Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics

The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion

and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.

Calibrated embeddings in the special Lagrangian and coassociative cases

Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.

Optical Control of Magnetic Feshbach Resonances by Closed-Channel Electromagnetically Induced Transparency

Optical control of interactions in ultracold gases opens new fields of research by creating ``designer" interactions with high spatial and temporal resolution. However, previous optical methods using single optical fields generally suffer from atom loss due to spontaneous scattering. This thesis reports new optical methods, employing two optical fields to control interactions in ultracold gases, while suppressing spontaneous scattering by quantum interference. In this dissertation, I will discuss the experimental demonstration of two optical field methods to control narrow and broad magnetic Feshbach resonances in an ultracold gas of $^6$Li atoms. The narrow Feshbach resonance is shifted by $30$ times its width and atom loss suppressed by destructive quantum interference. Near the broad Feshbach resonance, the spontaneous lifetime of the atoms is increased from $0.5$ ms for single field methods to $400$ ms using our two optical field method. Furthermore, I report on a new theoretical model, the continuum-dressed state model, that calculates the optically induced scattering phase shift for both the broad and narrow Feshbach resonances by treating them in a unified manner. The continuum-dressed state model fits the experimental data both in shape and magnitude using only one free parameter. Using the continuum-dressed state model, I illustrate the advantages of our two optical field method over single-field optical methods.