Collaborative Research: The Heisenberg--Weil Symmetries, their Geometrization and Applications

合作研究：海森堡-韦尔对称性、其几何化和应用

• 批准号: 1101660
• 负责人: Shamgar Gurevich
• 金额: $ 15.1万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101660&HistoricalAwards=false
• 关键词: Collaborative; Research; Heisenberg; Weil; Symmetries

Harmonic Analysis is the theory of the Fourier transform operator acting on complex valued functions on various fields, such as, finite fields, p-adic fields, reals and the complex numbers. Interestingly, the Fourier transform is a part of a family of operators, that satisfy relations with respect to one another that can be described by a group symmetry structure, strongly related to the symplectic group. This family is called the Weil representation. The Weil representation serves as a bridge that connects classical harmonic analysis with representation theory, it also gives a powerful fresh perspective about the very nature of the theory. Furthermore, the Weil representation is governed by an object from algebraic geometry, called the geometric Weil representation. The geometric Weil representation, developed by the authors, serves as a bridge that connects between harmonic analysis and algebraic geometry, hence enables to solve analytic problems using modern cohomological techniques. The present project revolves around the following themes: canonical model of the Weil representation and its geometrization, the Weil representation in characteristic two, the Weil representation of symplectic similitudes, and applications to various fields.This project will enhance our understanding of the representation theoretic and algebraic geometric structures that underlie harmonic analysis and their applications. It will also reveal new perspectives about classical statements from number theory and analysis. The PI and Co-PI have presented their work in numerous classes and seminars over the past years. They are also collaborating with people from other scientific areas, such as engineering and physics, on the applications of their results to questions outside pure mathematics. They strongly believe that their methods and results are of interest to the broader scientific community and has the potential to have radical impact on disciplines outside of mathematics.

调和分析是傅立叶变换算子作用于不同域上的复值函数的理论，如有限域、p-ady域、实数和复数。有趣的是，傅里叶变换是算子族的一部分，它满足彼此之间的关系，可以用群对称结构来描述，与辛群密切相关。这个家族被称为韦尔代表。Weil表象是连接经典调和分析与表象理论的桥梁，它也为理解表象理论的本质提供了一个强有力的新视角。此外，Weil表示由代数几何中的一个对象管理，称为几何Weil表示。作者发展的几何Weil表示是连接调和分析和代数几何的桥梁，从而使我们能够利用现代上同调技术来解决解析问题。本课题围绕以下主题展开：Weil表示及其几何化的规范模型、特征二中的Weil表示、辛相似的Weil表示以及在各个领域中的应用。本项目将加深我们对作为调和分析及其应用基础的表示理论和代数几何结构的理解。它还将从数论和分析的角度揭示经典陈述的新视角。在过去的几年里，国际和平协会和联合国际组织在许多课堂和研讨会上介绍了他们的工作。他们还与其他科学领域的人合作，如工程学和物理学，将他们的结果应用于纯数学以外的问题。他们坚信，他们的方法和结果引起了更广泛的科学界的兴趣，并有可能对数学以外的学科产生根本性的影响。