The principal investigator will study the relationship between the Laplacian of a compact manifold and the geometry of the manifold. He will attempt to determine the extent to which the spectrum of the manifold determines the class up to finitely many topological types. The principal investigator will also study the relationship between spectrum and other geometric quantities such as diameter and injectivity radius. This award will support research in the general area of differential geometry and global analysis. Differential geometry is the study of the relationship between the geometry of a space and analytic concepts defined on the space. Global analysis is the study of the overall geometric and topological properties of a space by piecing together local information. Applications of these areas of mathematics in other sciences include the structure of complicated molecules, liquid-gas boundaries, and the large scale structure of the universe.
