Inverse spectral problems ask how much informationon about an object is encoded in spectral data. In the setting of Riemannian manifolds, spectral data are provided by the eigenvalues of the associated Laplacian. We will consider methods for constructing Riemannian manifolds with the same eigenvalue spectrum and compare their geometry. We will also refer to related constructions on discrete and quantum graphs.
Originally published at math.nd.edu.