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

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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.