Quantum Ergodicity: from geodesic flow to Weyl's law

On compact Riemannian manifolds, a fascinating connection has been found
between geodesic flow and the eigenfunctions of the Laplace operator. If
the geodesic flow is ergodic, the eigenfunctions corresponding to large
eigenvalues will spread 'evenly' over the manifold. I will explain how
to make this precise, and in particular how the latter can be seen as a
stronger version of Weyl's law. This result from the '80s by Shnirelman,
Zelditch and Colin de Verdière was the start of the field that is now
called Quantum Ergodicity, and I will showcase some current conjectures
and developments in the area.