Ziwet Lecture I | Morrey's theory and direct methods of the calculus of variations

In the middle of the last century, Charles Morrey introduced the notions of poly-convexity, rank-one-convexity and quasi-convexity functions, to study variational problems. While the first two notions are local conditions, the third one which turned out to be the most important one, is a non local and poorly understood notion. In this talk we comment on Morrey's conjecture and Iwaniec's conjecture on the Beurling-Ahlfors Transform.