Special Lagrangian and quadratic Hessian equations

We survey some new and old, positive and negative results on a priori
estimate, regularity, rigidity, and constant rank results for special
Lagrangian and quadratic Hessian equations. These equations originate in

calibrated, convex, conformal, and complex geometries among other fields.

Unlike all other order Hessian equations such as the linear (Laplace) and top

order (Monge-Ampere) equations, the regularity/irregularity and a priori Hessian estimates

for these two equations have not been settled yet in five and higher dimensions.