Rectifiability for points with positive Alt-Caffarelli-Friedman limit

 In this talk we study the interface between two nonnegative subharmonic functions. Our interest concerns the differentiability of the interface. We will review the Alt-Caffarelli-Friedman (ACF) monotonicity formula which is a useful tool for studying the interface. We utilize a new quantitative version of the Alt-Caffarelli-Friedman (ACF) monotonicity formula. With this quantitative version available, we apply the recently developed techniques of Naber and Valtorta (which they applied to the singular sets of PDEs). We are able to conclude that the portion of the interface on which the ACF formula is positive is an H^{n-1} rectifiable set. Moreover, for H^{n-1} almost every such point, the interface has a unique tangent. This is joint work with Dennis Kriventsov and Robin Neumayer.