Length Spectrum vs Laplace Spectrum

Intuition drawn from quantum mechanics and geometric optics raises the following long-standing problem: is the length spectrum of a closed Riemannian manifold encoded in its Laplace spectrum? That is, can you hear the length spectrum of a manifold? The answer to this question is known to be positive for a generic (i.e. sufficiently “bumpy”) manifold; however, little seems to be known in general. This is especially true for spaces with large symmetry groups. In this talk we’ll report on the progress we’ve made answering this question for symmetric spaces of the compact type and other homogeneous spaces.