This work is funded through the Professional Opportunities for Women in Research and Education (POWRE) program as a Visiting Professorship Activity. In 1966, Mark Kac popularized the question, "Can one hear the shape of a drum?" Viewing a drumhead as a plane domain, the frequencies produced when the drum vibrates correspond to the eigenvalues of the Laplace operator. Thus the mathematical formulation of the question posed by Kac is: "Does the spectrum of a plane domain determine its geometry?" Inverse spectral geometry studies the generalization of this question to Riemannian manifolds. This proposal addresses three topics in inverse spectral geometry: (1) the classical trace formula and the length spectrum, (2) spectral rigidity and the Laplace spectrum on 1-forms,and (3) spectral analysis of hyperbolic 3-manifolds. The first two topics concern Riemannian nilmanifolds, which have played a vital role in demonstrating properties not determined by the spectrum. In the first project, wave invariants, constructed using the trace formula, motivate a new notion of length spectrum, which will be studied toward proving a necessary condition that closed geodesics on isospectral manifolds must satisfy. In the second project, one-dimensional invariant subspaces of the Laplacian will be examined to show that the 1-form spectrum detects the Sunada method. In the third topic, the tools of hyperbolic geometry and topology are brought in to elucidate the relationship between spectrum and geometry, particularly on geometrically infinite hyperbolic 3-manifolds that are realized as limits of geometrically finite ones.
