Geometry and Asymptotics of Schubert Polynomials, Graph Colorings, and Flows on Graphs

舒伯特多项式的几何和渐近、图着色和图流

• 批准号: 2154019
• 负责人: Alejandro Morales
• 金额: $ 20.56万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154019&HistoricalAwards=false
• 关键词: Geometry; Asymptotics; Schubert; Polynomials; Graph

Three fundamental and challenging problems in mathematics inspired from real-world activities are to count the number of ways of transporting goods through a network, sorting a list of tasks, and scheduling jobs to time slots. Some instances of these counting problems are difficult to count exactly and one can instead study bounds, asymptotics, and large-scale behaviors of these numbers as well as using geometric structures encoding the objects that are counted. Abstractly these problems can be studied with the following mathematical objects: "integer flows on a graph", "Schubert polynomials of permutations", and "graph vertex colorings", respectively that have connections to other fields of mathematics and other areas like computer science, and physics.More concretely, this project studies problems in enumerative, algebraic, and asymptotic combinatorics with connections to representation theory and geometry. The project has three parts. The first part is about finding a ¨q-analogue¨ of a constant term identity of Zeilberger to compute volumes of flow polytopes and establishing a connection between this identity and the famous Selberg integral. The second part is about studying the large-scale behavior of Schubert and Grothendieck polynomials using existing combinatorial models like rc-graphs, bumpless pipe dreams, and excited diagrams. The last part is about studying chromatic symmetric functions of Dyck paths, which are the object of the famous Stanley--Stembridge--Shareshian--Wachs conjecture. The PI will study the Newton polytope, Lorentzian property, and connections to q-rook theory of these symmetric functions. Students will be trained during the course of this project.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.

从现实世界的活动中得到启发，数学中有三个基本且具有挑战性的问题：计算通过网络运输货物的方式的数量，对任务列表进行排序，以及将工作安排到时间段。这些计数问题的一些实例很难精确计数，人们可以研究这些数字的边界、渐近性和大规模行为，以及使用几何结构编码被计数的对象。抽象地说，这些问题可以用以下数学对象来研究：“图上的整数流”、“排列的舒伯特多项式”和“图顶点着色”，它们分别与其他数学领域和计算机科学、物理学等其他领域有联系。更具体地说，这个项目研究与表示理论和几何有关的枚举、代数和渐近组合学中的问题。这个项目有三个部分。第一部分是寻找Zeilberger常数项恒等式的“q-模拟”来计算流多面体的体积，并建立该恒等式与著名的Selberg积分之间的联系。第二部分是利用现有的组合模型，如rc图、无碰撞白日梦和激发态图，研究Schubert多项式和Grothendieck多项式的大尺度行为。最后是对著名的Stanley- Stembridge- Shareshian- Wachs猜想的对象Dyck路径的色对称函数的研究。PI将研究牛顿多面体，洛伦兹性质，以及与这些对称函数的q-rook理论的联系。学生将在这个项目的过程中接受培训。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Generalized Pitman–Stanley flow polytopes

广义 Pitman—Stanley 流多面体

• 发表时间: 2023
• 期刊: Séminaire lotharingien de combinatoire
• 作者: Dugan, William T.;Hegarty, Maura;Morales, Alejandro H.;Raymond, Annie
• 通讯作者: Raymond, Annie

Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson

组合和代数枚举：Ian P. Goulden 和 David M. Jackson 工作综述

• DOI: 10.5802/alco.269
• 发表时间: 2022
• 期刊: Algebraic Combinatorics
• 作者: Foley, Angèle M.;Morales, Alejandro H.;Rattan, Amarpreet;Yeats, Karen
• 通讯作者: Yeats, Karen

Minimal skew semistandard Young tableaux and the Hillman–Grassl correspondence

最小倾斜半标准 Young 画面和 HillmanâGrassl 对应

• 发表时间: 2023
• 期刊: Séminaire lotharingien de combinatoire
• 作者: Morales, Alejandro H.;Panova, Greta;Park, GaYee
• 通讯作者: Park, GaYee