**What is more likely: Symmetric or asymmetric manifolds?**

For a metric space a symmetry is an isommetry on
the space. Geometrically the most interesting
spaces are Riemannian manifolds. Although most
Riemannian manifolds which occur "in nature"
admit non-trivial symmetries, one expects that
this is not typical. A much stronger expectation
was formulated about 30 years ago by Raymond and
Schulz, namely that a compact smooth manifold
picked at random is asymmetric, meaning that for
all Riemannian metrics the group of isometries
is trivial. I would like to report on the state
of the art. Until recently the only known
examples of asymmetric manifolds were certain
aspherical manifolds, where one uses the
fundamental group to show that these manifolds
are asymmetric. I want to report about new
results concerning simply connected manifolds.

*Matthias Kreck*

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