PhD comprehensive exam | Mikhail Panine, Infinitesimal and Finite Methods in Inverse Spectral Geometry

Infinitesimal and Finite Methods in Inverse Spectral Geometry

Abstract

   Spectral geometry is the study of the relationship between the shape of a compact Riemannian manifold and the spectra of differential operators, typically Laplacians, defined on it. Of particular interest is the attempt to recover the manifold from knowledge of the spectrum alone. This is known as inverse spectral geometry. It is well known that the program of inverse spectral geometry taken in its most general form is not possible due to the existence of counterexamples: manifolds with different shape but the same spectrum. Still, a working theory of inverse spectral geometry would have applications in various areas such as shape recognition, computer graphics and quantum gravity. It is thus of interest to circumvent those obstacles by restricting the scope of inverse spectral geometry. This can be done by reducing the studied class of shapes or by adding additional information to the spectrum. Alternatively, it is of interest to quantify the prevalence of counterexamples. In particular, it would be useful to show that they are negligible in some satisfactory sense.

Concretely, I study a linearized version of the inverse spectral problem by both numeric and analytic methods, I attempt to augment the information contained in the spectrum of the Laplacian by giving it a meaningful ordering, I try distinguishing pairs of isospectral non-isometric manifolds by the spectra of their neighboring shapes and, finally, I propose a method that could restrict or perhaps even rule out the existence of certain continuous families of isospectral non-isometric manifolds. I also discuss some tangentially related research projects pertaining to alternative expressions of integrals such as Fourier and Laplace transforms and ultraviolet cutoffs of quantum fields over generic space-times.