Weyl’s law gives an asymptotic formula for the number of eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold. In 2005, Elon Lindenstrauss and Akshay Venkatesh gave a proof of this law for quite general quotients of semisimple Lie groups. The proof crucially uses the fact that solutions of the corresponding wave equation propagate with finite speed. I will try to explain what they did in the simplest setting of the upper half plane. I will also try to explain why such eigenfunctions, also known as automorphic forms, are of central importance in number theory. The first 45 minutes of the talk should be accessible to students who have a knowledge of some basic complex analysis and calculus.

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Last updated: 03 Feb 2023