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

°Õ³ó¾±²õÌýtalk is based on work of Nabutovsky and Weinberger.