Representations of Reductive Groups and Étale Hessenberg Varieties

朗兰兹程序是当代数论的一个中心主题，它包括几个预测，这些预测提供了一种分析研究丢番图方程解集的方法，使用线性代数的技术来利用这些解集中的隐藏对称性。几何表示理论为研究线性代数中的问题提供了几何工具。本课题将应用几何表示论中的方法，首先将朗兰兹程序中某些积分的计算简化为组合几何中的离散问题，然后研究后一个问题的解。这些结果有望加深和扩展对数论和几何表示理论的理解，这两个数学领域与许多其他学科有关，包括网络安全和物理。在已经完成的工作中，研究人员已成功地将某些轨道积分的计算简化为计算有限域上的Hessenberg族的有理点的问题。该项目将扩展和改进先前的计算，以及研究Hessenberg品种本身，特别是仿射Springer纤维中的层，并将它们置于相对Springer理论的更广泛背景下。

The Langlands program, a central theme in contemporary number theory, comprises several predictions that provide an approach to the analytic study of solution sets of Diophantine equations using techniques from linear algebra to exploit hidden symmetries in these solution sets. Geometric representation theory brings geometric tools to bear in studying problems in linear algebra. This project will apply methods from geometric representation theory to first reduce the computation of certain integrals that arise in the Langlands program to a discrete problem in combinatorial geometry, and then to study the solution of this latter problem. The results are expected to deepen and extend understanding in number theory and geometric representation theory, areas of mathematics with connections to many other subjects, including cyber security and physics.In work already completed, the investigator has succeeded in reducing the computation of certain orbital integrals to the problem of counting rational points of étale Hessenberg varieties over finite fields. This project will extend and refine the earlier computation, as well as study étale Hessenberg varieties in their own right, specifically as strata in affine Springer fibers, and place them in a broader context of relative Springer theory.