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

代数组合学的研究旨在建立离散结构和代数对象之间的联系，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。这些工具用于研究多项式方程的复杂系统的解和简化计算，以便在其他复杂数据中破译模式。由此产生的见解为解决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结构化的途径。这个项目的研究部分是关于海森伯格品种的组合和几何结构。Hessenberg品种是flag品种的亚种，其上同环编码丰富的组合结构。从这些变体中获得的拓扑数据将为代数组合中的开放正性猜想及其在任意Weyl群中的推广开辟新的道路。只有在少数情况下，海森伯格变分的几何才被完全理解。PI将改变这一领域的传统研究路径，通过研究仅由几个基本性质联合起来的海森伯格品种家族，并证明这些品种的几何形状随着其他输入的变化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一个本科生研究项目，将那些高中背景不能让他们迅速上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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.