Combinatorics and its Applications

组合学及其应用

• 批准号: 2054129
• 负责人: Alexander Postnikov
• 金额: $ 30万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054129&HistoricalAwards=false
• 关键词: Combinatorics; its; Applications

Combinatorics is the area of mathematics that studies discrete structures. This area has close links with many other fields of pure and applied mathematics and with other sciences such as physics and biology. This project is dedicated to several problems in combinatorics and their applications to algebra, geometry, and theoretical physics. The project will lead to a better understanding of fundamental mathematical concepts and constructions and will have an impact in many other areas of research. The results will be disseminated among specialists in the field as well as to broader audiences. Special cases of some problems will be used for undergraduate and high school research projects. Graduate students will be involved in the research. This project will promote the general public knowledge and appreciation of mathematics.Many problems in this project revolve around special classes of polytopes and other polytope-like structures, such as permutohedra, root polytopes, positroids, and the positive Grassmannian. Several problems are concerned with generalized permutohedra. These problems involve an extension of the Tutte polynomial to hypergraphs and polymatroids, as well as mirror duality for generalized permutohedra. These problems have links with low-dimensional topology, specifically knot invariants, with tropical geometry, and with toric geometry. Several problems are related to the positive Grassmannian, which is a beautiful geometrical object with rich combinatorial structure. The combinatorial constructions and techniques developed in the study of the positive Grassmannian have surfaced in many other areas: inverse boundary problems, matroid theory, convex geometry, toric geometry, statistical mechanics, the theory of solitons, Fomin-Zelevinsky's cluster algebras, symmetric functions, affine Schubert calculus, Lusztig's canonical bases, matrix completion problems, and Schur positivity problems, as well as the study of scattering amplitudes of elementary particles. This project will study new links between the positive Grassmannian and geometry of polyhedral subdivisions. The project also includes problems on the purity phenomenon for oriented matroids, chip-firing games for root systems, algebras of Chern forms, and power ideals.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.

组合学是研究离散结构的数学领域。这个领域与纯数学和应用数学的许多其他领域以及物理学和生物学等其他科学有着密切的联系。本项目致力于研究组合学中的几个问题及其在代数、几何和理论物理中的应用。该项目将有助于更好地理解基本的数学概念和结构，并将对许多其他研究领域产生影响。研究结果将在该领域的专家之间以及向更广泛的受众传播。一些问题的特殊案例将用于本科和高中的研究项目。研究生将参与研究。这个项目将促进大众对数学的认识和欣赏。这个项目中的许多问题围绕着特殊类型的多面体和其他类似多面体的结构，如复面体、根多面体、正极体和正极格拉斯曼体。讨论了广义复面体的几个问题。这些问题涉及到Tutte多项式对超图和多拟阵的推广，以及广义复面体的镜像对偶性。这些问题与低维拓扑，特别是结不变量，热带几何和环面几何有联系。正格拉斯曼方程是一种具有丰富组合结构的美丽几何对象。研究正格拉斯曼问题的组合结构和技术已经出现在许多其他领域：逆边界问题、类阵理论、凸几何、环几何、统计力学、孤子理论、ffomin - zelevinsky的簇代数、对称函数、仿射舒伯特演算、Lusztig的正则基、矩阵补全问题、舒尔正性问题，以及基本粒子散射振幅的研究。这个项目将研究正格拉斯曼和多面体细分几何之间的新联系。该项目还包括定向拟阵的纯度现象，根系统的芯片发射游戏，陈氏形式代数和幂理想的问题。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

The maximum multiplicity of a generator in a reduced word

简化字中生成器的最大重数

• 发表时间: 2021
• 期刊: Seminaire lotharingien de combinatoire
• 作者: Gaetz, Christian;Gao, Yibo;Jiradilok, Pakawut;Nenashev, Gleb;Postnikov, Alexander
• 通讯作者: Postnikov, Alexander

Universal Tutte polynomial

通用塔特多项式

• DOI: 10.1016/j.aim.2022.108355
• 发表时间: 2022
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Bernardi, Olivier;Kálmán, Tamás;Postnikov, Alexander
• 通讯作者: Postnikov, Alexander