In differential geometry there have been two old conjectures both solved in spring 2012 within 1 month:

1. A proof of the Willmore conjecture by Marques and Neves (arXiv:1202.6036) which states that the Willmore minimizer amongs immersed tori in $S^3$ is the Clifford torus up to Moebius transformation.
2. A proof of the Lawson conjecture on less than 10 pages by Brendle (arXiv:1203.6597): The only embedded minimal torus in $S^3$ is the Clifford torus.

One should note that the proofs of the theorems rely on totally different methods.