Automorphic forms are solutions of systems of linear PDE on manifolds which carry extra arithmetic symmetries. We will introduce some of the fundamental conjectures and probabilistic models concerning the automorphic spectrum

For $\mathrm{SL}(2,\mathbb{R})$ automorphic forms are Laplace eigenfunctions on the upper-half plane invariant under a Fuchsian group of isometries. These were first studied by Poincaré using the theory of differential equations and Riemann surfaces, establishing the uniformization theorem in the process. Their existence was established by Selberg when he discovered the trace formula.

In works with S. W. Shin, P. Sarnak, J. Matz, we have generalized some of the classical results to higher rank groups. Consequences are a new understanding of families of L-functions and a result towards the Ramanujan conjecture on average for $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$.