Camillo DeLellis: Regular and singular minimal surfaces

Minimal surfaces are surfaces whose area is stationary under smooth perturbations: a well known example is given by minimizers of the area among those which span a given contour and the study of their shape and properties dates back at least to the work of Lagrange in the middle of the 18th century. It is known since long that such objects are, in general, not necessarily smooth and this very fact presents immediately an intriguing challenge: what should we understand with the words "surface" and "area"? The mathematical literature has seen quite a few different approaches, all leading to concepts of "generalized minimal surfaces" which have distinctive features. A pivotal question is when and where these objects are smooth, how large the sets of their singularities can be, and which behavior they can possibly display at the singular points. In my talk I will review a selection of classical works, recent results, and future challenges.
Slides: www.mathunion....