A still-open conjecture of Selberg predicts that on a noncompact hyperbolic surface of finite area, the Laplace-Beltrami operator has infinitely many $L^2$ eigenfunctions. While analytic in nature, the question has intimate links to geometry and number theory. Certain cases of Selberg's conjecture are known, but under natural deformations of the surface, most eigenfunctions are not preserved. Recent advances in other fields suggest that we should consider deformations in a "different" direction, which is obscured in the classical setting. We will explain why this might be the case, talk about some applications, and point to questions that such results raise.