Isospectrality: Length vs. Laplace Spectra and Isospectral Families

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Project Abstract - Ruth Gornet - DMS-0204648This proposal addresses several topics in inverse spectralgeometry. In the first project, the Principal Investigator willshow that the classical trace formula, which relates the Laplaceand length spectra for generic manifolds, provides lessinformation about isospectral manifolds than was previouslyspeculated. In joint work with P. Perry, the wave trace onHeisenberg manifolds will be explicitly calculated in order tounderstand the behavior of the length vs. Laplace spectra.Additionally, a new notion of length spectrum will be studied inorder to prove a necessary condition that lengths of closedgeodesics on isospectral manifolds must satisfy. A further project(joint with J. McGowan) studies the p-form spectrum on lensspaces. The Principal Investigator has constructed examples oflens spaces whose p-form spectra are equal for certain p but withunequal spectra on functions. This behavior will be furtherstudied toward constructing p-isospectral lens spaces with unequalabsolute length spectrum; i.e., different lengths of closedgeodesics. In the final project (joint with R. Brooks) tools fromrepresentation theory of the symmetric groups will be used toconstruct an explicit upper bound on the number of isospectralRiemann surfaces of a fixed genus that can be constructed from theSunada method. When this final project is completed, an explicitupper bound on the number of nonisomorphic number fields with agiven zeta function will result.In 1966, Mark Kac popularized the question, "Can one hear theshape of a drum?" The mathematical formulation of this questionis: ``What geometric information is contained in the spectrum of aRiemannian manifold?'' Isospectrality, i.e., the study ofisospectral families and/or the geometric properties they may ormay not share, impacts areas outside of spectral geometry; theresearch funded by this proposal thus supports thepure-mathematical foundations of these areas. The first examplesof closed isospectral manifolds, Milnor's flat tori, have appearedin string theory in physics (related to mirror symmetry). Theempirical science of spectroscopy has studied frequencies of atomsand molecules to provide information about vibrating objects.Inverse spectral problems also arise in medical imaging,geophysical prospection, and non-destructive testing.