## *Global Geometry and Analysis on Locally Pseudo-Riemannian
Homogeneous Spaces*.
JSPS-DST Asian
Academic Seminar 2013: Discrete Mathematics & its Applications. Graduate
School of Mathematical Sciences, the University of Tokyo, Japan, 3-10
November 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 of general signature, surprising
little is known about global properties of the geometry even if we
impose a locally homogeneous structure.
Taking anti-de Sitter manifolds, which are locally modelled on AdS^{n} as
an example, I plan to explain two programs:

1. (global shape) Exisitence problem of compact locally homogeneous
spaces, and defomation theory.

2. (spectral analysis) Construction of the spectrum of the Laplacian,
and its stability under the deformation of the geometric structure.

© Toshiyuki Kobayashi