Interfaces of Combinatorics and Physics

这个项目旨在解决组合学和物理学交界处的几个问题。更具体地说，它将应用代数和组合工具来解决排除过程、Macdonald多项式、Schubert簇的簇结构、簇簇的镜像对称性和幅值等问题。更具体地说，这个项目涉及多物种非对称简单排除过程(ASEP)的几个相互关联的问题，这些问题围绕着具有开放边界的格子和环上的多物种非对称简单排除过程。这个组合模型与Koornwinder多项式和Macdonald多项式密切相关；对ASEP的更好的理解应该会导致Koornwinder和Macdonald多项式的组合公式。第二个方向是了解Grassmanian中Schubert变种的簇结构。这应该适用于Grassmanian及其Schubert簇上的环形退化和镜像对称性。最后一个方向与振幅多面体有关，它是由物理学家在N=4超杨Mills理论的散射振幅的背景下引入的。据推测，振幅多面体的体积计算特定的散射振幅，这衡量了无质量粒子如何相互作用。理解这一猜想应该会带来关于积极格拉斯曼尼亚的新见解。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。

This project aims to address several questions at the interface of combinatorics and physics. More specifically, it will apply algebraic and combinatorial tools to questions involving the exclusion process, Macdonald polynomials, cluster structures in Schubert varieties, mirror symmetry in cluster varieties, and the amplituhedron.More concretely, this project concerns several interrelated questions surrounding the multispecies asymmetric simple exclusion process (ASEP) on both a lattice with open boundaries, and on a ring. This combinatorial model is intimately connected to both Koornwinder polynomials and Macdonald polynomials; and a better understanding of the ASEP should lead to combinatorial formulas for Koornwinder and Macdonald polynomials. A second direction is to understand the cluster structure on Schubert varieties in the Grassmannian. This should have applications to toric degenerations and mirror symmetry on the Grassmannian and its Schubert varieties. A final direction concerns the amplituhedron, which was introduced by physicists in the context of scattering amplitudes in N=4 super Yang Mills theory. Conjecturally, the volume of the amplituhedron computes certain scattering amplitudes, which measure how massless particles interact. Understanding this conjecture should lead to a new insights concerning the positive Grassmannian.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.