Geometric Measure Theory
The study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth. A fundamental notion in this subject is that of the "Rectifiability" of a set, which, roughly speaking, entails that the set be locally well approximated by lines or planes.
Recent developments in the field include results which have explored the connections between the rectifiability properties of the boundary of an open set, and the behavior of its associated harmonic measure, or between rectifiability and the behavior of singular integral operators and related tools from harmonic analysis.