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

Representation Theory and Moduli Spaces via Young Tableaux and Parking Functions

通过 Young Tableaux 和停车函数的表示理论和模空间

基本信息

批准号：
2054391
负责人：
Maria Gillespie
金额：
$ 19.19万
依托单位：
Colorado State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-15 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054391&HistoricalAwards=false
关键词：
Representation Theory Moduli Spaces via

项目摘要

This research focuses on the use of combinatorial tools, such as orderings of numbers or algorithms for parking cars, to provide a more concrete understanding of abstract mathematical concepts in geometry and algebra. In modern-day Schubert calculus, many questions in enumerative geometry - counting intersections of lines, planes, curves - have been translated into discrete combinatorial problems or algorithms that a computer can then analyze. The aim of this project is to come up with these types of combinatorial rules in related areas of geometry and algebra, in order to both increase computational efficiency and to make the geometric constructions more accessible to scientists in other disciplines. The geometric spaces and algebraic structures that will be studied are of central importance to quantum physics and string theory. The project will involve graduate students in the research.This work will particularly focus on subvarieties and generalizations of flag varieties, Grassmannians, and moduli spaces of curves, all three of which are important geometric spaces whose cohomology rings are graded S_n-modules. We aim to give combinatorial rules, in terms of Young tableaux, parking functions, and other combinatorial objects, that govern computational aspects of their cohomology rings, and use them to resolve open questions about the corresponding geometric spaces. These new combinatorial rules also will be used to approach long-standing open problems in symmetric function theory, including the Macdonald positivity conjecture and the problem of determining equality of skew Schur Q functions.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.

本研究的重点是使用组合工具，如排序的数字或算法的停车汽车，以提供一个更具体的理解抽象的数学概念的几何和代数。在现代舒伯特微积分中，许多枚举几何中的问题--计算直线、平面、曲线的交点--已经被转化为离散的组合问题或算法，然后计算机可以分析。该项目的目的是在几何和代数的相关领域提出这些类型的组合规则，以提高计算效率，并使其他学科的科学家更容易获得几何结构。几何空间和代数结构，将被研究是至关重要的量子物理学和弦理论。本项目将邀请研究生参与研究，重点研究曲线的旗簇、Grassmannian和模空间的子簇和推广，这三个空间都是重要的几何空间，它们的上同调环都是分次S_n-模。我们的目标是组合规则，在杨tableaux，停车功能，和其他组合对象，管理计算方面的上同调环，并使用它们来解决相应的几何空间的开放问题。这些新的组合规则也将用于解决对称函数理论中长期存在的问题，包括麦克唐纳正性猜想和确定斜Schur Q函数的平等性问题。该奖项反映了NSF的法定使命，并被认为值得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来支持。