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

几何结构共振的频率集合称为该结构的频谱。 光谱中编码了大量的几何形状信息，这些信息很难提取。 有人可能会问：钟的声音如何决定它的形状，反之亦然？ 一种新的方法来解决问题的几何关系的频谱，基于一个概念，称为亚椭圆拉普拉斯算子，已显示出很大的希望。本计画的目的是为次椭圆拉普拉斯建立一个新的理论基础，进而发展其在调和分析及其他领域的应用。 预期成果将包括对亚椭圆拉普拉斯算子有更清晰和更深入的全面了解，并扩大其可能适用的应用范围。 将有显着的培训和指导机会，研究生和博士后研究员在几何和谐波分析，分布在三个网站参与该项目。更详细地说，这个项目将为Jean-Michel Bismut的次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，研究人员将使用以前在非交换几何中开发的技术，特别是最初开发用于解决非交换几何中局部指数问题的伪微分算子理论。转向应用，在原则上的亚椭圆拉普拉斯算子提供了一个新的方法，哈里什-钱德拉的Plancherel公式的真实的还原群，早期的优先事项将是进一步探讨这一应用。 新建立的麦基双射在约化群的表示理论（发现在非交换几何）将同时进行研究。在非对易几何中还有许多其他潜在的应用，这些将在项目过程中进行仔细研究。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.