Mathematical Sciences: Spectral Theory of Riemannian Manifolds

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The principal investigator will study the eigenfunctions of the Laplacian operator on complete Riemannian manifolds. When the space is compact, he will provide upper and lower bounds for the Hausdorff measure of the nodal set of these eigenfunctions. In the noncompact situation, he will derive lower bounds for the decay of these eigenfunctions. Eigenfunctions of the Laplacian operator are the mathematical representations of harmonics of bells and musical instruments. Peculiar shaped bells can have an intricate assortment of harmonics; often the third or seventh harmonics dominate. In higher dimensions, the shapes of these bells are represented mathematically by Riemannian manifolds. Were these bells to flare out to infinity, the Riemannian manifold would have to be noncompact. The principal investigator will study the decay or damping out of such harmonics.