The sigma-2 equation in dimension four

The sigma-k equations are fully nonlinear elliptic PDEs with sigma-1 the Laplacian and sigma-n the Monge-Ampere equation.  Pogorelov's result in the 1970's established that sigma-3+ equations have singular solutions.  With Yu Yuan, we find a Hessian estimate and interior regularity for solutions of the sigma-2 equation in dimension four.  In higher dimensions, we show partial regularity and derive a Hessian estimate for solutions satisfying dynamic semi-convexity.  The dimension two case of the Monge-Ampere equation was done by Heinz in the 1950's.  The dimension three case was done 15 years ago by Warren and Yuan using its minimal surface structure, which is absent in higher dimensions.  Our approach synthesizes several PDE ingredients.