Interfaces of Combinatorics and Physics

这个项目旨在解决组合学和物理学之间的几个问题。更具体地说，它将应用代数和组合工具来解决涉及排斥过程、麦克唐纳多项式、舒伯特簇结构、簇簇簇的镜像对称和振幅面体的问题。更具体地说，该项目涉及围绕开放边界晶格和环上的多物种不对称简单不相容过程（ASEP）的几个相互关联的问题。该组合模型与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。第二个方向是了解Grassmannian中Schubert变种的簇结构。这应该可以应用于格拉斯曼及其舒伯特变种的环退化和镜像对称。最后一个方向与振幅面体有关，它是物理学家在N=4超杨米尔斯理论中散射振幅的背景下引入的。据推测，振幅面体的体积可以计算一定的散射振幅，从而测量无质量粒子如何相互作用。理解了这个猜想，我们就会对正格拉斯曼方程有一个新的认识。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 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.