Hyperbolic surfaces, spectral gaps, and random matrices

It was understood long ago that the first nonzero eigenvalue of the Laplacian on a closed hyperbolic surface cannot exceed that of the hyperbolic plane, asymptotically as the diameter goes to infinity. But until recently, it was not known whether there exist hyperbolic surfaces that attain this bound. Remarkably, it turns out that such surfaces notonly exist, but they are everywhere! I will aim to explain how the understanding of such phenomena has been made possible by new interactions between hyperbolic geometry, spectral theory, and random matrix theory. (Based on joint work with Michael Magee and Doron Puder.)