Number Theoretic Methods in Differential Geometry 

Geodesics are an important concept in differential geometry. In non-Euclidean model geometries, it is possible to have closed geodesics, and naturally a question arises. Does the Geodesic intersect itself before it completes its path? Such a geodesic is called non-simple. In this talk, we will explore a construction of real hyperbolic manifolds all of whose closed geodesics are simple. Almost all of the techniques used are number theoretic in nature, utilizing quaternion algebras formed over number fields or (characteristic 0) local fields. Time permitting, we will also discuss current developments in translating this project to complex hyperbolic space.