Master's defence | Explorations of Infinitesimal Inverse Spectral Geometry

It has recently been proposed that inverse spectral geometry could be the missing mathematical link between quantum field theory and general relativity needed to unify those theories into a single theory of quantum gravity. Unfortunately, this proposed link is not well understood. Most of the efforts in inverse spectral geometry were historically concentrated on the generation of counterexamples to the most general formulation of inverse spectral geometry and the few positive results that exist are quite limited. In order to remedy that, it has been proposed to linearize the problem, and study an infinitesimal version of inverse spectral geometry.

In this thesis, we begin by reviewing the spectral theorem for elliptic operators. We then survey positive and negative results in inverse spectral geometry. Finally, we introduce a formulation of inverse spectral geometry adapted for numerical implementations and apply it to the inverse spectral geometry of a particular class of star-shaped domains in R^∧2.