Multivariate Hypergeometric Functions: Combinatorics and Algebra

多元超几何函数：组合学和代数

• 批准号: 1500832
• 负责人: Laura Matusevich
• 金额: $ 15万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500832&HistoricalAwards=false
• 关键词: Multivariate; Hypergeometric; Functions; Combinatorics; Algebra

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.

单变量超几何函数是数学、科学和工程中广泛应用的基本对象。几个变量的超几何函数具有同样的重要性。例如，五次或更高次的多项式方程不能用根来解，但它们总是可以用多变量超几何函数来解，而不考虑次。然而，在几个变量中工作带来了巨大的挑战。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；改变参数会导致实质性的变化，在大多数情况下，这些影响和控制它们的机制都没有被完全理解。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。在二十世纪后期，Gelfand, Graev， Kapranov和Zelevinsky介绍了基于环面变分的超几何函数和微分方程的广义理论。这些作者在超几何环境中开发了强大的代数组合工具，它们提供了关于单变量超几何函数的一些非常经典的陈述的广泛而优雅的推广。本项目的目标是利用多面体几何、交换代数、d模理论和复分析等技术来研究这些超几何函数和微分方程。本研究的新工具的开发也激励和启发了组合交换代数中的特定项目。另一个主要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。