Historically, it is geometry which led to important developments in several areas of mathematics including number theory. But recently there have been several instances of number theory being applied to settle important questions in geometry. I will talk about two problems in whose solution number theory has been used in a crucial way.
The first one, settled in collaboration with Sai-Kee Yeung, is classification of fake projective planes and their higher dimensional analogs. (I recall that fake projective planes are smooth projective complex surfaces with same Betti numbers as the complex projective plane, but which are not isomorphic to the complex projective plane. The first such surface was constructed by David Mumford.)
The second problem concerns compact Riemannian manifolds and it has the following very interesting formulation due to Mark Kac: “Can one hear the shape of a drum?”. In precise mathematical terms, the question asks whether two compact Riemannian manifolds with same spectrum (i.e., the set of eigenvalues counted with multiplicities) are isometric. The answer is in general “no”. However, Andrei Rapinchuk and I investigated Kac’s question, using number theoretic results and tools, for a particularly nice class of manifolds, namely locally symmetric spaces. The answer turned out to be very interesting and has led to several other developments which, if time permits, I will mention.