What is it?

Geometric measure theory is concerned with investigating the structure of surfaces from a measure-theoretic viewpoint. Since the notion of a surface (in an appropriately general sense) appears in many different settings in mathematics, it is unsurprising that GMT has applications in many areas of modern mathematics including: PDEs, Harmonic analysis and variational problems.

GMT is traditionally considered to be a hard subject. This is primarily because, although many of the ideas involved are simple, in order to work at the level of generality that we do, a lot of technicalities need to be considered and understood in order to prove useful results.

To gain a brief overview of the subject, you may like to look at an article that  the paper online from Springer.