In-Person and Online Talk 

Given a closed manifold, how can one measure its geometric size? Can one construct from it a corresponding interesting metric space?  After surveying related results, I will discuss a recent approach to these questions involving the notion of spherical volume. A version of it was defined by Besson-Courtois-Gallot in their work on entropy rigidity, and it can be understood as a geometric relative of the simplicial volume of Thurston-Gromov. This invariant naturally leads to the construction of certain ``volume minimizing cycles'' in a space of infinite dimension, called Plateau solutions. As an example, we will see that for a closed 3-manifold M, the spherical volume essentially coincides with its hyperbolic volume, and any Plateau solution for M is given by its hyperbolic part. I will try to explain how the spherical volume fits into a broader framework of volumes of unitary representations, and I will mention some open problems.