The overall goals of the project will be clearer, simpler and more powerful understanding of the algebraic and functional analytic foundations of hypoelliptic Laplacian; the extensive development of its applications to tempered representation theory, including the Harish-Chandra Plancherel formula and the Mackey bijection; and a deepened understanding of the geometric and topological invariants of singular spaces. Specific goals for the proposed project period will include:

• A new analytic foundation for the hypoelliptic Laplacian, using techniques borrowed from non- commutative geometry to obtain subelliptic and spectral estimates.

• A new approach Harish-Chandra’s spherical Plancherel formula, and an analysis of its correspondence with both the classical approaches and other geometric approaches.

• Analysis and characterization of the tempered duals of real reductive groups from the noncommutative geometric point of view, using higher orbital integrals to determine the Hochschild complex of Harish-Chandra’s Schwartz space via (higher) orbital integrals.

• Development of the noncommutative geometry and spectral theory of coefficient systems, constructible and coherent sheaves as they arise in the theory of real and complex singular spaces.

PIs: Nigel Higson (Penn State) Yanli Song (Wash U) Xiang Tang (Wash U) Zhizhang Xie (Texas A&M)

We gratefully acknowledge funding from the National Science Foundation under the aegis of DMS-1952557, DMS-1952551, DMS-1952669 and DMS-1952693.