Multivariate Hypergeometric Functions: Combinatorics and Algebra

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Hypergeometric functions in one variable are fundamental objects of widespread use in mathematics, science, and engineering. Hypergeometric functions in several variables share this importance. For instance, polynomial equations of degrees five or higher cannot be solved in terms of radicals, but they can always be solved using multivariate hypergeometric functions, regardless of the degree. Working in several variables, however, presents substantial challenges. This project seeks to overcome these challenges by developing new combinatorial and algebraic techniques. An important attribute of all hypergeometric functions and differential equations is that they depend on parameters; varying the parameters can cause substantial changes, and in most cases, neither these effects nor the mechanisms that control them are completely understood. The specific questions addressed in this project involve the investigation of the parametric behavior of hypergeometric functions and differential equations.In the late twentieth century, Gelfand, Graev, Kapranov, and Zelevinsky introduced a generalized theory of hypergeometric functions and differential equations based on toric varieties. Powerful algebro-combinatorial tools of independent interest were developed by these authors in the hypergeometric context, which have provided vast and elegant generalizations of some very classical statements about hypergeometric functions in one variable. The goal of this project is to use techniques drawn from polyhedral geometry, commutative algebra, D-module theory and complex analysis to study these hypergeometric functions and differential equations. The development of new tools for this study also motivates and inspires specific projects within combinatorial commutative algebra. Another major theme is to use hypergeometric tools and intuition to obtain results beyond the hypergeometric world.