A couple years ago Atiyah published a claimed proof that $S^6$ has no complex structure. I've heard murmurs and rumors that there are problems with the argument, but just a couple months ago he apparently published a follow-up which fleshes out the details. He writes:
I gave a proof of the long-standing conjecture that the 6-dimensional sphere has no complex structure. In this paper I will present the proof in a more transparent manner. I use the example of the 6-sphere to shed new light on many problems of physics. In the future I expect these ideas will provide a different perspective, with substantial benefits in all areas.
Unfortunately I can't find a version of the new paper in front of a pay wall, and even if I could I would wonder if he had addressed any of the possibly existent problems with the original. Does anyone have more information (or at least references)?