Algebraic and Combinatorial Aspects of Generalized Hypergeometric Functions

本课题围绕A-超几何函数的研究展开。这些由Gel‘Fand和他的同事在80年代的S中作为编码多面体组合数据的偏微分方程组的解而被引入。这个项目要解决的具体问题包括：有理超几何函数的分类及其与环面剩余的关系；完整秩和有理秩界；作为Calabi-Yau流形周期产生的超几何函数。这项工作的潜在应用包括更深入地理解a-判别式的性质和计算全余项的新算法--计算代数几何中相当感兴趣的多项式方程组系数的有理表达式--以及最终发展多项式方程的求解算法。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，很明显，解决这些问题的最好方法是那些结合了数值方法和符号方法的方法。这个项目的工作试图回答一类与多项式方程组密切相关的偏微分方程组的符号研究中的一些基本问题，目的是开发新的算法。同样的微分方程组也出现在镜面对称性中，这是高能物理的基本理论，在纯数学中具有显著的序列。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。

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.