Inverse spectral problems for Dirichlet-to-Neumann operators

Inverse spectral problems ask the extent to which spectral data encode geometric information.
The Dirichlet-to-Neumann operator of a compact Riemannian manifold M with boundary is a
linear map $C^\infty(\partial M)\to C^\infty(\partial M)$ that maps the Dirichlet boundary values
of each harmonic function f on M to the Neumann boundary values of f. The spectrum of this
operator is discrete and is called the Steklov spectrum. The Dirichlet-to-Neumann operator also
generalizes to the setting of orbifolds. We will compare the behavior of the Steklov spectrum
on smooth surfaces with that of two-dimensional orbifolds and ask whether the spectrum detects
the presence of singularities. In arbitrary dimensions, we will adapt to the setting of the Steklov
spectrum techniques that were originally developed for constructing Laplace isospectral manifolds.