We aim to study geometric aspects of sub-Riemannian structures and relations to the spectral analysis of induced differential operators (e.g. sub-Laplace operators).
The project is divided into three parts that have close relations.
Existence and construction of sub-Riemannian geometries.
We wish to study the existence of sub-Riemannian structures (also under additional conditions such as regular or trivializable) on symmetric spaces of compact and noncompact type, Lie groups and their homogeneous spaces. In particular, the notion and analysis of sub-Riemannian curvature will be of interest since it is an object which might be detected from (or is related to) the spectral data as it happens in the Riemannian setting.
Spectral analysis of geometrically defined sub-elliptic operators.
Specific questions related to the spectral analysis and geometry of (sub)-Laplace operators concern:
- (Sub)-Laplace operators on exotic spheres
- (Sub)-Laplace operator on pseudo-H-type Lie groups and compact quotients
- Path integrals and heat kernels for "higher step" nilpotent Lie groups
Heat kernel for the Laplacian on differential forms and sub-Riemannian limit
We plan to study the (sub)-Laplacian on differential forms and its heat kernel and trace in concrete cases. We plan to consider the behaviour of such objects under taking sub-Riemannian limits. Applications to \(L^2\)-invariants may be within reach.
- Heat kernel of the form Laplacian on nilpotent Lie groups
- Novikov-Shubin invariants