Multivariate Hypergeometric Functions and Equations

多元超几何函数有许多不同的定义。这些包括Horn函数（由Appell和Horn在1880年左右定义），矩阵参数的超几何函数（由Bochner和Herz在1950年左右引入，用于多元统计），与根相关的超几何函数（由于Opdam和Heckman在20世纪80年代中期，与动力系统和表示理论有关），以及Gelfand， Kapranov和Zelevinsky的a -超几何函数（可追溯到20世纪80年代后期，用于代数几何）。尤其是镜像对称)。虽然这些函数乍一看似乎彼此无关，除了每个函数都包含单变量高斯超几何函数作为特殊情况这一明显事实之外，它们之间已经发现了一些重要的联系。该项目旨在研究上述的每一个多元超几何理论，并将其发展为数学研究人员的工具；为了概括它们之间已知的联系并找到新的联系，可能在每个超几何类所属的区域之间建立新的桥梁；并推进代数d模理论（研究超几何方程的自然框架），以及离散几何和环面几何（它们提供了使超几何系统特殊的额外结构）。单变量超几何函数的研究始于两百多年前。这本身就是一个美丽而有趣的理论，但它的重要性在于它在纯数学和应用数学以及物理学、工程学和统计学中的许多应用。大多数熟悉的函数，从基本的正弦和余弦到更复杂的贝塞尔函数，都是超几何的。多变量的超几何函数在许多情况下也会自然出现。例如，一个著名的问题是，要找到一个五次多项式的根的系数表达式，它只使用根号，将ax^2 + bx + c = 0的根的二次公式推广到a， b和c。阿贝尔在1824年第一次完整地证明了这一点，几年后，伽罗瓦准确地发现了哪些多项式可以用根号解。但是，问题仍然存在：在一般情况下，我们需要使用什么样的函数来代替基？梅林在1921年通过证明多项式的根是系数的超几何函数回答了这个问题。该项目的目标是进一步研究多变量的超几何函数，这是目前在许多领域研究的前沿，但对其知之甚少比在一个变量的情况下。

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.