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 这项研究涉及将特征滑轮方法从有限域上的群的情况推广到 p 进数域上的群。 这项工作涉及两个部分。 第一部分是设计 ind 方案上的滑轮概念，模仿 p-adic 变体上的局部常数函数。 第二部分是推广Lefschetz-Verdier迹公式及其在基于反常滑轮粘合的有限域上还原群离散级数表示的构建中的应用。 这是代数几何领域的研究。 代数几何是现代数学最古老的部分之一，但在过去的四分之一个世纪里得到了革命性的发展。 在它的起源中，它处理可以通过最简单的方程（即多项式）在平面上定义的图形。 如今，该领域不仅使用代数方法，还使用分析和拓扑方法，相反，它正在这些领域以及物理学、理论计算机科学和机器人技术中得到应用。