9704369 Gordon This project lies in the area of Riemannian geometry. More specifically, the investigator is to consider the structure of isopectral sets of metrics, the possible existence of metrics in Euclidean space with the same scattering behavior, constructions of isopectral metrics whose geometric properties differ in certain ways, and the extent to which the spectrum of a Schrodinger operator determines the potential. The spectral geometry of orbifolds, minimal hypersurfaces and their applications to the geometry of Ricci curvature in dimension three, and the Ricci flow on manifolds with boundary are also to be pursued. Riemannian manifolds are higher dimensional generalizations of curved surfaces. Such manifolds have well-known applications in theoretical physics. A Riemannian manifold possesses a notion of distance, or a metric. And by taking the derivative of the metric one can measure its curvature. The Laplace operator and its spectrum are a fundamental object associated to any Riemannian manifold. Much of this research project is concerned with the question "To what extent is the geometry of a Riemannian manifold determined by the spectrum of its Laplace operator?"
