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 spacetime 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 colloquim, I plan to discuss two topics.

Global geometry : Existence problem of compact manifolds modelled 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