CAREER: Hessenberg Varieties, Symmetric Functions, and Combinatorial Representation Theory

职业：Hessenberg 簇、对称函数和组合表示论

• 批准号: 2237057
• 负责人: Martha Precup
• 金额: $ 46.9万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2028-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2237057&HistoricalAwards=false
• 关键词: CAREER; Hessenberg; Varieties; Symmetric; Functions

Research in algebraic combinatorics seeks to build connections between discrete structures and algebraic objects, with broad applications in mathematics and other sciences. This project develops tools to organize discrete data in a way that reflects key structural properties. These tools are applied to study solutions of complicated systems of polynomial equations and streamline computation in order to decipher patterns in otherwise complex data. The resulting insights yield new approaches to important unsolved problems in both geometry and combinatorics. The educational component of this project will train the next generation of scientists through a targeted mentoring program that includes research opportunities for traditionally underrepresented students. It also creates structured pathways for outreach to students in local schools.The research component of this project is concerned with the combinatorial and geometric structure of Hessenberg varieties. Hessenberg varieties are subvarieties of the flag variety whose cohomology rings encode rich combinatorial structure. Topological data obtained from these varieties will be used to make new in-roads toward open positivity conjectures in algebraic combinatorics and their extension to arbitrary Weyl groups. The geometry of Hessenberg varieties is completely understood in only a few cases. The PI will transform the traditional path of research in this area by studying families of Hessenberg varieties united by only a few essential properties, and prove the geometry of these varieties fluctuates in predictable ways as other inputs vary. This project also expands an existing undergraduate research program at Washington University in St. Louis to include students whose high school background does not prepare them to take advanced courses quickly. Funds will be used to establish a new math outreach seminar.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.

代数组合学的研究试图在离散结构和代数对象之间建立联系，在数学和其他科学中有着广泛的应用。该项目开发工具，以反映关键结构属性的方式组织离散数据。这些工具被应用于研究复杂的多项式方程组的解和流水线计算，以便破译原本复杂的数据中的模式。由此产生的洞察力为几何和组合学中尚未解决的重要问题提供了新的方法。该项目的教育部分将通过一项有针对性的指导计划培训下一代科学家，该计划包括为传统上代表性不足的学生提供研究机会。它还为当地学校的学生创造了有组织的联系途径。这个项目的研究部分涉及Hessenberg变种的组合和几何结构。Hessenberg品种是FLAG品种的亚种，其上同调环编码丰富的组合结构。从这些簇得到的拓扑数据将被用来为代数组合学中的开正性猜想及其推广到任意Weyl群开辟新的道路。Hessenberg簇的几何在少数情况下是完全理解的。PI将通过研究仅由几个基本性质结合起来的Hessenberg变种家族来改变这一领域的传统研究路径，并证明这些变种的几何形状随着其他投入的变化而以可预测的方式波动。该项目还扩大了圣路易斯华盛顿大学现有的本科生研究项目，将那些高中背景不足以迅速攻读高级课程的学生纳入其中。资金将用于建立一个新的数学外展研讨会。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。