## *Global Analysis of Locally Symmetric Spaces with
Indefinite-metric*.
Colloquium,
National University of Singapore. (online), 13 August 2021.

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 pseudo-Riemannian
geometry, familiar to us as the space-time of relativity theory, and
more generally in pseudo-Riemannian geometry of general signature,
surprisingly little is known about global properties of the geometry
even if we impose a locally homogeneous structure.
In this colloquium, I plan to discuss two topics.

Global geometry: Existence problem of compact manifolds modelled locally
on homogeneous spaces, and their deformation theory.

Spectral analysis: Construction of periodic eigenfunctions for the
(indefinite) Laplacian, and stability question of eigenvalues under
deformation of geometric structure by using representation theory.

© Toshiyuki Kobayashi