Mathematical Sciences: Spectral Theory of Riemannian Manifolds

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The principal investigator will study eigenfunctions of the Laplacian operator on complete manifolds. When the space is compact and there is a differentiable metric, sharp bounds on the Hausdorff measure of the nodal set will be computed. The decay rate of these eigenfunctions will be studied when the space is not compact. Techniques will be developed to handle the general geometry and topology which exists for such manifolds at infinity. The eigenfunctions of the Laplacian operator correspond to the dominant frequencies one might hear if one could hear a surface. If an object in flight emits sound, some of the shape of the object can be recovered by careful analysis of these sound waves. Key to the understanding of such emissions is an analysis of the dominant wave fronts. The principal investigator will build on earlier mathematical work on these problems which applied only to very special cases.