## *Global Geometry and Analysis on Locally Homogeneous Spaces* (2 lectures).
Workshop d'analyse harmonique. Reims, France, 2 November 2012.

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 Lorentz geometry, familiar to us
as the space-time of relativity theory, and more generally in
pseudo-Riemannian geometry, surprising little is known about global
properties of the geometry even if we impose a locally homogeneous
structure. Further, almost nothing is known on global analysis on such spaces.
Taking anti-de Sitter manifolds, which are locally modelled on AdS^n as
an example, I plan to explain two programs:
1. (global shape) Is a locally homogeneous space closed?

2. (spectral analysis) Does spectrum of the Laplacian vary when we deform the geometric structure?

© Toshiyuki Kobayashi