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, surprising little wa known about global properties of the geometry even if we impose a locally homogeneous structure. This theme has been developed rapidly in the last three decades.In the series of lectures, I plan to discuss two topics by the general theory and some typical examples.

1. Global geometry: Properness criterion for the action of discrete groups of isometries on reductive homogeneous spaces, existence problem of compact manifolds modeled on homogeneous spaces, and their deformation theory.

2. Spectral analysis: Construction of periodic $L^2$ eigenfunctions for the (indefinite) Laplacian, stability question of eigenvalues under deformation of geometric structure, and spectral decomposition.

References for the first topic include:

References for the second topic include:

- [1a] T. Kobayashi, Proper actions on homogeneous spaces of reductive type, Math. Ann. 1989,
- [1b] --, Deformation of compact Clifford-Klein forms of indefinite- Riemannian homogeneous manifolds. Math. Ann., 1998.
- [1c] T. Kobayashi-T.Yoshino, Compact Clifford--Klein forms of symmetric spaces---revisited. Pure and Appl. Math. Quarterly, 1, 2005. (Special Issue: In Memory of Armand Borel)

- [2a] F. Kassel-T. Kobayashi, PoincarĂ© series for non-Riemannian locally symmetric spaces. Adv. Math. 2016.
- [2b] Kobayashi. Global analysis by hidden symmetry. Mathematics 323, 2017. (Special Issue in honor of Roger Howe)

© Toshiyuki Kobayashi