Collaborative Research: Catalan Function and Schubert Calculus

组合数学是一个数学领域，涉及分析，组织和优化离散数据。 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。 这个项目将开发用于攻击代数，几何和对称函数理论中的问题的组合方法，这是一个应用于概率和统计力学的领域。一个互利的组成部分是SAGE开源数学软件的进一步发展，这是一个重要的工具，这项调查。这个项目跨越组合问题的表示论，代数几何和物理。 灵感来自于结构，如tableaux和Bruhat偏序集，这是经典数学的核心，如舒伯特微积分和复杂线性群的表示理论。 在我们的调查的基本对象是舒尔函数和推广被称为k-舒尔函数，这是密切相关的（仿射）舒伯特演算。 这个项目将利用PI最近的工作，将k-Schur函数与某些Catalan对称函数连接起来，以解决涉及Macdonald多项式，抛物线Hall-Littlewood多项式，Gromov-Witten不变量，和co（同源性）和K-该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查评估的支持的搜索.

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.