The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particulary 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, as an example, I plan to explain two programs :
1.(global shape) existence problem of compact locally homogeneuous spaces, and deformation theory;
2. (spectral analysis) construction of the spectrum of the Laplacian, and its stability under the deformation of the geometric structure.
[ poster ]
© Toshiyuki Kobayashi