Mathematical Sciences: Algebraic, Geometric and Combinatorial Structures Related to Multivariate Hypergeometric Functions

数学科学：与多元超几何函数相关的代数、几何和组合结构

• 批准号: 9625511
• 负责人: Andrei Zelevinsky
• 金额: --
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625511&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Algebraic; Geometric; Combinatorial

Zelevinsky This award provides funding for a project that will mainly be conducted in the following two areas: (A) Multivariate discriminants, resultants, and hyperdeterminants. (B) Representations of quantum groups and piecewise- linear combinatorics. Both of these subjects have deep relations with the theory of multivariate hypergeometric functions. Another unifying feature is that in both areas an essential role is played by new kinds of combinatorics whose main objects of study are lattice points in convex cones and polytopes and their piecewise-linear transformations. The principal investigator is going to continue his work on the following topics: (1) explicit polynomial formulas for multivariate discriminants and resultants, (2) singular loci of discriminant varieties, (3) algebraic and combinatorial properties of canonical bases in representations of quantum groups, (4) geometry of totally positive matrices, (5) Chow varieties of 0-cycles and symmetric functions in several vector variables, (6) secondary polytopes and maximal minor polytopes. This research explores connections between Combinatorics, Algebra, and Geometry. One of the goals of Combinatorics is to find efficient methods to study how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.

Zelevinsky 该奖项为一个项目提供资金，该项目主要在以下两个领域进行：（A）多元判别式、结果和超决定因素。 (B) 量子群和分段线性组合的表示。这两门学科都与多元超几何函数理论有着深厚的关系。另一个统一的特征是，在这两个领域中，新型组合学发挥着重要作用，其主要研究对象是凸锥体和多面体中的格点及其分段线性变换。首席研究员将继续从事以下主题的工作：（1）多元判别式和结果的显式多项式公式，（2）判别簇的奇异位点，（3）量子群表示中规范基的代数和组合性质，(4) 全正矩阵的几何，(5) 0 循环的 Chow 变体和几个向量变量中的对称函数，(6) 二次多面体和最大次数多胞体。 这项研究探讨了组合学、代数和几何之间的联系。组合学的目标之一是找到有效的方法来研究如何排列离散的对象集合。离散系统的行为对于现代通信极其重要。例如，大型网络的设计（例如电话系统中的网络）以及计算机科学中处理离散对象集的算法设计，都利用了组合研究。