Algebraic and Combinatorial Aspects of Generalized Hypergeometric Functions

广义超几何函数的代数和组合方面

• 批准号: 0099707
• 负责人: Eduardo Cattani
• 金额: $ 9.9万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0099707&HistoricalAwards=false
• 关键词: Algebraic; Combinatorial; Aspects; Generalized; Hypergeometric

This project is centered around the study of A-hypergeometricfunctions. These have been introduced in the 80's by Gel'fandand his coworkers as solutions of a system of partialdifferential equations encoding combinatorial data of polytopes. Specific problems to be addressed in this project include: theclassification of rational hypergeometric functions and theirrelationship with toric residues; bounds for the holonomic rankand rational rank; hypergeometric functions arising as periods ofCalabi-Yau manifolds. Potential applications of this workinclude a deeper understanding of the properties of theA-discriminant and new algorithms for the computation of totalresidues -a rational expression on the coefficients of a systemof polynomial equations which is of considerable interest incomputational algebraic geometry- and, ultimately, thedevelopment of algorithms for solving polynomial equations. The study of polynomial equations is of fundamental importance onalmost all branches of science and technology. In the last fewyears it has become clear that the best approaches to theirsolution are those that combine numerical and symbolic methods. The work on this project attempts to answer some basic questionson the symbolic study of a class of partial differentialequations closely related to systems of polynomial equations withthe goal of developing new algorithms. This same system ofdifferential equations arise also in Mirror Symmetry, afundamental theory in high-energy physics, with remarkableconsequences in pure mathematics. The investigator hopes to beable to relate the symbolic approach to the classical algebraicgeometric (Hodge Theory) approach.

本项目围绕A-超几何函数的研究展开。 这些已经介绍了在80年代的凝胶fand和他的同事作为解决方案的一个系统的偏微分方程编码组合数据的多面体。具体问题包括：有理超几何函数的分类及其与复曲面残数的关系;完整秩和有理秩的界;作为Calabi-Yau流形周期的超几何函数。 这项工作的潜在应用包括更深入地了解的A-判别式和新的算法的计算总残留物的属性-一个合理的表达的系数系统的多项式方程是相当大的兴趣在计算代数几何-并最终，thedevelopment的算法求解多项式方程。 多项式方程的研究在几乎所有的科学和技术分支中都是非常重要的。 在过去的几年里，人们已经清楚地认识到，解决这些问题的最佳方法是将联合收割机的数值方法和符号方法结合起来。本项目的工作试图回答与多项式方程组密切相关的一类偏微分方程的符号研究中的一些基本问题，并以发展新的算法为目标。 同样的微分方程系统也出现在镜像对称中，镜像对称是高能物理学的基础理论，在纯数学中有着可解释的推论。 研究者希望能够将符号方法与经典的代数几何（霍奇理论）方法联系起来。