Recent progress on the Dirichlet problem for the minimal surface system and minimal cones

Abstract: This is a very brief report on recent developments on the Dirichlet problem for the minimal surface system and minimal cones in Euclidean spaces. We shall mainly focus on two directions: (1) Further systematic developments after Lawson-Osserman's paper \cite{l-o} on the Dirichlet problem for minimal graphs of high codimensions. Aspects including non-existence, non-uniqueness and irregularity properties of solutions have been explored from different points of view. (2) Complexities and varieties of area-minimizing cones in high codimensions. We shall mention interesting history and exhibit some recent results which successfully furnished new families of minimizing cones of different types.