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

调和分析是傅里叶变换算子作用于各种域上的复值函数的理论，例如有限域，p-adic域，实数和复数。有趣的是，傅里叶变换是一个算子家族的一部分，这些算子之间的关系可以用一个群对称结构来描述，与辛群密切相关。这个族称为Weil表示。Weil表示是连接经典调和分析和表示论的桥梁，也为理解表示论的本质提供了一个强有力的新视角。此外，韦伊表示是由一个对象从代数几何，称为几何韦伊表示。几何Weil表示，作者开发的，作为一个桥梁，连接调和分析和代数几何，因此，能够解决分析问题，使用现代上同调技术。本项目围绕以下主题：Weil表示的正则模型及其几何化，特征2的Weil表示，辛相似的Weil表示及其在各个领域的应用，本项目将提高我们对调和分析基础的表示论和代数几何结构及其应用的理解。它还将揭示数论和分析经典陈述的新视角。PI和Co-PI在过去几年中在许多课程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，如工程和物理，将他们的成果应用于纯数学之外的问题。他们坚信，他们的方法和结果对更广泛的科学界感兴趣，并有可能对数学以外的学科产生根本性的影响。

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.