The main purpose of the talk is to explain a recent classification scheme of fake projective planes by Gopal Prasad and myself, together with the work of Cartwright and Steger.  A fake projective plane is a smooth complex surface with the same Betti numbers as the complex projective plane but not biholomophic to it.  We would introduce some applications to problems in complex geometry and algebraic geometry.  We would also mention some higher dimensional analogue by considering arithmetic fake compact Hermitian symmetric spaces of higher dimensions.

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