Algebraic Combinatorics

福明9700927 该研究项目致力于探索组合学和代数之间的新关系，特别是表示论和代数几何。 一种基于广义伪线排列的全正性方法提出了新的发展和应用，从量子群中的正则基的组合数学到线性代数的数值和符号算法。 量子上同调理论中的组合问题提供了另一个有前途的研究方向。该提案概述了详细研究量子舒伯特多项式的计划。 本研究关注组合学和代数几何之间的相互作用。组合数学的目标之一是找到有效的方法来研究如何安排离散的对象集合。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，如电话系统中的网络设计，以及计算机科学中的算法设计，都要处理离散的对象集，这就需要使用组合研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。如今，该领域不仅使用代数方法，而且还使用分析，拓扑学和组合学方法，相反，这些方法在这些领域以及物理学，理论计算机科学和机器人学中得到应用。

Fomin 9700927 The research project is devoted to exploring new relations between combinatorics and algebra, in particular representation theory and algebraic geometry. An approach to total positivity based on generalized pseudo-line arrangements suggests new developments and applications ranging from combinatorics of canonical bases in quantum groups to numerical and symbolic algorithms of linear algebra. The combinatorial problems arising in the theory of quantum cohomology provide another promising direction of research. The proposal outlines a plan for a detailed study of quantum Schubert polynomials. This research concerns the interplay between combinatorics and algebraic geometry. One of the goals of combinatorics is to find efficient methods of studying 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. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis, topology, and combinatorics, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.