Combinatorics and its Applications

组合学是研究离散结构的数学领域。这一领域与纯数学和应用数学的许多其他领域以及物理和生物学等其他科学有着密切的联系。这个项目致力于组合数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。结果将在该领域的专家以及更广泛的受众中传播。一些问题的特殊案例将用于本科和高中的研究项目。研究生将参与这项研究。这个项目将促进公众对数学的了解和欣赏。这个项目中的许多问题都围绕着特殊类别的多面体和其他类似多面体的结构，如置换多面体、根多面体、正多面体和正Grassmanian。关于广义置换面体的几个问题。这些问题涉及将Tutte多项式推广到超图和多面体，以及广义置换面体的镜像对偶。这些问题与低维拓扑，特别是纽结不变量，与热带几何和环面几何有联系。几个问题与正Grassman有关，它是一个美丽的几何对象，具有丰富的组合结构。在研究正Grassman过程中发展起来的组合结构和技术已经出现在许多其他领域：反边界问题、拟阵理论、凸几何、环面几何、统计力学、孤子理论、Fomin-Zlevinsky簇代数、对称函数、仿射Schubert演算、Lusztig正则基、矩阵补全问题、Schur正性问题，以及基本粒子散射幅度的研究。本项目将研究正Grassmanian与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净现象、根系的烧片游戏、陈形式的代数和功率理想。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.