This lecture reports on joint work with Christopher Judge and David
 Determinants of Laplacians are nonlocal spectral invariants which carry
 important geometric information about Riemannian manifolds and have played
a crucial role in spectral geometry. Osgood, Phillips, and Sarnak defined
 and analyzed determinants of Laplacians for compact surfaces and planar
 domains, and used them to partially characterize the set of all
 surfaces or plane domains with a given spectrum. They use a spectral zeta
 function, formed from the eigenvalues of the Laplacian, to define the
determinant, and use heat kernel asymptotics to prove some its key
  In this lecture I will describe how determinants of Laplacians are being
 used to study inverse scattering problems on non-compact Riemann surfaces
 and certain perturbations. The discrete data analogous to the eigenvalues
of a compact surface are the scattering poles for the Laplacian,
 a discrete set of complex numbers which solve a non-selfadjoint eigenvalue problem. We
use a Green's function approach to define the determinant and use recent
 work of Patterson and Perry to connect the determinant for convex
 co-compact Riemann surfaces with Selberg's zeta function. Perturbation
 theory, entire function theory, and fine estimates on the resolvent and
 scattering operator for two-dimensional Riemann surfaces due to Guillope
and Zworski enable us to elucidate the geometric information contained in
 the determinant. Using the determinant as an `isopolar' invariant, we are
able to prove a compactness theorem for `isopolar' manifolds analogous to
the one proved by Osgood, Phillips, and Sarnak for compact surfaces

 1. David Borthwick, Christopher Judge, and Peter Perry.
 Determinants of Laplacians and isopolar metrics on surfaces of infinite
area. Submitted to Duke Math. J.; available as a preprint at

2. Laurent  Guillope, Maciej Zworski. Scattering asymptotics for
Riemann surfaces. Ann. Math. 145 (1997), 597-660.

3. S. J. Patterson, Peter Perry. The divisor of the Selberg zeta function
 for Kleinian groups. Duke Math. J. 106 (2001), 321-390.