First I'm going to talk about why there isn't an algorithm to recognize any closed n-dimensional manifold for n above 4. This forces the variational landscape of the moduli space of all Riemannian metrics on a given n-manifold to be much more complicated than in lower dimensions. For example, the diameter functional on the (suitably normalized) moduli space has local minima of arbitrarily large depth. A crucial role in all this is played by special round-sphere lookalikes which come from number theory. If I have time I hope to sketch an approach to constructing Einstein metrics based on these ideas.

This talk is based on work of Nabutovsky and Weinberger.