appear in many different areas of mathematics and have recently received attention in Combinatorics. We will talk about how they appear naturally in the solutions to open problems and conjectures in Combinatorics, Matrix Theory and Complex Analysis, and how they come up in the study of immanants and in the "Permanent-on-top" conjecture. This talk is based on work joint with Julius Borcea and Boris Shapiro.