Isospectrality: Length vs. Laplace Spectra and Isospectral Families

Texas Tech University, Lubbock TX

Investigators

Abstract

Project Abstract - Ruth Gornet - DMS-0204648 This proposal addresses several topics in inverse spectral geometry. In the first project, the Principal Investigator will show that the classical trace formula, which relates the Laplace and length spectra for generic manifolds, provides less information about isospectral manifolds than was previously speculated. In joint work with P. Perry, the wave trace on Heisenberg manifolds will be explicitly calculated in order to understand the behavior of the length vs. Laplace spectra. Additionally, a new notion of length spectrum will be studied in order to prove a necessary condition that lengths of closed geodesics on isospectral manifolds must satisfy. A further project (joint with J. McGowan) studies the p-form spectrum on lens spaces. The Principal Investigator has constructed examples of lens spaces whose p-form spectra are equal for certain p but with unequal spectra on functions. This behavior will be further studied toward constructing p-isospectral lens spaces with unequal absolute length spectrum; i.e., different lengths of closed geodesics. In the final project (joint with R. Brooks) tools from representation theory of the symmetric groups will be used to construct an explicit upper bound on the number of isospectral Riemann surfaces of a fixed genus that can be constructed from the Sunada method. When this final project is completed, an explicit upper bound on the number of nonisomorphic number fields with a given zeta function will result. In 1966, Mark Kac popularized the question, "Can one hear the shape of a drum?" The mathematical formulation of this question is: ``What geometric information is contained in the spectrum of a Riemannian manifold?'' Isospectrality, i.e., the study of isospectral families and/or the geometric properties they may or may not share, impacts areas outside of spectral geometry; the research funded by this proposal thus supports the pure-mathematical foundations of these areas. The first examples of closed isospectral manifolds, Milnor's flat tori, have appeared in string theory in physics (related to mirror symmetry). The empirical science of spectroscopy has studied frequencies of atoms and molecules to provide information about vibrating objects. Inverse spectral problems also arise in medical imaging, geophysical prospection, and non-destructive testing.