  The unification of general relativity with quantum theory will also require a
coming together of the two quite different mathematical languages of general
relativity and quantum theory, i.e., of differential geometry and functional
analysis respectively. Of particular interest in this regard is the field of
spectral geometry, which studies to which extent the shape of a Riemannian
manifold is describable in terms of the spectra of differential operators
defined on the manifold. Spectral geometry is hard because it is highly
nonlinear, but linearized spectral geometry, i.e., the task to determine small
shape changes from small spectral changes, is much more tractable, and may be
iterated to approximate the full problem. Here, we generalize this approach,
allowing, in particular, non-equal finite numbers of shape and spectral degrees
of freedom. This allows us to study how well the shape degrees of freedom are
encoded in the eigenvalues. We apply this strategy numerically to a class of
planar domains and find that the reconstruction of small shape changes from
small spectral changes is possible if enough eigenvalues are used. While
isospectral non-isometric shapes are known to exist, we find evidence that
generically shaped isospectral non-isometric shapes, if existing, are
exceedingly rare.
