Fernando Coda-Marques
Together with Andre Neves he proved the Willmore conjecture. This states that the Clifford torus in the 3-sphere minimizes the Willmore energy, with energy $2\pi^2$. To prove this conjecture, they show that for 5-parameter sweepouts of $S^3$ by surfaces, with certain boundary conditions, the min-max area is $2\pi^2$. This theorem has other consequences besides the Willmore conjecture.
