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

该奖项为一个项目提供资金，该项目将主要在以下两个领域进行：(a)多元判别、结果和超决定因素。(B)量子群的表示和分段线性组合。这两门学科都与多元超几何函数理论有着深刻的联系。另一个统一的特征是，在这两个领域中，新的组合学发挥了重要作用，其主要研究对象是凸锥和多面体中的点阵点及其分段线性变换。主要研究方向为：(1)多元判别式和结果式的显式多项式公式，(2)判别型的奇异座，(3)量子群表示中的正则基的代数和组合性质，(4)全正矩阵的几何，(5)若干向量变量中0环和对称函数的Chow变，(6)次多交体和极大次多交体。这项研究探索了组合学、代数和几何之间的联系。组合学的目标之一是寻找有效的方法来研究如何排列离散的对象集合。离散系统的行为对现代通信极为重要。例如，大型网络的设计，比如那些出现在电话系统中的网络，以及计算机科学中处理离散对象集的算法设计，这就利用了组合研究。

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.