It is well-known that submanifolds of least area for a fixed boundary (Plateau problem) or in a fixed homology class (homological Plateau problem) shall not be smoothly embedded in general, but rather exhibit a singular set (as first noted by Simons and then justified by Bombieri-De Giorgi-Giusti half a century ago). The first singular example(s) of minimizers were in fact extremely rigid: cones with an isolated singularity at the origin. As it is now clear, the occurrence of singularities is an intriguing and partly elusive pathology that may be imputable to diverse causes, ranging from topological obstructions (related e.g. to pioneering work by Thom) to basic complex-analytic phenomena. But how wild may the singular set possibly be, and how frequently will it be observable as one varies the boundary in question or, respectively, the background metric? Over the past five years we have witnessed striking advances on both fronts. In this lecture I will present the general state of the art and my contributions to the latter question(s), known as the generic regularity problem, as well as some surprising geometric applications. Based on joint works with Yangyang Li and Zhihan Wang.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006