The principal investigator will study the relationship between the spectrum of the Laplacian of a manifold and the geometry of that manifold. This will include an investigation of the relationship between the spectrum and geometric properties such as diameter and injectivity radius, as well as a trace formula for Riemannian manifolds. The project will include a study of eigenvalue estimates in the context of obtaining finiteness and compactness results for isospectral manifolds. The problems to be addressed in this project are extensions of Mark Kac's famous question, "Can you hear the shape of a drum?" Mathematicians would ask if the spectrum of sound frequencies radiating from a drum can be used to distinguish between different shaped drums. In two dimensions these frequencies can be used to make such distinctions, but in higher dimensions they can not. The principal investigator will study whether or not other mathematical distinctions between objects other than their exact shapes, can be determined by an analysis of their sound spectrums.
