Hypergeometric functions are power series (in one or several variables) whose coefficients satisfy a two-term recursion. Their very definition makes hypergeometric functions amenable to treatment by analytic and combinatorial methods. The study of their differential equations from a D-module theoretic point of view is the subject of this project. A breakthrough in the late 1980s due to Gelfand, Graev, Kapranov and Zelevinsky provided a new connection between hypergeometric functions (in several variables) and the theory of toric varieties, a subfield of algebraic geometry with a strong combinatorial flavor. This link has made available a variety of algebraic, homological and combinatorial tools to study hypergeometric differential equations and their solutions. The goal of this project is to advance the theory of hypergeometric functions and differential equations.
The solutions of hypergeometric differential equations comprise a rich and interesting class of functions. Familiar functions, such as the trigonometric functions, belong to this class. The importance of these functions lies in their usefulness, not only within mathematics, but in physics and engineering as well. The present research will provide advances in the theory of hypergeometric differential equations; its underlying philosophy is to view hypergeometric functions as bridges between different areas of mathematics.
