Algebraic Combinatorics

该研究项目致力于探索组合学与代数之间的新关系，特别是表示理论和代数几何。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范围，从量子群的正则基组合到线性代数的数值和符号算法。量子上同调理论中出现的组合问题提供了另一个有前途的研究方向。该提案概述了一个详细研究量子舒伯特多项式的计划。本研究涉及组合学与代数几何之间的相互作用。组合学的目标之一是找到研究离散对象集合如何排列的有效方法。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。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。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅利用代数的方法，还利用分析、拓扑和组合学的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

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.