New developments in hypergeometric equations

超几何函数是幂函数(一个或几个变量)，其系数满足两项递归。它们的定义使超几何函数易于用解析和组合方法处理。从D-模理论的角度研究它们的微分方程式是本课题的主题。20世纪80年代末，Gelfand，Graev，Kapranov和Zlevinsky的突破为超几何函数(多元)和具有强烈组合色彩的代数几何的子域环变簇理论之间提供了新的联系。这一链接提供了各种代数、同调和组合工具来研究超几何微分方程及其解。这个项目的目标是推进超几何函数和微分方程式的理论。超几何微分方程式的解包含丰富而有趣的函数类别。常见的函数，如三角函数，都属于这一类。这些函数的重要性在于它们的有用性，不仅在数学中，而且在物理和工程中也是如此。目前的研究将提供超几何微分方程式理论的进展；其基本哲学是将超几何函数视为不同数学领域之间的桥梁。

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.