Abstract: Given a manifold X and a manifold M that is embedded in Euclidean space R^N, we can consider functions in some Sobolev space from X to R^N whose values lie in M almost everywhere. These are called Sobolev maps from X to M. They can arise naturally in geometry, and can be thought of as maps from X to M that may have singularities. I will give an expository talk on what has been learned about these maps, from results of Schoen and Uhlenbeck from the 80s to recent results by Brezis and Mironescu.
One fundamental question concerns smooth approximation. While Sobolev maps can always be approximated by smooth maps from X to R^N, they might or might not be able to be approximated by smooth maps from X to M. Whether or not we have smooth approximation turns out to depend both on the strength of the Sobolev space and on the topology of the target space M. This question is deeply connected to the question of whether taking a limit can "break" the topology. Namely, we know that continuous maps from X to M induce maps between the homology groups of X and M. Sobolev maps will do so as well if the Sobolev space is strong enough. But as the Sobolev space becomes weaker, Sobolev maps may only induce maps on homology below a certain degree.
Host: Xiang Tang