\documentclass{article}
\begin{document}
This will be a two part lecture series. In the first lecture, we give an introduction to arithmetic Kleinian groups. In the second lecture, we show that if $M$ is an arithmetic hyperbolic $3$-manifold, then the set of all rational multiples of lengths of closed geodesics of $M$ both determines and is determined by the commensurability class of $M$. This implies that the spectrum of the Laplace operator of $M$ determines the commensurability class of $M$.
\end{document}