Consider the family of Laplace-Beltrami operators Δ(g) on a given smooth, closed manifold M, parametrized by the choice of a smooth metric g on M. How big is the set of metrics g such that Δ(g) admits an eigenvalue of multiplicity m or higher?

An analogy with the space of nxn matrices, first observed by Wigner and von Neumann in 1929, suggests that this set should have codimension m(m+1)/2-1. This analogy can be made precise and shown to hold for all metrics away from an exceptional set of infinite codimension. In the case m=2, this is due to Karen Uhlenbeck, whose proof however does not extend to m>2. The analysis of higher multiplicities revolves around an overdetermined spectral problem akin to that of critical metrics for Laplace-Beltrami eigenvalue functionals. Time permitting, we will discuss the analogous question for the Dirichlet Laplacian and the curl operator on smoothly bounded domains.