**Abstract:** The local to global study of geometries was a major trend of 20th
century geometry, with remarkable developments achieved particularly in
Riemannian geometry. In contrast, in areas such as pseudo-Riemannian
geometry, familiar to us as the space-time of relativity theory, and
more generally in pseudo-Riemannian geometry of general signature,
surprising little is known about global properties of the geometry even
if we impose a locally homogeneous structure. In this colloquium, I plan to discuss two topics. Global geometry: Existence problem of compact manifolds modeled
locally on homogeneous spaces, and their deformation theory. Spectral analysis: Construction of periodic eigenfunctions for the (
indefinite) Laplacian, and stability question of eigenvalues under
deformation of geometric structure. |