Collaborative Research: Catalan Function and Schubert Calculus

组合学是一个涉及分析、组织和优化离散数据的数学领域。它是许多科学领域的基本工具，如基因组学、计算机科学、统计学和物理学。这个项目将开发组合方法来解决代数、几何和对称函数理论中的问题，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，这是这项研究的关键工具。这个项目涵盖了表示论、代数几何和物理中的组合问题。灵感来自于位于经典数学核心的Tableaux和Bruhat偏序集等构造，如舒伯特微积分和复线性群的表示理论。我们研究的基本对象是Schur函数和称为k-Schur函数的推广，它们与(仿射)Schubert演算密切相关。这个项目将利用PIS最近的工作，将k-Schur函数与某些加泰罗尼亚对称函数联系起来，以解决涉及Macdonald多项式、抛物型Hall-Littlewood多项式、Gromov-Witten不变量以及仿射Grassmannians的上(同调)和K-理论的问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Combinatorics is an area of mathematics concerned with analyzing, organizing, and optimizing over discrete data. It is a fundamental tool in many scientific areas such as genomics, computer science, statistics, and physics. This project will develop combinatorial methods for attacking problems in algebra, geometry, and 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, a crucial tool for this investigation.This project spans combinatorial problems in representation theory, algebraic geometry, and physics. The inspiration comes from constructions such as tableaux and Bruhat posets which lie at the heart of classical mathematics such as Schubert calculus and the representation theory of the complex linear group. Fundamental objects in our investigations are the Schur functions and generalizations known as k-Schur functions, which are closely tied to (affine) Schubert calculus. This project will capitalize on PIs' recent work connecting k-Schur functions with certain Catalan symmetric functions to solve problems involving Macdonald polynomials, parabolic Hall-Littlewood polynomials, Gromov-Witten invariants, and the co(homology) and K-theory of affine Grassmannians.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.