"What is typical: symmetric or asymmetric manifolds"
For a metric space a symmetry is an isometry of 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.