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。频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：钟声如何决定它的形状，反之亦然？一种基于亚椭圆拉普拉斯概念的将几何与光谱联系起来的新方法已经显示出巨大的前景。这个项目的目的是为亚椭圆拉普拉斯建立一个新的理论基础，进而发展它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方程有更清晰和更深入的整体理解，以及扩大其可能适用的应用范围。将为研究生和博士后提供几何和调和分析方面的重要培训和指导机会，分布在参与该项目的三个地点。更详细地说，这个项目将为Jean-Michel Bismut的亚椭圆拉普拉斯建立一个基础理论，因为它出现在对称和局部对称的空间，以及其他地方。为此，研究人员将使用以前在非对易几何中发展起来的技术，特别是伪微分算子理论，最初是为了解决非对易几何中的局部指数问题而开发的。关于应用，原则上，亚椭圆拉普拉斯为实约群的Harish-Chandra的Plancerel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。我们将同时研究约化群表示理论中新建立的Mackey双射(发现于非对易几何中)。非对易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被仔细研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.