Roughly, Geometric Measure Theory is the study of geometric properties of sets with measure theoretic tools. This course provides an introduction to Geometric Measure Theory. In large parts, we will adopt a metric viewpoint, which turns out to be often simpler than the classical Euclidean viewpoint, despite its bigger generality. Lipschitz functions play the role of smooth functions in the context of metric spaces. In the first part of the course we will study extendability and differentiability of Lipschitz functions and mappings, as well as their relationship with Hausdorff measures. The second part of the course will be devoted to Federer-Fleming's theory of currents in Euclidean space and to the more general theory of currents in metric spaces developed by Ambrosio-Kirchheim's and by Lang. This provides a suitable and powerful theory of surfaces in the generality of metric spaces. It extends the classical theory of currents in Euclidean space, first developed by Federer-Fleming in the 1960's with the aim to solve the generalized Plateau problem of finding $k$-dimensional surfaces of minimal volume with prescribed boundary. Nowadays, currents (both in Euclidean space and in metric spaces) have applications going far beyond the context of area minimization problems.
- Enseignant·e: Stefan Wenger