There are many different definitions for multivariate hypergeometric functions. These include the Horn functions (defined by Appell and Horn around 1880), the hypergeometric functions of matrix arguments (introduced by Bochner and Herz around 1950, used in multivariate statistics), the hypergeometric functions associated to root systems (due to Opdam and Heckman in the middle 1980s, connected to dynamical systems and representation theory), and the A-hypergeometric functions of Gelfand, Kapranov and Zelevinsky (dating from the late 1980s, used in algebraic geometry, especially mirror symmetry). Although these functions seem at first glance to be unrelated to each other beyond the obvious fact that each contains the one-variable Gauss hypergeometric functions as a special case, some nontrivial connections among them have been found. This project aims to study each of the aforementioned multivariate hypergeometric theories, developing them as tools for researchers across mathematics; to generalize known connections between them and to find new ones, potentially building new bridges between the areas each hypergeometric class belongs to; and to advance the theory of algebraic D-modules (the natural framework in which to study hypergeometric equations), as well as discrete and toric geometry (which provide the extra structure that makes hypergeometric systems special).
The study of hypergeometric functions in one variable was started over two hundred years ago. This is a beautiful and interesting theory in its own right, but its importance lies in its many uses in both pure and applied mathematics, as well as in physics, engineering and statistics. Most familiar functions, from the elementary sine and cosine to the more sophisticated Bessel functions, are hypergeometric. Hypergeometric functions in several variables also arise naturally in many contexts. For example, it was a famous problem to find an expression for the roots of a polynomial of degree five in terms of their coefficients which uses only radicals, generalizing the quadratic formula for the roots of ax^2 + bx + c = 0 in terms of a, b and c. The first complete proof that this cannot be done was given by Abel in 1824, and a few years later, Galois found exactly which polynomials admitted solutions by radicals. Still, the question remained: what kind of functions do we need to use instead of radicals in the general case? Mellin answered this question in 1921 by showing that the roots of polynomials are hypergeometric functions of the coefficients. The goal of this project is to further the study of hypergeometric functions in several variables, which are now at the forefront of research in many areas, but for which much less is known than in the one variable case.
