Combinatorics of Macdonald Polynomials and Schubert Calculus

组合学是纯数学和应用数学的一个活跃的中心分支。由于该领域涉及分析、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中都是必不可少的，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组合结果。这项研究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，这是一门应用于概率和统计力学的学科。这项研究将利用计算符号代数系统，并将进一步开发SAGE开源数学软件套件。这项研究项目涉及表示论、代数几何和物理学中的组合问题。其灵感来自于在复线性群表示和Schubert演算的经典研究中，Tableaux和Bruhat偏序集等结构的核心重要性。通过对称函数代数的一个独特的Schur基，组合学的精度被带到了更抽象的问题上。该项目涉及经典理论的变体，这些经典理论解决了几何Gromov-Witten不变量、参数域上的对称函数、双级表示和量子可积系统等微妙问题。当务之急是为当代的一系列问题开发必要的组合框架。该项目与对与组合学相关的精细物理、几何和代数理论的研究齐头并进。由此得到的组合结构将为相关应用领域提供计算便利。

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 essential 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 a combinatorial result in modular arithmetic. This research project is devoted to developing combinatorial techniques for attacking problems that connect to algebraic and geometric areas such as symmetric function theory, a subject with applications to probability and statistical mechanics. The investigation will make use of a computational symbolic algebra system and will further develop the SAGE open-source mathematics software suite. This research project spans combinatorial problems in representation theory, algebraic geometry, and physics. The inspiration comes from the central importance of constructions such as tableaux and Bruhat posets in the classical studies of representations of the complex linear group and Schubert calculus. The precision of combinatorics is carried to more abstract problems by a distinguished Schur basis for the algebra of symmetric functions. The project concerns variations on the classical theories that address subtle questions about geometric Gromov-Witten invariants, symmetric functions over a field of parameters, bi-graded representations, and quantum integrable systems. A priority is to develop the necessary combinatorial framework for the contemporary set of problems. The venture goes hand in hand with an investigation of refined physical, geometric, and algebraic theories attached to the combinatorics. The resulting combinatorial constructions will lend computational facility to related application areas.