# Lecture Series on “Arithmetic in Geometry" - #1

# Lecture Series on “Arithmetic in Geometry" - #1

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, Algebraic Groups and Automorphic Forms. 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). We have also determined higher dimensional analogues of the fpp's. These works have required considerable amount of number theoretic bounds and computations and also inputs about the cohomology of Shimura varieties. My first two talks will be devoted to this topic. In the next two talks, I will discuss 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 (in Publ Math IHES) with Andrei Rapinchuk. For the solution, we introduced a notion of "weak commensurability" of arithmetic, and more general Zariski-dense, subgroups and derive very strong consequences of weak commensurability.