This interpolation preserves certain spectral quantities. On very rigid man- ifolds like tori or on locally symmetric spaces, the spectrum of the original elliptic Laplacian remains rigidly embedded in the spectrum of the hypoelliptic deformation. Generalized Poisson formulas like Selberg’s trace formula can be obtained as a consequence of this interpolation.
In the lecture, I will explain the construction of the hypoelliptic Laplacian on simple examples, and describe the associated stochastic processes. The spectral properties of the hypoelliptic deformation will be emphasized.