权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Combinatorics in Geometry and Physics

几何和物理中的组合数学

基本信息

批准号：
1953852
负责人：
Thomas Lam
金额：
$ 36万
依托单位：
Regents of the University of Michigan - Ann Arbor
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953852&HistoricalAwards=false
关键词：
Combinatorics Geometry Physics

项目摘要

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.

这个项目位于代数组合学和粒子物理学的接口。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。代数组合学试图将这两门学科联合收割机结合起来，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，这样的计算，称为“散射振幅”，已涉及到组合结构，称为“正几何”，出现在代数组合。粗略地说，正几何的组合学控制着基本粒子的特殊位置。该项目旨在通过进一步提高我们对正几何的组合理解，并与物理学建立更强大的关系来推进该领域。该项目为研究生提供研究培训机会。该项目是在代数组合学。PI旨在进一步研究最近定义的正几何和这些空间上的积分函数。这些功能有应用程序的散射振幅和镜像对称性，并包括特殊的功能，如贝塔功能和贝塞尔函数。正几何的一个主要例子是完全正空间，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，将朗兰兹互易性与Fano品种的镜像对称性联系起来。最后，PI将研究簇的上同调，这在组合学（有理Catalan数）和表示论（Verma模的更高扩展）中有应用。该奖项反映了NSF的法定使命，并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，被认为值得支持。