We present joint work with Ravi Shankar on Hessian estimate and interior regularity for the sigma-2 equation in four dimensions. Our method also provides respectively a Hessian estimate for smooth solutions satisfying a dynamic semi-convexity condition in higher dimensions, and a new proof for the corresponding three dimensional results, which were obtained in the 2007 joint work with Warren. The two dimensional result for the sigma-2 or Monge-Ampere equation was achieved by Heinz in the 1950s. Irregular solutions to the sigma-3 (Monge-Ampere) equation in dimension three were first constructed by Pogorelov in the 1970s.