In this talk, we will discuss several mathematical and computational questions around the representation of geometric information distributed in some ambient space and the underlying Riemannian structure of the space of such representations induced by the use of diffeomorphic transport. Most of the mathematical developments have been fuelled by the need to analyze biological shape variations in neuroimaging leading to the new field of computational anatomy. However, current descriptions of brain diseases usually need to put together several orders of magnitude ranging from the millimeter scale for tissues in standard imaging devices to the micron or even nano scale for neural cells and molecules. I will present some of the new challenges for mathematicians and our recent attempt to make a step in that direction.