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.

代数组合学是一个研究领域，旨在在离散结构和代数对象之间建立连接，并在计算，统计，生物学和其他数学学科中进行广泛应用。组合问题问题中的一个中心主题是以反映关键结构属性的方式组织离散数据，从而更容易分析。该项目将代数组合学的工具应用于研究复杂方程系统的解决方案，称为代数品种。 PI将开发复杂的计数技术，以简化原本复杂数据中的计算和解密模式。然后，PI将使用代数品种的几何特性来发现代数和组合学中未解决问题的新方法。此外，该项目还为研究生提供了研究培训机会。该项目中涉及的具体研究涉及Hessenberg品种和扩展的Springer纤维的组合和几何结构。赫森伯格品种是标志品种的亚变化，其共同体环编码丰富的代数结构。 PI将使用从Hessenberg品种获得的拓扑数据来概述一种新方法，以组合长期存在的Stanley-STAMBRIDGE猜想。仅在少数情况下，完全了解了黑森伯格品种的几何形状和拓扑结构。使用组合不变性和仿射铺路，PI将表征Hessenberg品种的几何特性。格雷厄姆（Graham）定义了施普林格（Springer）决议的类似物，称为“扩展的施普林格分辨率”。 PI将使用此地图的纤维为广义施普林格通讯开发一个新的几何框架，从而将通常的方法转化为这项开创性的工作。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子来评估的支持的。和更广泛的影响审查标准。