MIT Combinatorics Seminar Hypergeometric Series and Binomial Primary Decomposition Ezra Miller (University of Minnesota and Ann Arbor) Wednesday November 8, 2006   4:15 pm    Room 2-136 ABSTRACT Erd\'elyi's work on multivariate hypergeometric functions in the 1950's raises fundamental questions about the number of holomorphic solutions to classical Horn systems. Techniques developed by Gelfand, Kapranov, Zelevinsky, and others in the 1980's and 1990's successfully deal with series solutions having full support (where the set of monomials with nonzero coefficients fills a cone of the maximum possible dimension). In work with Alicia Dickenstein and Laura Matusevich, we show that dealing with all holomorphic solutions is intimately tied to precise combinatorial descriptions, in terms of lattice points, of primary components of binomial ideals in polynomial rings. The talk will paint a more detailed picture of the historical development as well as the recent advances.