Algebraic Combinatorics

首席研究人员正在进行研究，将组合数学与代数、表示理论和代数几何的不同主题联系起来。具体内容包括：全正性理论；半单李代数表示论的组合方面；Coxeter/Weyl群的组合学；以及Schubert演算。此外，提出者正在开发自由软件来协助这些领域的研究。这项研究是在组合学的一般领域，即离散结构的研究。组合学的目标之一是找到有效的方法来操作和枚举离散的对象集合。离散系统的行为对现代通信极其重要，是计算机科学数学基础的重要组成部分。组合技术在代数、几何、概率和数学物理等较老的数学分支中也越来越有价值，因为显式计算通常需要更好地理解基本的离散结构。

The Principal Investigators are conducting research connecting combinatorics with various topics in algebra, representation theory, and algebraic geometry. The specific areas include: the theory of total positivity; combinatorial aspects of representation theory of semisimple Lie algebras; combinatorics of Coxeter/Weyl groups; and Schubert calculus. In addition, the proposers are developing free software to assist research in these areas.This research is in the general area of combinatorics, the study of discrete structures. One of the goals of combinatorics is to find efficient methods for the manipulation and enumeration of discrete collections of objects. The behavior of discrete systems is extremely important to modern communications, and is an essential part of the mathematical foundations of computer science. Combinatorial techniques are also of increasing value in older branches of mathematics such as algebra, geometry, probability and mathematical physics, since explicit computations often require a better understanding of the underlying discrete structures.