*Special Seminar*

In this talk, I will give an overview of the regularity theory for minimal hypersurfaces which are stable (on their smoothly embedded part). We will start with the theory of Schoen—Simon and end with more recent developments. The regularity theory of stable minimal hypersurfaces is particularly important for establishing existence theories, and indeed the theory of Wickramasekera forms an important cornerstone in the Allen—Cahn existence theory of minimal hypersurfaces in closed Riemannian manifolds (as an alternative to the Almgren—Pitts theory). Furthermore, in recent years similar ideas have been used by Bellettini—Wickramasekera to establish a powerful existence theory for CMC and PMC hypersurfaces. 

The first half of the talk will be focused on the background, including the results of Schoen—Simon, and Wickramasekera. The remainder of the talk will then be focused on recent work by the speaker extending the regularity theory to allow for certain types of so-called classical singularities. One key application of some of these results is to understanding the structure of area minimising hypercurrents modulo p about their branch points. Some of these results are joint with Neshan Wickramasekera.