Arithmetic Aspects of Hyperbolic Knots and their Dehn Surgeries

A consequence of Mostow-Prasad rigidity is that the trace-field of an orientable finite volume hyperbolic 3-manifold M is a number field. In addition, there is a quaternion algebra over this number field that is also a natural invariant associated to M. This talk will review this background, and survey recent advances in trying to understand the trace-field and this quaternion algebra, especially in the setting of Dehn surgery on a hyperbolic knot. If time permits, we will discuss an intriguing connection concerning the existence of an Azumaya algebra over "the canonical curve" and the Alexander polynomial of the knot.