Regularity of area-minimizing surfaces in higher codimension: old and new

Camillo De Lellis, Institut fuer Mathematik Universitaet Zuerich
Fine Hall 314

The theory of integral currents, developed by Federer and Fleming in the 60s, gives a powerful framework to solve the Plateau's problem in every dimension and codimension. The interior and boundary regularity theory for the codimensionone case is  rather well understood, thanks to the work of several mathematicians in the 60es, 70es and 80es.
In codimension higher than one the phenomenon of branching causes instead very serious problems. A celebrated monograph of Almgren provided a far-reaching interior regularity theorem. However, his original (typewritten) manuscript was more than 1700 pages long. Four years ago, in a series of works with Emanuele Spadaro  we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program. This has allowed us to go beyond Almgren's result in several directions: in this talk I will discuss some of the most recent achievements.