Combinatorics in algebra, geometry, and physics

这个建议跨越了表示论、拓扑学、代数几何和物理学中的组合问题。一个共同的线索是由每个连接到弱和强（Bruhat）秩序上的A型仿射外尔群。 其基本思想是，有一般的方法来攻击各种各样的问题，出现在这个仿射设置。 灵感来自于tableaux，Bruhat偏序集和晶体基在对称函数，舒伯特微积分和量子代数等理论中的作用。 其目的是建立一个组合的骨干，以同样的方式支持新的理论。 重点是应用Gromov-Witten不变量，共形场论，舒伯特微积分和麦克唐纳多项式。 一个指路明灯将是这些应用程序中的每一个连接到k-Schur函数。 组合数学是纯数学和应用数学的一个活跃的中心分支。 由于该领域关注的是分析，组织和安排离散数据的工具的开发，组合方法在许多科学领域，如基因组学，计算机科学，统计学和物理学中是基本的。 这些方法通常可以追溯到受代数和几何问题启发的研究。 例如，RSA公钥加密算法基于模运算中的初等组合结果。 该项目致力于开发组合技术，用于解决与代数和几何领域相关的问题，如对称函数理论，该领域应用于概率和统计力学。 一个互利的组成部分是进一步开发SAGE开放源码数学软件，因为使用计算机代数系统对这项调查至关重要。

This proposal spans combinatorial problems in representation theory, topology, algebraic geometry, and physics. A common thread is drawn by the connection of each to the weak and strong (Bruhat) orders on the type-A affine Weyl group. The underlying idea is that there are general methods for attacking the wide variety of problems that arise in this affine setting. Inspiration comes from the role that tableaux, Bruhat posets, and crystal bases play in theories such as symmetric functions, Schubert calculus and quantum algebras. The aim is to establish a combinatorial backbone that supports new theories in the same way. An emphasis is on applications to Gromov-Witten invariants, conformal field theories, Schubert calculus, and Macdonald polynomials. A guiding light will be the connection of each of these applications to the k-Schur functions. Combinatorics is an active and central branch of pure and applied mathematics. Because the field is concerned with the development of tools for analyzing, organizing, and arranging discrete data, combinatorial methods are rudimentary in many scientific areas such as genomics, computer science, statistics, and physics. The methods can often be traced back to research inspired by problems in algebra and geometry. For example, the RSA public-key encryption algorithm is based on an elementary combinatorial result in modular arithmetic. This project is devoted to developing combinatorial techniques for attacking problems that connect to algebraic and geometric areas such as symmetric function theory, an area with applications to probability and statistical mechanics. A mutually beneficial component is the further development of the SAGE open-source mathematics software, as the use of a computer algebra system is crucial for this investigation.