Spectral and geometric problems in global analysis

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Inverse spectral geometry is the study of the extent to which the geometry of a Riemannian manifold can be recovered from spectral data. The principal investigator, along with her research collaborators, will study Inverse spectral problems in both the compact and noncompact settings. For compact Riemannian manifolds, the natural spectral data are the eigenvalues of the Laplacian. To obtain information concerning Riemannian invariants that are not spectrally determined, constructions of isospectral manifolds will be investigated. Continuing joint work with E. Makover and D. Webb, the principal investigator will study the extent to which the spectrum of a Riemann surface determines its Jacobian. She and Webb, along with E. Dryden and S. Greenwald, will also consider inverse spectral results on orbifolds. For noncompact manifolds, natural spectral data include the scattering resonances and scattering phase. The principal investigator and Webb, along with P. Perry, will study of obstacles with the same scattering resonances and scattering phase. They will also study isopolar and isophasal potentials for the Schrodinger operator.In spectroscopy, one attempts to understand the chemical composition of an object such as a star from the characteristic frequencies of light emitted. Analogously, the question phrased by Mark Kac as "Can one hear the shape of a drum?" asks whether one can determine the shape of a vibrating membrane such as a drumhead from its characteristic frequencies of vibration. In earlier work, the principal investigator, along with D. Webb and S. Wolpert, constructed examples of polygonal shaped "membranes" (bounded domains in the plane) that have exactly the same characteristic frequencies, thus answering Kac's question in the negative. Kac's question generalizes to nonplanar surfaces and higher dimensional objects (Riemannian manifolds), with the analog of the characteristic frequencies being the Laplace spectrum. The principal investigator, along with her collaborators, will continue her investigation of the extent to which spectral data determines the geometry of a Riemannian manifold. Constructions of manifolds with the same spectrum will be studied to identify geometric properties that are not spectrally determined. Singular spaces (orbifolds) will also be considered.