## *Global Geometry and Analysis on Locally Pseudo-Riemannian
Homogeneous Spaces*.
Colloquium.
University of Gothenburg, Sweden, 20 May 2013.

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 the spectrum of the Laplacian vary when we deform the geometric structure?

© Toshiyuki Kobayashi