Applications of Lie Theory: Combinatorial Algebraic Geometry and Symmetric Functions

代数组合学是一个研究领域，旨在建立离散结构和代数对象之间的联系，在计算、统计学、生物学和其他数学学科中有着广泛的应用。组合问题的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方程组的解，称为代数变分。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的模式。然后，PI将使用代数变体的几何性质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训练的机会。本项目的具体研究涉及海森伯格品种和延伸施普林格纤维的组合和几何结构。Hessenberg变异体是旗变异体的子变异体，其上同调环编码了丰富的代数结构。PI将使用从海森伯格变种中获得的拓扑数据来概述一种新的方法来解决组合学中长期存在的Stanley-Stembridge猜想。只有在少数情况下，海森伯格变分的几何和拓扑才被完全理解。使用组合不变量和仿射铺装，PI将表征海森伯格变量的几何性质。Graham定义了施普林格分辨率的类似物，称为扩展施普林格分辨率。PI将使用这张图的纤维为广义施普林格对应开发一个新的几何框架，改变这项开创性工作的通常方法。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。

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.