The Willmore conjecture
  The Willmore conjecture, proposed in 1965, concerns the quest to find the best torus of all. This problem has inspired a lot of mathematics over the years, helping bringing together ideas from subjects like conformal geometry, partial differential equations, algebraic geometry and geometric measure theory.   In this article we survey the history of the conjecture and our recent solution through the min-max approach. We finish with a discussion of some of the many open questions that remain in the field. 
Fernando C. Marques and Andr\'e Neves
math.DG math.AP math.GT