Abstract: Harmonic maps and hyperbolic surfaces are among the most studied special objects in Differential Geometry. Harmonic maps into Riemannian manifolds are a nonlinear generalization of harmonic functions. Hyperbolic surfaces are surfaces with constant Gaussian curvature equal to -1. In this talk, I will describe a phenomenon connecting the two notions: often, “random” harmonic maps from surfaces to Euclidean spheres have images which are almost hyperbolic surfaces with high probability. Among other ingredients, this connection relies on a new invariant for unitary representations of surface groups, and on the concept of strong convergence appearing in random matrix theory.