Geometric Measure Theory studies the geometric properties of the measure of sets and generalizes concepts of differential geometry to surfaces with singularities. The birth of the subject is usually set by the pioneering works of Besicovitch on the 1-dimensional sets with finite 1-dimensional Hausdorff measure. However the subject boomed in the fifties and the sixties thanks to the genius of mathematicians like De Giorgi, Federer, Fleming, Reifenberg and Almgren, who used methods from measure theory to settle long standing questions in the calculus of variations.
A central question in geometric measure theory is that of minimizing the area among all surfaces with a given contour. This famous question, first raised by Plateau in the eigtheenth century, is the main focus of the works of the five mathematicians mentioned above. Geometric measure theory was developed further by their students and has found beautiful applications in geometric analysis and in the analysis of partial differential equations (see for instance the works of Simon, White, Preiss and Ambrosio).Some ideas have had profound influences in the study of singularities in geometry and analysis.
The field has many "internal" long-standing questions. However, its many applications to other fields of mathematics has stimulated new ideas and generated other interesting open problems. Geometric measure theory is either a field of investigation or a tool in the vast majority of my works.