Rosemary Guzman (UIUC) Quantitative Mostow Rigidity: Relating volume to topology for hyperbolic 3manifolds Abstract: A celebrated result of Mostow states that if M, N are two closed, con nected, orientable, hyperbolic nmanifolds which are homotopy equivalent in dimensions $n\geq 3$, then M, N are equivalent up to isometry. This unique geometrictopological relationship has been the framework for many important results in the field, including notable results providing lower bounds on the volume of M, and results relating volume to homology (CullerShalen). In this talk, we will focus on the case where the fundamental group of M has a property, $k$free, for $k\geq5$, and discuss current work toward an improvement on the volume bound from the current known bound of 3.44 which holds for $k\geq 4$. This is joint work with Peter Shalen. 
