Mathematical Sciences: Sheaves on Witt Schemes and Trace Formula with Application to Representation Theory

数学科学：维特方案和迹公式及其在表示论中的应用

• 批准号: 9700458
• 负责人: Alexander Polishchuk
• 金额: $ 8.25万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700458&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Sheaves; Witt; Schemes

Polishchuk 9700458 This research is concerned with generalizing the approach of character sheaves from the case of groups over finite fields to groups over p-adic fields. The work involves two parts. The first part is to devise a notion of sheaves on ind-schemes mimicking that of locally-constant functions on p-adic varieties. The second part is to generalize the Lefschetz-Verdier trace formula and its application to the construction of the discrete series representation of a reductive group over a finite field based on the gluing of perverse sheaves. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.

Polishchuk 9700458本研究致力于将特征标方法从有限域上的群推广到p-ady域上的群。这项工作包括两个部分。第一部分是设计一个关于内模的层的概念，它模仿了p-进系簇上的局部不变函数的概念。第二部分推广了Lefschetz-Verdier迹公式，并将其应用于构造有限域上基于倒数组粘合的约化群的离散级数表示。这是在代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。