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

基本信息

批准号：
1952669
负责人：
Nigel Higson
金额：
$ 42.82万
依托单位：
Pennsylvania State Univ University Park
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=1952669&HistoricalAwards=false
关键词：
FRG Collaborative Research Hypoelliptic Laplacian

项目摘要

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.

几何结构共振的频率集合称为该结构的频谱。频谱中编码了大量关于几何形状的信息，而这些信息很难提取。有人可能会问：钟的声音如何决定其形状，反之亦然？一种基于亚椭圆拉普拉斯概念的解决几何与频谱相关问题的新方法已显示出巨大的前景。该项目的目的是为亚椭圆拉普拉斯建立新的理论基础，然后开发其在调和分析和其他方面的应用。预期成果将包括对亚椭圆拉普拉斯算子有更清晰、更深入的整体理解，并扩大其应用范围。几何和调和分析方面的研究生和博士后研究员将获得重要的培训和指导机会，分布在该项目涉及的三个地点。更详细地说，该项目将为让-米歇尔·比斯穆特 (Jean-Michel Bismut) 的亚椭圆拉普拉斯算子创建一个基础理论，因为它出现在对称和局部对称空间以及其他地方。为此，研究人员将使用先前在非交换几何中开发的技术，特别是最初为解决非交换几何中的局部索引问题而开发的伪微分算子理论。谈到应用，原则上，亚椭圆拉普拉斯算子为真实还原群的 Harish-Chandra Plancherel 公式提供了一种新方法，早期的首要任务是进一步探索该应用。同时研究还原群表示论中新建立的麦基双射（在非交换几何中发现）。非交换几何中的许多其他潜在应用也出现了，这些将在项目过程中得到仔细研究。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。