题目：Introduction to Hypoelliptic Laplacian of Bismut on Symmetric Spaces

报告人：刘冰萧 博士 （德国科隆大学 数学研究院）

地点：腾讯会议室

时间：2021年11月 17日（星期三） 16:30 -- 18:30

摘要：In this series of talks, I would like to give a brief introduction to the theory of the hypoelliptic Laplacian developed by Bismut for symmetric spaces. This theory is a geometric approach to evaluate explicitly the semisimple orbital integrals associated with the heat kernels of the Casimir operator on a symmetric space. As a consequence, Bismut obtained a trace formula for the heat trace of Bochner-like Laplacians on compact locally symmetric spaces. The basic idea of Bismut is to construct a smooth family of hypoelliptic differential operators which deforms in proper sense the elliptic

Laplacians like Casimir operator to the Lie derivative of the generator of the geodesic flow. Moreover, such deformation preserves certain spectral invariance, and in particular, it preserves the semisimple orbital integrals in the case of symmetric spaces. Then by a method inspired by local index theory, Bismut managed to work out his explicit geometric formula for the orbital integrals.

There are three types of hypoelliptic Laplacians with different origins: de Rham - Hodge theory, the one for symmetric spaces, and Dolbeault theory for complex manifold.  The talks will cover only the first two of them with the following main points:

1. A quick review on the spectral geometry of the Riemannian manifold.One of the motivations for Bismut's theory of the hypoelliptic Laplacian is to understand the connection between the spectrum of the Laplacian operator and the length spectrum defined by the closed geodesics. I will explain briefly the construction of  hypoelliptic Laplacians on Riemannian manifold using an exotic Hodge theory.

2. The evaluation of heat trace has a formal approach by an analogue to the local index theory. This inspires the construction of hypoelliptic Laplacian for the symmetric space.

3. Bismut's construction of hypoelliptic Laplacians for a symmetric space G/K, where G is a real reductive Lie group. Moreover, I will explain how they deform the Casimir operator in proper sense.

4. Selberg trace formula for compact locally symmetric spaces and a geometric interpretation of the semisimple orbital integrals. In particular, the semisimple orbital integrals are preserved under the hypoelliptic deformation.

5. Introduction on the explicit evaluation of the semisimple orbital integrals using the hypoelliptic deformation. I will take the identity orbital integral in the case of hyperbolic half-plane (G=SL(2,R)) as an explanatory example.

平台：腾讯会议号码：11月17日 648634473，11月24日 651244555，11月26日 756577241

欢迎各位参加！