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（如基因组学、计算机科学、统计学和物理学）都是必不可少的。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组合结果。该研究项目致力于开发组合技术，以解决与代数和几何领域相关的问题，例如对称函数理论，这是一个应用于概率论和统计力学的主题。该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。这个研究项目涵盖了表示理论、代数几何和物理中的组合问题。灵感来自于复杂线性群表示和舒伯特微积分的经典研究中，诸如tableaux和Bruhat poset等结构的核心重要性。通过对称函数代数的独特的舒尔基，组合学的精度被用于更抽象的问题。该项目涉及经典理论的变化，解决几何Gromov-Witten不变量、参数域上的对称函数、双梯度表示和量子可积系统等微妙问题。一个优先事项是为当前的一系列问题发展必要的组合框架。这一冒险与对精细的物理、几何和代数理论的研究密切相关，这些理论与组合学密切相关。由此产生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.