（転載）
http://rio2016.5ch.io/test/read.cgi/math/1783377448/189
多変数解析函数論6
189 ：１３２人目の素数さん[]：2026/08/26(水) 22:01:14.79 ID:yBc7uIlp
78年間解決されなかった世界的な難題が、たった3日で解かれてしまった！
1948年から今日に至るまで、六次元球面上に複素構造が存在するかどうかという問題について、今や答えは「存在する」である。
この回答を提出したのは、ハーバード大学の数学者レヴェント・アルポゲとクロードである。

(参考)
AI Solves 78-Year-Old Mathematical Conjecture on the ...
KuCoin
https://www.kucoin.com › News
3 日前 — The question of whether a complex structure exists on the six-dimensional sphere has been asked since 1948, and now the answer is yes. The ...

六次元球面上に複素構造が存在するか
(google訳)
Does a complex structure exist on the 6-dimensional sphere?
＜検索結果＞
AI による概要
complex structure exists on the 6-dimensional sphere (\(S^{6}\)) is historically one of the most famous open problems in differential geometry, and its status is currently a subject of intense active verification.

Hopf problem—remained entirely unsolved. However, in August 2026, mathematician Levent Alpöge announced a purported proof asserting that a complex structure does exist on \(S^{6}\). Because this 100+ page construction relied heavily on AI assistance (specifically using Claude), and given the problem's long history of flawed or retracted proofs, the mathematical community is currently rigorously scrutinizing the paper to see if it holds up.

The Context and Core Challenge
To understand why this problem is so difficult, it helps to break down the distinction between an almost complex structure and a true complex structure:

・The Topological Constraint \(S^{2}\) and \(S^{6}\) only): In 1953, Armand Borel and Jean-Pierre Serre proved that among all even-dimensional spheres, only \(S^{2}\) and \(S^{6}\) can admit an almost complex structure (a smoothly varying way to multiply tangent vectors by \(\sqrt{-1}\)).

・The Octonion Connection on \(S^{6}\): The standard almost complex structure on \(S^{6}\) is beautifully constructed using the multiplication rules of octonions (8-dimensional numbers).

・The Integrability Failure:For an almost complex structure to be a true complex structure (allowing \(S^{6}\) to be treated as a complex manifold with holomorphic coordinates), it must be integrable. This requires its Nijenhuis tensor to vanish. The famous octonional almost complex structure on \(S^{6}\) is not integrable, leaving the core question open.

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