In 1966, Mark Kac popularized the question, ``Can one hear the
shape of a drum?'' Viewing a drumhead as a plane domain, the
frequencies produced when the drum vibrates correspond to the
collection of eigenvalues of the Laplace operator acting on smooth
functions. We call the collection of eigenvalues the spectrum of
the domain. Thus the question actually posed by Kac is: ``Does
the spectrum of a plane domain determine its shape?'' Only recently
has this question been answered in the negative. An analogous
question for Riemannian manifolds, generalizations of plane domains,
is: ``What geometric information is contained in the spectrum of
a Riemannian manifold?'' That is, how much geometry can be ``heard?''
Despite considerable research in the area, only a few geometric
properties are known to be spectrally determined: dimension, volume,
and total scalar curvature being prime examples. The only way to
show a specific geometric property cannot be ``heard'' is by
constructing pairs of isospectral manifolds that differ by this
property. The primary obstacle is that the spectrum is rarely
computable, forcing us to rely on other methods to produce examples
of isospectral manifolds. My interests lie in the areas of constructing
isospectral manifolds, the Laplace spectrum on differential forms,
and comparing the geometry of isospectral manifolds, particularly
the length spectrum and marked length spectrum.
The Laplace--Beltrami operator of a Riemannian manifold M may be extended to act on smooth p-forms, where p lies between 1 and the dimension of M. We call its eigenvalue spectrum the p-form spectrum. The length spectrum of a Riemannian manifold is the set of lengths of closed geodesics, counted with multiplicity. The marked length spectrum is similar, but also records the free homotopy classes of loops in which the geodesics occur.