The theory of currents, developed in the '70 by Federer and Fleming, provides weak solutions (area-minimizing currents) to Plateau's problem with no restriction on dimension and codimension. The regularity theory of area-minimizing currents, besides its intrinsic interest, has been the source of inspiration for many regularity theorems in elliptic and parabolic partial differential equations even in a non-geometric context. The regularity theory of area-minimizing currentsstarted with the seminal work of De Giorgi for codimension one currents, namely weak hypersurfaces, and culminated in a monumental work (even in terms of size) by F.J. Almgren, who established an optimal result for currents of arbitrary codimension. In the last few years Almgren's work has been revisited, improved and streamlined in a series of papers by De Lellis and Spadaro. The seminar will describe these recent developments, emphasizing the key technical ideas.

[After Almgren-De Lellis-Spadaro]