## *Existence Problem of Compact Locally Symmetric Spaces*.

colloquium. Louisiana State University, Baton Rouge, USA, 27 March
2008.

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, as well as in various other kinds of
geometry (symplectic, complex geometry, ...), surprising little is known
about global properties of the geometry even if we impose a locally
homogeneous structure.
I will give a survey on the recent developments regarding the question
about how the local geometric structure affects the global nature of
non-Riemannian manifolds, with emphasis on the existence problem of compact
forms, rigidity and deformation.

© Toshiyuki Kobayashi