Research in Algebraic Combinatorics

代数组合学研究

• 批准号: 2207337
• 负责人: Michelle Wachs
• 金额: $ 21万
• 依托单位: University of Miami
• 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: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2207337&HistoricalAwards=false
• 关键词: Research; Algebraic; Combinatorics

The research supported by this grant is in algebraic combinatorics, an area of mathematics that seeks to develop connections between combinatorics (the science of counting, arranging and analyzing concrete discrete configurations) and fields of pure mathematics that involve sophisticated abstract algebraic structures. The idea is to use these connections to gain deeper insights and solve problems in combinatorics and in the other fields. The discrete configurations that are studied in combinatorics arise in various fields of mathematics, computer science, physics, biology and engineering; DNA sequences, phylogenetic trees, and communications networks are all examples of such discrete configurations. Combinatorial methods are playing an increasing role in these fields. The proposed research is comprised of three related projects involving generalizations and variations of classical combinatorial objects such as chromatic polynomials, Eulerian polynomials, partition lattices, and free Lie algebras. The first project deals with a refinement of Stanley's chromatic symmetric functions that was introduced in work of John Shareshian and the PI. The refinement, the chromatic quasisymmetric functions, generalizes Eulerian polynomials as well as chromatic polynomials. An algebro-geometric approach to settling the longstanding Stanley-Stembridge e-positivity conjecture, involving a connection between the chromatic quasisymmetric functions and the Hessenberg varieties of De Mari, Procesi, and Shayman, was presented in this work. This approach initiated significant interaction between combinatorialists working with symmetric functions and algebraic geometers working on Hessenberg varieties. Further study of the chromatic quasisymmetric functions, as well as its connection with Hessenberg varieties is proposed. The second project involves a new polynomial graph invariant and a more general symmetric function graph invariant introduced by the PI and her former student Rafael Gonzalez D'Leon. These graph invariants generalize variations of Eulerian polynomials such as the classical Narayana polynomials and Haiman's parking function symmetric functions. This project is focused on settling unimodality conjectures, e-positivity conjectures and other concrete conjectures for these graph invariants. The third project arose in theoretical physics. It deals with an n-ary generalization of a Lie algebra known as a Filippov algebra. The study of the representation of the symmetric group on the multilinear component of the free Filippov algebra was initiated in work of Friedmann, Hanlon, Stanley, and the PI. Although this work has already produced some significant results, there is much that remains to be done.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.

该补助金支持的研究是代数组合学，这是一个数学领域，旨在发展组合学（计数，安排和分析具体离散配置的科学）与涉及复杂抽象代数结构的纯数学领域之间的联系。我们的想法是利用这些联系来获得更深入的见解，并解决组合数学和其他领域的问题。在组合学中研究的离散构型出现在数学、计算机科学、物理学、生物学和工程学的各个领域; DNA序列、系统发育树和通信网络都是这种离散构型的例子。组合方法在这些领域中发挥着越来越重要的作用。拟议的研究是由三个相关的项目，涉及的经典组合对象，如色多项式，欧拉多项式，分区格和自由李代数的推广和变化。第一个项目涉及改进斯坦利的色对称函数，介绍了工作的约翰Shareshian和PI。色拟对称函数的精化推广了欧拉多项式和色多项式。 一个代数几何的方法来解决长期存在的斯坦利-Stembridge e-阳性猜想，涉及色拟对称函数和赫森伯格品种的德马里，Procesi，和Shayman之间的连接，在这项工作中提出。 这种方法开始了显着的相互作用combinatorialists工作与对称函数和代数geometers工作的海森堡品种。 进一步研究了色拟对称函数及其与Hessenberg簇的关系。第二个项目涉及一个新的多项式图不变量和一个更一般的对称函数图不变量介绍了PI和她以前的学生拉斐尔冈萨雷斯德莱昂。 这些图不变量推广了欧拉多项式的变化，如经典的纳拉亚纳多项式和海曼的停车函数对称函数。这个项目的重点是解决这些图不变量的单峰性、正性和其他具体的不变量。第三个项目产生于理论物理学。它涉及一个n元推广的李代数称为菲利普代数。自由Filippov代数的多线性分量上的对称群表示的研究始于Friedmann、Hanlon、Stanley和PI的工作。 虽然这项工作已经产生了一些显著的成果，还有很多工作要做。这个奖项反映了NSF的法定使命，并已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标准进行评估的支持。