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) Existence problem of compact locally homogeneous spaces, and deformation theory.
2. (spectral analysis) Construction of the spectrum of the Laplacian, and its stability under the deformation of the geometric structure.
 F. Kassel and T. Kobayashi, Stable spectrum for pseudo-Riemannian locally symmetric spaces. C. R. Math. Acad. Sci. Paris, 349, (2011), pp. 29-33.
 F. Kassel and T. Kobayashi, Discrete spectrum for non-Riemannian locally symmetric spaces, I. construction and stability, preprint, 95 pp. arXiv: 1209.4075.
 T. Kobayashi. Proper action on a homogeneous space of reductive type. Math. Ann., 285, (1989), pp.249-263.
 T. Kobayashi. Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds. Math. Ann., 310, (1998), pp. 395-409.
 T. Kobayashi and T. Yoshino. Compact Clifford-Klein forms of symmetric spaces-revisited. Pure and Appl. Math. Quarterly, 1 (2005), pp. 603-684. Special Issue: In Memory of Armand Borel.
© Toshiyuki Kobayashi