Regularity in the calculus of variations and the minimal surface system

Hilbert's 19th problem asks whether solutions of regular variational problems are always analytic. It was solved independently by De Giorgi and Nash in the 1950s. However, variants of this problem continue to have a profound impact. One such variant is the vector-valued analogue, which arises in many geometric and physical applications (elasticity, minimal surfaces of higher codimension, harmonic maps, etc.). I will first review Hilbert's 19th problem. I will then discuss the vector-valued analogue. Finally, I will specialize to the case of the area functional and the minimal surface system. Joint works with J. Hirsch, R. Tione, and O. Savin will be mentioned along the way.