Spaces arising from variational problems oftentimes carry very complicated singularities. Even worse, the regular portions of these spaces, and also the typical analysis on them, degenerate towards these singular set in a still more twisted way. In this talk we introduce a new set of tools, the skin structures, making it, nevertheless, possible to approach many aspects of the delicate geometric analysis near the singularities.

Area minimizing hypersurfaces play an important role in scalar curvature geometry and general relativity. The singularities of such hypersurfaces inhibited to use them in dimensions beyond seven. By means of skin structures this classical dimensional hurdle can be removed. We indicate how to derive results like the Riemannian, the space-time positive mass theorem or the fact that enlargeable manifolds do not admit positive scalar curvature metrics, in arbitrary dimensions.