Analysis of sub-Riemannian structures and related operators

本项目将集中在以下相关主题上：a.在特殊紧流形和非紧流形及其齐次空间上强意义上的次黎曼结构的分析。相应的次椭圆算子的谱数据的显式确定及其热核的积分或几何形式的构造。谱量和几何量之间的关系。B.广义西格尔-巴格曼空间的Berezin-Toeplitz量子化的分析(参见下面的描述)和特殊类型的实流形。C.我们计划研究次黎曼结构与Berezin-Toeplitz量子化之间的关系。作为应用，我们期望找到由底层结构的几何诱导的经典特殊函数(如theta函数、椭圆函数、超几何函数···)之间的显式关系。D.我们打算研究自旋流形上经典Dirac算子的次黎曼类似。由于通常没有与次黎曼结构规范相关的联系，我们将分析给出额外数据的具体例子。

The project will focus on the following related topics: A. Analysis of sub-Riemannian structures in the strong sense on special compact and non-compact manifolds and their homogeneous spaces. Explicit determination of the spectral data for corresponding hypo-elliptic operators and construction of their heat kernels in an integral or a geometric form. Relations between the spectral and geometric quantities. B. Analysis of the Berezin-Toeplitz quantization for generalized Segal-Bargmann spaces (cf. the description below) and for real manifolds of special type. C. We plan to study relations between sub-Riemannian structures and Berezin-Toeplitz quantization. As an application we expect to find explicit relations between classical special functions (such as theta functions, elliptic functions, hypergeometric functions • • • ) which are induced by the geometry of the underlying structures. D. We intend to study sub-Riemannian analogs of the classical Dirac operator on spinmanifolds. Since in general there is no connection canonically associated with a sub- Riemannian structure we will analyze specific examples which give additional data.