Regularity vs. singularity for the special Lagrangian equation

The special Lagrangian equation (SLE) is a fully nonlinear elliptic PDE that originates in the work of Harvey and Lawson on calibrated geometries. The question whether a viscosity solution to the SLE is smooth (or at least has minimal gradient graph) is delicate, and the answer depends on the Lagrangian phase and on the convexity properties of the solution. We will discuss some examples of singular solutions to the SLE, as well as some sharp conditions that guarantee regularity. This is based on joint works with O. Savin and R. Shankar.