Nonvariational singular elliptic equations and their geometries

In this talk I will discuss fine geometric properties of solutions to fully nonlinear elliptic equations with singular potentials. Singularities occurs along the (unkown) free surface {u=0} and the degree of the singularity determines optimal regularity estimates for u though the singular level set. These classes of equations are related to important free boundary problems that were previously accessed only through variational considerations.