Bernoulli Lecture III by Jean-Michel Bismut - Hypoelliptic Laplacian, classical mechanics and the Langevin process   21 May 2015

17:15 - 18:15


Jean-Michel Bismut, Université Paris-Sud

If X" role="presentation">X is a compact Riemannian manifold, the hypoelliptic Laplacian Lb|b>0" role="presentation">Lb|b>0 is a family of operators acting on the total space X" role="presentation"> of the tangent bundle TX" role="presentation">TX, that interpolates between the Laplace-Beltrami operator and the geodesic flow. The associated diffusion is a geometric Langevin process on X" role="presentation">X, whose dynamics interpolates between Brownian motion and the geodesic flow.
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.

Name University Dates of visit
Jean-Michel Bismut Université Paris-Sud 21/05/2015 - 21/05/2015
Total Guests : 1
Name University Dates of visit
Total Guests : 0