Combinatorics in Geometry and Physics

这个项目处于代数组合学和粒子物理学的交叉点。组合学是研究离散结构（如排列）及其分类和枚举的学科。代数是研究方程及其解的学科。代数组合试图将这两门学科结合起来，用组合学解决代数问题，反之亦然。在粒子物理学中，人们试图模拟和预测涉及基本粒子碰撞的实验结果。近年来，这种被称为“散射振幅”的计算与代数组合学中出现的被称为“正几何”的组合结构有关。粗略地说，正几何的组合控制着基本粒子的特殊位置。该项目旨在进一步提高我们对正几何的组合理解，并与物理学建立更牢固的关系，从而推动该领域的发展。本项目为研究生提供研究训练机会。这个项目是关于代数组合学的。PI旨在进一步研究这些空间上最近定义的正几何和积分函数。这些函数应用于散射振幅和镜像对称，并包括特殊函数，如β函数和贝塞尔函数。正几何的一个主要例子是全正空间，PI将研究这些空间的组合学和拓扑学。PI将进一步开发一个程序，将朗兰兹互易与法诺品种的镜像对称联系起来。最后，PI将研究簇变异的上同调，这在组合学（有理加泰罗尼亚数）和表示理论（Verma模的高级扩展）中有应用。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 lies at the interface of algebraic combinatorics and particle physics. Combinatorics is the study of discrete structures (such as permutations) and their classification and enumeration. Algebra is the study of equations and their solutions. Algebraic combinatorics seeks to combine the two subjects, solving algebraic problems using combinatorics, and vice versa. In particle physics, one tries to model and predict outcomes of experiments involving collisions of elementary particles. In recent years, such calculations, called "scattering amplitudes", have been related to combinatorial structures, called "positive geometries", appearing in algebraic combinatorics. Roughly speaking, the combinatorics of positive geometries controls special positions for elementary particles. This project aims to advance the field by further improving our combinatorial understanding of positive geometries, and establishing more robust relations with physics. The project provides research training opportunities for graduate students.The project is in algebraic combinatorics. The PI aims to further study the recently defined positive geometries and integral functions on these spaces. These functions have applications to scattering amplitudes and to mirror symmetry, and include special functions such as Beta functions and Bessel functions. One of the main examples of positive geometries are totally positive spaces and the PI will study the combinatorics and topology of these spaces. The PI will further develop a program relating Langlands reciprocity to mirror symmetry for Fano varieties. Finally, the PI will study the cohomology of cluster varieties, which has applications to combinatorics (rational Catalan numbers) and representation theory (higher extensions of Verma modules).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.