In this lecture, we will introduce some classical techniques from analytic number theory and show how they can be used to count quaternion algebras over number elds subject to various constraints. Because of the correspondence between maximal subelds of quaternion algebras and geodesics on arithmetic hyperbolic manifolds, these counts can be used to produce quantitative results in spectral geometry.

In this lecture, we sketch a proof that there are innitely many k-tuples of arithmetic, hyperbolic 3-orbifolds which are pairwise non-commensurable, have certain prescribed geodesic lengths, and have volumes lying in an interval of bounded length. One of the key ideas stems from the breakthrough work of Maynard and Tao on bounded gaps between primes. We will introduce the Maynard-Tao approach and then discuss how it can be applied in a geometric setting. This talk is based on joint work with B. Linowitz, D. B. McReynolds, and P. Pollack.