The goal of this mathematical research is to understand a partial differential operator, the Laplace-Beltrami operator, defined on compact two-dimensional manifolds. The central role of this operator lies in its geometric significance, or rather, the geometric significance of the associated Dirichlet boundary problem. The differential operator is determined not by the manifold alone, but by the manifold equipped with a specific metric. If the metric is changed, so will the operator and then (possibly) the spectrum. A collection of metrics all giving rise to the same spectrum is called an isospectral set. This work is concerned with the problem of showing that isospectral sets are finite. Progress on this question has been encouraging. While finiteness is still to be established (or negated), recent results have determined that the isospectral sets are compact. In all cases the main tool is a function, the logarithm of the determinant of the differential operator, which is called the height of the metric. What is needed at this stage of the investigation is some form of rigidity theorem for manifolds. There are a number of deep results of this type, but they all fall short of the type needed for this investigation - suggesting that the final resolution is going to be difficult. Another direction this work will take concerns Riemannian manifolds of infinite volume. The Laplacian may have continuous spectrum. Two immediate goals under consideration are that of establishing a Hodge theory for k-forms and the appearance of point spectra embedded in the continuous spectrum. Under fairly general conditions for the two particle Schroedinger operator, this cannot happen. However, for the helium atom, there are such occurrences and the extent of the corresponding eigenfunctions is not well understood.
