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:

In

`[1]`

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)?

Foundations of Mathematics and Physics One Century After Hilbert: New Perspectivesedited by Joseph Kouneiher. Sadly, copyright law does not permit me to give you a copy or even link to a site where it can be found, but this Wikipedia page can get you started if you wish to follow this path to the Dark Side. – Gro-Tsen Jul 1 at 23:21almostcomplex structure on the $6$-sphere, which, of course, is false. While I haven't gone through it in detail, there are several other unjustified claims that appear, on the face of them, to be false. For example, he supposedly constructs a nontrivial finite group $\Gamma$ that acts on the $6$-sphere and then claims that, because the group $\Gamma$ is 'intrinsic to the $6$-sphere', any (almost) complex structure on the $6$-sphere must be invariant under $\Gamma$. – Robert Bryant Jul 2 at 0:16