Inserted: 2 jul 2013
Last Updated: 2 jul 2013
Journal: Proceedings of the International Congress of Mathematicians.
In a pioneering work written 30 years ago, Almgren developed a far-reaching regularity theory for area-minimizing currents in codimension higher than $1$. Building upon Almgren's work, Chang proved later the optimal regularity statement for $2$-dimensional currents. In some recent papers the author, in collaboration with Emanuele Spadaro, has simplified and extended some results of Almgren's theory, most notably the ones concerning $\D$-minimizing multiple valued functions and the approximation of area-minimizing currents with small cylindrical excess. In this talk I will give an overview of our contributions and illustrate some possible future directions.