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Geometry of polynomials and applications

Wednesday 23rd January 2008 - 16:30 to 17:30
INI Seminar Room 1

We discuss closely related geometric properties of zero sets of multivariate polynomials such as real stability, the half-plane property, (Gårding) hyperbolicity and the strong Rayleigh property. In particular, using the latter we define a class of probability measures (called strongly Rayleigh) that contains uniform random spanning tree measures, determinantal measures (for contractions) and distributions for symmetric exclusion processes. We develop a theory of negative dependence for this class of measures and settle several conjectures of Liggett, Pemantle and Wagner. Examples and applications include extensions of Lyons' recent results on determinantal measures and the half-plane property for certain matroids studied by Sokal, Wagner et al.

This is joint work with Petter Brändén (KTH) and Thomas M. Liggett (UCLA).

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons