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Applications of Lie Theory: Combinatorial Algebraic Geometry and Symmetric Functions

李理论的应用：组合代数几何和对称函数

基本信息

批准号：
1954001
负责人：
Martha Precup
金额：
$ 19.99万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954001&HistoricalAwards=false
关键词：
Applications Lie Theory Combinatorial Algebraic

项目摘要

Algebraic combinatorics is an area of research that seeks to build connections between discrete structures and algebraic objects, with broad applications in computing, statistics, biology, and other subjects of mathematics. A central theme in combinatorics problems is to organize discrete data in a way that reflects key structural properties, thereby making it easier to analyze. This project applies the tools of algebraic combinatorics to study solutions of complicated systems of equations, called algebraic varieties. The PI will develop sophisticated counting techniques to streamline computations and decipher patterns in otherwise complex data. The PI then will use geometric properties of algebraic varieties to uncover new approaches to unsolved problems in algebra and combinatorics. In addition this project also provides research training opportunities for graduate students.The specific research addressed in this project concerns the combinatorial and geometric structure of Hessenberg varieties and extended Springer fibers. Hessenberg varieties are subvarieties of the flag variety whose cohomology rings encode rich algebraic structure. The PI will use topological data obtained from Hessenberg varieties to outline a new approach to the long-standing Stanley-Stembridge conjecture in combinatorics. The geometry and topology of Hessenberg varieties is completely understood in only a few cases. Using combinatorial invariants and an affine paving, the PI will characterize geometric properties of Hessenberg varieties. Graham has defined an analogue of the Springer resolution, called the extended Springer resolution. The PI will use the fibers of this map to develop a new geometric framework for the generalized Springer correspondence, transforming the usual approach to this seminal work.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.

代数组合学是一个寻求在离散结构和代数对象之间建立联系的研究领域，在计算、统计学、生物学和其他数学学科中有着广泛的应用。组合学问题的一个中心主题是以一种反映关键结构属性的方式组织离散数据，从而使其更容易分析。这个项目应用代数组合学的工具来研究复杂的方程组的解，称为代数族。PI将开发复杂的计数技术，以简化计算并破译其他复杂数据中的模式。然后，PI将利用代数簇的几何性质来揭示代数和组合学中未解决的问题的新方法。此外，本项目还为研究生提供了研究培训机会。本项目所涉及的具体研究涉及Hessenberg品种和扩展Springer纤维的组合和几何结构。Hessenberg簇是FLAG簇的亚簇，其上同调环具有丰富的代数结构。PI将使用从Hessenberg变体获得的拓扑数据来勾勒出一种新的方法来解决组合学中长期存在的Stanley-Stembridge猜想。Hessenberg簇的几何和拓扑在少数情况下是完全理解的。使用组合不变量和仿射铺路，PI将表征Hessenberg簇的几何性质。格雷厄姆定义了斯普林格分辨率的类似物，称为扩展斯普林格分辨率。PI将利用这张地图的纤维为广义斯普林格通信开发一个新的几何框架，改变这项开创性工作的通常方法。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。