FRG: Collaborative Research: The Hypoelliptic Laplacian, Noncommutative Geometry, and Applications to Representations and Singular Spaces

FRG：合作研究：亚椭圆拉普拉斯、非交换几何以及在表示和奇异空间中的应用

• 批准号: 1952557
• 负责人: Yanli Song
• 金额: $ 15.39万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1952557&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Hypoelliptic; Laplacian

The collection of frequencies at which a geometric structure resonates is called spectrum of that structure. Encoded in the spectrum is a great deal of information about geometric form, which is difficult to extract. One might ask: How does the sound of a bell determines its shape, or vice versa? A new approach to the problem of relating geometry to the spectrum, based on a concept called the hypoelliptic Laplacian, has shown great promise. The purpose of this project is to build a new theoretical foundation for the hypoelliptic Laplacian, and then develop its applications in harmonic analysis and elsewhere. Expected outcomes will include a clearer and deeper overall understanding of the the hypoelliptic Laplacian, and a broadening of the range of applications to which it may be applied. There will be significant training and mentoring opportunities for graduate students and postdoctoral fellows in geometric and harmonic analysis, distributed across the three sites involved in the project. In more detail, this project will create a foundational theory for Jean-Michel Bismut's hypoelliptic Laplacian as it arises in symmetric and locally symmetric spaces, and elsewhere. For this purpose the investigators will use techniques previously developed in noncommutative geometry, especially the pseudodifferential operator theory originally developed to tackle the local index problem in noncommutative geometry. Turning to applications, in principle the hypoelliptic Laplacian offers a new approach to Harish-Chandra's Plancherel formula for real reductive groups, and an early priority will be to explore this application further. The newly established Mackey bijection in the representation theory of reductive groups (discovered in noncommutative geometry) will be investigated simultaneously. Many other potential applications in noncommutative geometry present themselves, and these will be studied carefully during the course of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

几何结构共鸣的频率的集合称为该结构的光谱。 在频谱中编码是有关几何形式的大量信息，这很难提取。 有人可能会问：铃铛的声音如何确定其形状，反之亦然？ 基于一种称为低纤维化拉普拉斯式的概念，将几何形状与光谱相关的问题的新方法表现出了很大的希望。该项目的目的是为低纤维化拉普拉斯人建立新的理论基础，然后在谐波分析和其他地方开发其应用。 预期的结果将包括对低纤维化拉普拉斯人的更清晰，更深入的总体理解，以及对可能应用的应用范围的扩展。 在几何和谐波分析中，将为研究生和博士后研究员提供大量的培训和指导机会，分布在该项目的三个站点上。更详细地，该项目将为Jean-Michel Bismut的低纤维化拉普拉斯（Laplacian）创建基础理论，因为它是在对称和局部对称空间中以及其他地方产生的。为此，研究人员将使用先前在非交通性几何形状中开发的技术，尤其是最初开发的伪差操作者理论，该理论最初是为了解决非交通性几何形状中局部索引问题的问题。转向申请，原则上，低纤维化拉普拉斯人为Harish-Chandra的Plancherel公式提供了一种新的方法，用于真正的还原群体，而早期的优先级将是进一步探索该应用程序。 在还原群体的表示理论（在非共同的几何形状中发现）中，新建立的Mackey Berivation将同时研究。在项目过程中将仔细研究非交通性几何形状中的许多其他潜在应用。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估值得支持。