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.

代数组合学的研究旨在在离散结构和代数对象之间建立联系，并在数学和其他科学中进行广泛应用。该项目开发了以反映关键结构属性的方式组织离散数据的工具。这些工具用于研究多项式方程式复杂系统和简化计算的解决方案，以便在原本复杂的数据中破译模式。产生的见解为几何和组合学中重要的未解决问题提供了新的方法。该项目的教育组成部分将通过有针对性的指导计划培训下一代科学家，其中包括传统代表性不足的学生的研究机会。它还为当地学校的学生提供了结构化的途径。该项目的研究组成部分与Hessenberg品种的组合和几何结构有关。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组合结构。从这些品种中获得的拓扑数据将用于为代数组合中的开放性阳性猜想提供新的道路，并将其扩展到任意Weyl基团。仅在少数情况下，完全了解了黑森伯格品种的几何形状。 PI将通过研究Hessenberg品种的家族仅具有少数基本特性来改变这一领域的传统研究路径，并证明这些品种的几何形状会以可预测的方式波动，因为其他投入都会有所不同。该项目还扩大了圣路易斯华盛顿大学现有的本科研究计划，其中包括他们的高中背景没有准备好快速参加高级课程的学生。资金将用于建立新的数学外展研讨会。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优点和更广泛影响的评论标准来评估值得支持的。