  Two years ago, F.C. Marques and A.A. Neves implemented, in the framework of
closed rectifiable 2-dimensional currents of the 3-dimensional sphere, a
min-max method in geometric measure theory due to F. Almgren and J. Pitts.
Using this approach they succeeded in proving that the famous Clifford torus
minimizes the area among all closed minimal surfaces of non-zero genus in
$S^3$. Another spectacular consequence of their work is to provide a proof of
the Willmore conjecture. The goal of this talk is to discuss first the general
framework of these two theorems of Marques and Neves. We shall then present the
structures and some key details of their proofs. We will then address the scope
of this remarkable contribution to the calculus of variations on surfaces in a
3-dimensional space.
