Fachvortrag in englischer Sprache

Quotients of symmetric spaces of semi-simple Lie groups by torsion-free arithmetic subgroups are particularly nice Riemannian manifolds which can be studied by using diverse techniques coming from the theories of Lie Groups, Lie Algebras and Algebraic Groups. One such manifold is a »fake projective plane« which is, by definition, a smooth projective complex algebraic surface with same Betti-numbers as the complex projective plane but which is not isomorphic to the latter. The first example of a fake projective plane (fpp) was constructed by David Mumford in 1978, and it has been known that there are only finitely many of them. In the theory of algebraic surfaces, it was an important problem to construct them and determine their geometric properties. In a joint work with Sai-Kee Yeung, I have classified them and given an explicit way to construct them all (it turns out that there are exactly 100 of them). This work, and the determination of higher dimensional analogues of the fpp's, have required considerable amount of number theoretic bounds and computations. If the time permits, I will describe another well-known problem which was very nicely formulated by Mark Kac as »Can one hear the shape of a drum?«, and its solution, for arithmetic quotients of symmetric spaces, obtained in a joint work with Andrei Rapinchuk.
 

Speaker: Prof. Dr. Gopal Prasad, University of Michigan, USA