Representations of Reductive Groups and Étale Hessenberg Varieties

还原群和 ätale Hessenberg 簇的表示

• 批准号: 1751940
• 负责人: Cheng-Chiang Tsai
• 金额: $ 9.13万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1751940&HistoricalAwards=false
• 关键词: Representations; Reductive; Groups; tale; Hessenberg

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.

兰兰兹计划是当代数字理论中的一个中心主题，包括几种预测，使用线性代数的技术来利用这些解决方案集中的隐藏对称性，为溶液方程解决方案的分析研究提供了一种方法。几何表示理论为研究线性代数中的问题提供了几何工具。该项目将使用几何表示理论的方法首先将Langlands计划中某些积分的计算减少到组合几何形状中的离散问题，然后研究以后问题的解决方案。预计该结果将加深和扩展数量理论和几何表示理论，与许多其他主题的数学领域，包括网络安全和物理学的联系。该项目将扩展和完善早期的计算，并以其本身的质量研究Hessenberg品种，特别是作为仿射弹簧纤维中的阶层，并将其放置在相对施普林格理论的更广泛背景下。