Constant rank theorems for elliptic PDEs

One of the most important and useful features of linear elliptic equations is the maximum principle which asserts that solutions attain their maximum values on the boundary of the domain. Solutions of fully nonlinear equations enjoy a similar feature for their Hessians provided the equation satisfies a convexity condition. Results of this type are known as constant rank theorems. In this talk we will introduce the constant rank theorem of Caffarelli-Guan-Ma, discuss a new approach to proving this result, and mention its application to the special Lagrangian equation.