Abstract: The sigma-2 equation is the remaining equation in the Monge-Ampere / sigma-n family to be understood. It unclear whether solutions are smooth inside their domains.

With Yu Yuan, we confirm the interior regularity in dimension four. In higher dimensions, we find an interior estimate under a weak condition. The dimension two case is by Heinz in the 1950’s, and dimension three is by Warren and Yuan in the 2000’s.

Our method is pointwise and combines several overlooked ingredients from the past two decades. The idea is to propagate partial regularity using a three-sphere inequality. The method also gives new pointwise proofs of the Monge-Ampere and special Lagrangian equation results.

Biography: Ravi Shankar is an instructor at Princeton University. He obtained his PhD at the University of Washington in 2021 under the supervision of Gunther Uhlmann and Yu Yuan.  His recent research focuses on aspects of fully nonlinear elliptic PDEs.

Zoom: https://us02web.zoom.us/j/83631990429?pwd=TDRNdkMwekI4YWtWR0dDTi9vcllsUT09

Meeting ID: 836 3199 0429    Passcode: 072570