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

几何结构共振频率的集合称为该结构的频谱。在光谱中编码有大量的几何形状信息，这些信息很难提取。有人可能会问：铃铛的声音是如何决定它的形状的，反之亦然？基于准椭圆拉普拉斯算子的概念，一种解决几何与谱之间关系问题的新方法显示出了巨大的希望。本课题的目的是为拟椭圆拉普拉斯建立一个新的理论基础，进而发展其在谐波分析等方面的应用。预期的结果将包括对准椭圆拉普拉斯的更清晰和更深入的全面理解，以及它可能应用的应用范围的扩大。在几何和谐波分析方面，研究生和博士后将有重要的培训和指导机会，分布在项目涉及的三个地点。更详细地说，这个项目将为Jean-Michel Bismut的准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中，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，特别是伪微分算子理论，最初是为了解决非交换几何中的局部指标问题而开发的。至于应用，原则上，准椭圆拉普拉斯算子为实约群的Harish-Chandra的Plancherel公式提供了一种新的方法，早期的优先事项将是进一步探索这一应用。同时研究了在非交换几何中发现的约化群表示理论中新建立的麦基双射。在非交换几何中有许多其他潜在的应用，这些将在项目过程中仔细研究。该奖项反映了美国国家科学基金会的法定使命，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.