# Spaces of manifolds

# Spaces of manifolds

**Online Talk **

***Please note the change in time***

The space of compact smooth submanifolds of infinite-dimensional euclidean space diffeomorphic to a fixed manifold W is a central object in geometric topology. It is to smooth W-bundles as Grassmannians are to vector bundles: it carries the universal W-bundle, and serves as geometric models for the classifying space of the diffeomorphism group of W. As such it is crucial to understand the topology of these spaces.

I will first explain some developments over the last ten years, which have led to a quite satisfactory picture when W is even-dimensional and in the limit as the complexity of W goes to infinity. I will then explain more recent work concerning a very un-complex manifold, namely the disc, where completely different methods are required. I will try to point to an emerging conjectural picture in this case.