Interactions of Combinatorics and Physics

近年来，组合数学和物理学之间有许多非常富有成效的相互作用。 该项目旨在使用组合技术来解决物理学各个领域的问题，包括统计物理学（包括不对称简单排斥过程的粒子过程），量子场论（N=4超级杨米尔斯理论中的散射振幅）和镜像对称性。 涉及的数学主题和工具包括特殊函数、振幅面体、簇代数和簇变种。研究生将作为该项目的一部分进行培训。更具体地说，该项目涉及非对称简单排除过程（ASEP）与代数组合学中重要多项式族之间的关系，包括舒伯特多项式，麦克唐纳多项式，Askey-Wilson多项式和Koornwinder多项式。 它试图使用ASEP的结构来获得这些多项式的洞察力和新公式。 第二个方向是研究组合学的amplituhedron，一个推广的积极格拉斯曼，特别是集群邻接猜想，并枚举猜想连接其tilings平面分区。最后一个方向是使用集群代数的工具来理解镜像对称的旗帜品种和相关的spaces.This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。

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.