Combinatorics in Algebra, Geometry, and Physics

这个项目致力于代数和几何组合中的几个问题。这项研究的一个关键主题是一个被称为格拉斯曼年的几何对象，它出现在许多其他数学主题的研究中，但也出现在物理学中，在那里它与某些量子场论有关。尽管它是技术性的，但项目中探索的一些问题适合本科生和高中生，研究者将通过各种项目与这些学生一起工作，例如他所在机构的数学、工程和科学研究项目（PRIMES）和本科生研究机会项目（UROP）。该项目解决了关于正面格拉斯曼尼亚的几个问题，格拉斯曼尼亚是一个美丽的几何物体，具有丰富的组合结构。它与矩阵理论、舒伯特微积分、对称函数、簇代数以及基本粒子散射振幅的研究密切相关。几个子项目涉及弱分离集合和纯度现象向取向拟阵的扩展。这一领域的研究涉及到分区平铺和簇代数的组合学研究。该项目探讨了有关根系芯片发射游戏的问题。这一领域有许多引人入胜的猜想和尚未解决的问题。广义复面体是一类凸多面体，其体积和整数格点的个数产生许多经典组合序列。广义复面体的研究涉及到超图的Tutte多项式的一个新的模拟和一个显著的对偶性，它与结理论和热带几何有联系。该项目还包括关于陈氏形式的代数和相关权力理想的问题。这些对象来自交换代数，涉及图、森林、超平面排列、停车函数和拟阵的组合。在这项工作中研究的许多结构都与凸多面体和其他类似多面体的结构有关，如复面体、幅面体、宇宙多面体和根多面体。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。

This project is dedicated to several questions in algebraic and geometric combinatorics. A key topic of the research is a geometric object, called the Grassmannian, that appears in the study of a wide variety of other mathematical topics, but also in physics, where it is related to certain quantum field theories. Despite its technical nature, some questions explored in the project are suitable for undergraduate and high school students, and the investigator will work with these students through the various programs, such as the Program for Research in Mathematics, Engineering, and Science for High School Students (PRIMES) and the Undergraduate Research Opportunities Program (UROP), available at his institution.The project addresses several questions about the positive Grassmannian, a beautiful geometrical object with a rich combinatorial structure. It is closely related to matroid theory, Schubert calculus, symmetric functions, and cluster algebras, as well as the study of scattering amplitudes of elementary particles. Several sub-projects are concerned with an extension of weakly separated collections and purity phenomenon to oriented matroids. This area of research is related to the study of combinatorics of zonotopal tilings and cluster algebras. The project explores questions about the chip-firing game for root systems. There are many fascinating conjectures and open problems in this area. Several questions concern generalized permutohedra, which are certain convex polytopes whose volumes and numbers of integer lattice points produce many classical combinatorial sequences. The research on generalized permutohedra involves a new analogue of the Tutte polynomial for hypergraphs and a remarkable duality, with links to the theory of knots and tropical geometry. The project also includes questions regarding the algebra of Chern forms and related power ideals. These are objects from commutative algebra involving combinatorics of graphs, forests, hyperplane arrangements, parking functions, and matroids. Many constructions that will be investigated in this work are related to convex polytopes and other polytope-like structures, such as permutohedra, amplituhedra, cosmological polytopes, and root polytopes.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.