## Global Analysis of Locally Symmetric Spaces with Indefinite-
metric.
Zariski Dense
Subgroups, Number Theory and Geometric Applications. ICTS, Bangalore, India,
6-17 April 2020.

* Conference was cancelled. Re-rescheduled, April 17-28, 2023.

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:

- [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)

References for the second topic include:
- [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