Multivariate Hypergeometric Functions and Equations

多元超几何函数有很多不同的定义。它们包括Horn函数(由Appell和Horn在1880年左右定义)，矩阵变元的超几何函数(由Bochner和Herz在1950年左右引入，用于多元统计)，与根系统相关的超几何函数(由Opam和Heckman在20世纪80年代中期定义，与动力系统和表示理论有关)，以及Gelfand，Kapranov和Zlevinsky的A超几何函数(始于20世纪80年代末，用于代数几何，特别是镜像对称)。虽然乍一看，这些函数似乎彼此不相关，但除了每个函数都包含作为特例的一元高斯超几何函数这一明显的事实之外，它们之间也存在着一些不平凡的联系。这个项目的目的是研究上述每一种多元超几何理论，将它们发展为研究人员跨越数学的工具；推广它们之间的已知联系并寻找新的联系，潜在地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(研究超几何方程的自然框架)以及离散和环面几何(它提供了使超几何系统特殊的额外结构)的理论。对一元超几何函数的研究始于200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用途。最常见的函数，从初等正弦和余弦到更复杂的贝塞尔函数，都是超几何函数。在许多情况下，多变量的超几何函数也是自然产生的。例如，根据系数求五次多项式的根的表达式是一个著名的问题，它只使用根，从而推广了用a、b和c表示的ax^2+bx+c=0的根的二次公式。1824年，Abel给出了第一个不能做到这一点的完整证明，几年后，Galois准确地找到了哪些多项式可以用根来求解。尽管如此，问题仍然是：我们需要使用什么样的函数来取代一般情况下的部首？1921年，梅林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。这个项目的目标是进一步研究多个变量的超几何函数，这些超几何函数目前在许多领域处于研究的前沿，但对这些超几何函数的了解比单变量情况要少得多。

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.