Almgren’s Q-Valued Functions Revisited

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 Dir-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. Mathematics Subject Classification (2000). Primary 49Q20 ; Secondary 35J55, 54E40, 53A10 .