Combinatorics in algebra, geometry, and physics

该建议涉及表示论、拓扑学、代数几何和物理学中的组合问题。通过A类仿射Weyl群上的弱和强(Bruhat)阶的连接，画出了一条公共的线。潜在的想法是，有一般的方法来解决在这种仿射环境中出现的各种问题。灵感来自Tableaux、Bruhat偏序集和水晶基在对称函数、舒伯特微积分和量子代数等理论中所扮演的角色。其目的是建立一个以同样的方式支持新理论的组合骨干。重点介绍了Gromov-Witten不变量、共形场理论、Schubert微积分和Macdonald多项式的应用。这些应用中的每一个都连接到k-Schur函数，这将是一个指导灯。组合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、组织和排列离散数据的工具的开发，因此组合方法在许多科学领域都是基本的，如基因组学、计算机科学、统计学和物理学。这些方法通常可以追溯到受到代数和几何问题启发的研究。例如，RSA公钥加密算法基于模算术中的基本组合结果。这个项目致力于开发组合技术来攻击与代数和几何领域有关的问题，例如对称函数理论，这是一个应用于概率和统计力学的领域。一个互惠互利的部分是SAGE开源数学软件的进一步开发，因为计算机代数系统的使用对这项研究至关重要。

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.