Title: Optimal smooth approximation of integral cycles

Speaker: Camillo De Lellis (Institute for Advanced Study, Princeton)

Abtract: Integer rectifiable currents without boundary were introduced in the 60es as a good variational generalization of smooth cycles (smooth oriented submanifolds without boundary) of Riemannian manifolds. In the presentation I will give an idea of the foundational paper of Federer and Fleming, who introduced the concept which later became a powerful tool to tackle variational problems like the existence of submanifolds of least area in every integral homology class. One of the most important points of their work is the so-called Deformation Theorem, which shows that integral cycles can be suitably approximated by polyhedral chains, the classical objects used to define integral homology groups. In particular it is possible to use the theory of integer rectifiable currents directly to define the integral homology of a Riemannian manifold (and of more general objects).

The talk will address to which extent it is possible to approximate integral cycles with smooth submanifolds. A celebrated discovery by Thom in the fifties is that there are integral homology classes which have no smooth representatives. In the eighties Almgren and Browder announced two rather interesting results: integral cycles can be approximated by smooth cycles when there are no topological obstructions, while in general they can always be approximated by cycles with a closed singular set of small dimension. In a joint work with Browder and Caldini we provide a proof of these facts, completing the program sketched in unpublished notes of Almgren and Browder.