Interfaces of Combinatorics and Physics

这个项目旨在解决组合学和物理学接口的几个问题。 更具体地说，它将应用代数和组合工具的问题，涉及排斥过程，麦克唐纳多项式，簇结构的舒伯特品种，镜像对称的簇品种，和amplituhedron.More具体地说，这个项目涉及到几个相互关联的问题，围绕多物种的不对称简单排斥过程（ASEP）的两个开放边界的格子，并在一个环。 这种组合模型与Koornwinder多项式和Macdonald多项式密切相关;更好地理解ASEP应该会导致Koornwinder和Macdonald多项式的组合公式。 第二个方向是理解Grassmannian中Schubert簇的簇结构。 这应该有应用程序的环面退化和镜像对称的格拉斯曼及其舒伯特品种。最后一个方向涉及振幅面，这是由物理学家在N=4的超级杨米尔斯理论中的散射振幅的背景下引入的。从理论上讲，振幅面体的体积计算了某些散射振幅，这些振幅测量了无质量粒子如何相互作用。 理解这一猜想应该导致一个新的见解有关的积极格拉斯曼。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.