The general statement in regularity theory for geometric variational problems is of the form: Outside a small singular set, a solution of such a problem is a submanifold. As a specific example the notion of almost area minimizing sets was introduced by Almgren and Taylor subsequently showed that such sets have the structure of soap films as predicted by Plateau in the 19th century.
In this talk we first review part of the history and framework of regularity theory related to almost area minimizing surfaces and later discuss some general strategies used in proofs of such regularity results