Colloquium

An exotic structure on a differentiable manifold is one which is homeomorphic but not diffeomorphic to the manifold. A fake structure on a complex manifold is one which has the same Betti numbers but not biholomorphic to the manifold. The main purpose of the talk is to explain the joint work of Gopal Prasad and myself on classification of fake projective planes. Some observations related to exotic structures will be mentioned. Fake projective plane was first introduced by David Mumford. It has the smallest possible Euler number among all smooth surfaces of general type. Related work of Klingler, myself, Prasad and Cartwright-Steger will also be discussed.