Seminar

It is a longstanding problem to determine whether the length spectrum of a Riemannian manifold can be recovered from its Laplace spectrum. This is known to be true for sufficiently ``bumpy'' manifolds and there are no known counterexamples; however, this problem is largely unexplored for manifolds possessing ``large'' isometry groups. I will demonstrate that the Laplace spectrum of a generic bi-invariant metric determines its length spectrum and the rank of the underlying compact Lie group. Specifically, I show that the so-called Poisson relation is an equality for generic bi-invariant metrics. Furthermore, I will exhibit a large collection $\mathscr{G}$ of compact Lie groups with the property that for any U in $\mathscr{G}$ and any bi-invariant metric g supported by U, the length spectrum of g and the rank of U are encoded in the Laplace spectrum of g. The collection $\mathscr{G}$ includes groups that are either simple, simply-connected, tori or products thereof. Together with the CROSSes, the spaces considered in this talk are the only globally symmetric spaces for which it is known that the length spectrum can be recovered from the Laplace spectrum.