I'll give two talks on some recent results of mine with Peter Doyle and John Voight.

In 1980 Vigneras used quaternion orders to exhibit pairs of hyperbolic manifolds which were isospectral but not isometric, thereby proving that one cannot hear the shape of a 'hyperbolic drum'. The example which appeared in Vigneras' paper turned out to be incorrect for a technical reason, and was subsequently corrected in her book on quaternion orders. The corrected manifolds turn out to have enormous genus: 100801! The goal of these talks will be to exhibit two substantially simpler examples: a pair of isospectral hyperbolic manifolds of genus 6 and a pair of isospectral orbifolds whose underlying surfaces have genus 0. The key to constructing these examples is the arithmetic of orders in quaternion algebras.

In the first talk I will provide a friendly introduction to orders in quaternion algebras over number fields. In particular I won't presuppose any sort of background knowledge beyond what is normally covered in a graduate algebra course (module theory, field theory..) The goal will be to define and gain familiarity with quaternion orders and then use them in order to produce hyperbolic surfaces. In a certain sense this talk may be seen as providing the background necessary to read and understand Vigneras' Annals paper "Varietes riemanniennes isospectrales et non-isometriques".

In the second talk we will consider a surface constructed from a quaternion order and show how its length spectrum can be studied by examining the rank 2-commutative orders which embed into a quaternion order. This provides a very strong connection between the surface's geometry and the embedding theory of quaternion orders. Indeed, it will be this connection which we will leverage in order to exhibit our nice examples of isospectral but not isometric surfaces. If time permits I will explain why these examples cannot be obtained via Sunada's method