Interactions of Combinatorics and Physics

组合学和物理学的相互作用

• 批准号: 2152991
• 负责人: Lauren Williams
• 金额: $ 44万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2152991&HistoricalAwards=false
• 关键词: Interactions; Combinatorics; Physics

In recent years there have been a number of extremely fruitful interactions between combinatorics and physics. This project aims to use combinatorial techniques to address questions from various fields of physics, including statistical physics (particle processes including the asymmetric simple exclusion process), quantum field theory (scattering amplitudes in N=4 super Yang Mills theory), and mirror symmetry. The mathematical topics and tools involved include special functions, the amplituhedron, cluster algebras, and cluster varieties. Graduate students will be trained as part of this project.More concretely, this project concerns the relationship between the asymmetric simple exclusion process (ASEP) and important families of polynomials in algebraic combinatorics, including Schubert polynomials, Macdonald polynomials, Askey-Wilson polynomials, and Koornwinder polynomials. It seeks to use the structure of the ASEP to obtain insight and new formulas for these polynomials. A second direction is to study the combinatorics of the amplituhedron, a generalization of the positive Grassmannian, and in particular the cluster adjacency conjecture, and an enumerative conjecture linking its tilings to plane partitions. A final direction is to use tools from cluster algebras to understand mirror symmetry for flag varieties and related spaces.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.

近年来，组合学和物理学之间出现了许多卓有成效的相互作用。这个项目旨在使用组合技术来解决来自不同物理领域的问题，包括统计物理(包括非对称简单排斥过程的粒子过程)、量子场论(N=4个超杨-米尔斯理论中的散射幅度)和镜像对称性。所涉及的数学主题和工具包括特殊函数、幅值多面体、簇代数和簇变元。作为本项目的一部分，将对研究生进行培训。具体地说，本项目研究了非对称简单排除过程(ASEP)与代数组合数学中重要的多项式族之间的关系，包括Schubert多项式、Macdonald多项式、Askey-Wilson多项式和Koornwinder多项式。它试图利用ASEP的结构来获得这些多项式的洞察力和新的公式。第二个方向是研究振幅多面体的组合学，正Grassman猜想的推广，特别是簇邻接猜想，以及将其拼接与平面划分联系起来的计数猜想。最后一个方向是使用集群代数中的工具来了解旗帜品种和相关空间的镜像对称性。这一奖项反映了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。