Multivariate Hypergeometric Functions: Combinatorics and Algebra

单变量超几何函数是数学、科学和工程中广泛使用的基本对象。多元超几何函数也具有这种重要性。例如，五次或更高次的多项式方程不能用根式求解，但它们总是可以用多元超几何函数求解，而不管次数如何。然而，在几个变量中工作提出了巨大的挑战。该项目旨在通过开发新的组合和代数技术来克服这些挑战。所有超几何函数和微分方程的一个重要属性是它们依赖于参数;改变参数可以引起实质性的变化，并且在大多数情况下，这些效应和控制它们的机制都没有完全理解。在这个项目中解决的具体问题涉及超几何函数和微分方程的参数行为的调查。在世纪后期，Gelfand，Graev，Kapranov和Zelevinsky介绍了一个广义理论的超几何函数和微分方程的基础上环面品种。这些作者在超几何背景下开发了独立感兴趣的强大代数组合工具，这些工具为有关一元超几何函数的一些非常经典的陈述提供了大量而优雅的概括。这个项目的目标是使用多面体几何，交换代数，D-模理论和复分析的技术来研究这些超几何函数和微分方程。这项研究的新工具的开发也激励和启发组合交换代数中的具体项目。另一个主要的主题是使用超几何工具和直觉来获得超几何世界之外的结果。

Hypergeometric functions in one variable are fundamental objects of widespread use in mathematics, science, and engineering. Hypergeometric functions in several variables share this importance. For instance, polynomial equations of degrees five or higher cannot be solved in terms of radicals, but they can always be solved using multivariate hypergeometric functions, regardless of the degree. Working in several variables, however, presents substantial challenges. This project seeks to overcome these challenges by developing new combinatorial and algebraic techniques. An important attribute of all hypergeometric functions and differential equations is that they depend on parameters; varying the parameters can cause substantial changes, and in most cases, neither these effects nor the mechanisms that control them are completely understood. The specific questions addressed in this project involve the investigation of the parametric behavior of hypergeometric functions and differential equations.In the late twentieth century, Gelfand, Graev, Kapranov, and Zelevinsky introduced a generalized theory of hypergeometric functions and differential equations based on toric varieties. Powerful algebro-combinatorial tools of independent interest were developed by these authors in the hypergeometric context, which have provided vast and elegant generalizations of some very classical statements about hypergeometric functions in one variable. The goal of this project is to use techniques drawn from polyhedral geometry, commutative algebra, D-module theory and complex analysis to study these hypergeometric functions and differential equations. The development of new tools for this study also motivates and inspires specific projects within combinatorial commutative algebra. Another major theme is to use hypergeometric tools and intuition to obtain results beyond the hypergeometric world.