Symmetric spaces are generalizations of the classical
two-dimensional geometries (Euclidean, Spherical, Non-Euclidean) to
higher dimensions. Locally symmetric spaces are quotients of
symmetric spaces by discrete groups of isometries acting on the
symmetric spaces. Considering fundamental domains of these actions
they can be viewed as tiles in a tiling with suitable pieces of the
boundary identified with each other - visualize Escher type tilings
of a disk.
Locally symmetric spaces occur naturally in number theory, but they
are also model cases for differential geometry, ergodic theory and
mathematical physics. This is so because in many contexts the large
degree of symmetry allows to use group theoretic methods to obtain
more specific results than for generic Riemannian manifolds.
The focus of this talk will be on the interplay between the spectral
theory of invariant differential operators and the geometry of
locally symmetric spaces. The methods employed come from
non-commutative harmonic analysis, microlocal analysis and dynamical
systems.
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