Speaker: Jean-Michel Bismut (Professor of Mathematics, University of Paris XI, France, Member of the French Academy of Sciences)

Time: 17:00—18:00, October10, 2013

Place: Lecture Hall, Second Floor, Jia Yi Bing Building, 82 Jing Chun Yuan, PKU

Abstract: If X is a Riemannian manifold, the Laplacian is a second order elliptic operator on X. The hypoelliptic Laplacian L_b is an operator acting on the total space X of the tangent bundle of X, that is supposed to interpolate between the elliptic Laplacian (when b→0) and the geodesic flow (when b →+∞). Up to lower order terms, L_b is a weighted sum of the harmonic oscillator along the fibre TX and of the of the geodesic flow. One expects that, in the deformation, there are conserved quantities. In the talk, I will describe three applications of the hypoelliptic Laplacian:
_ The case of the circle.
_ Selberg's trace formula and the evaluation of orbital integrals.
_ A Riemann-Roch-Grothendieck theorem in Bott-Chern cohomology.