Analysis of sub-Riemannian structures and related operators

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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.