Analysis of sub-Riemannian structures and related operators

该项目将侧重于以下相关主题：A。特殊紧和非紧流形及其齐性空间上强意义下的次黎曼结构分析。显式确定相应的次椭圆算子的谱数据，并以积分或几何形式构造其热核。光谱量和几何量之间的关系。B。广义Segal-Bargmann空间的Berezin-Toeplitz量子化的分析（参见下面的描述）和特殊类型的真实的流形。C.我们计划研究亚黎曼结构和Berezin-Toeplitz量子化之间的关系。作为一个应用，我们期望找到由底层结构的几何所诱导的经典特殊函数（如θ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狄拉克算子的次黎曼类似物。由于一般来说，没有连接规范与一个子黎曼结构，我们将分析具体的例子，提供额外的数据。

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.