Hopf problem

The question of whether the 6-dimensional sphere S⁶ has a complex structure is known as the Hopf problem, after Heinz Hopf.^[1] S⁶, when considered as the set of imaginary octonions, with unit norm, inherits an almost complex structure from octonion multiplication, but this is not a complex structure.

In August 2026, Levent Alpöge posted a 108-page paper generated using an internal version of the AI large language model Claude, outlining a direct geometric construction of a complex structure on S⁶. Boris Alexeev formalized parts of the proof using the Lean proof assistant, and peer reconstruction efforts have found the construction plausible, though community scrutiny continues due to the history of failed attempts on this problem since 1947.^[2]

References

↑ Agricola, Ilka; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". Differential Geometry and Its Applications. 57: 1–9. arXiv:1708.01068. doi:10.1016/j.difgeo.2017.10.014. S2CID 119297359.
↑ Manon Bischoff; Joseph Howlett; Daisy Yuhas (September 24, 2026). "AI solves 79-year-old mathematical mystery". Scientific American.

[1] Agricola, Ilka; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". Differential Geometry and Its Applications. 57: 1–9. arXiv:1708.01068. doi:10.1016/j.difgeo.2017.10.014. S2CID 119297359.

[2] Manon Bischoff; Joseph Howlett; Daisy Yuhas (September 24, 2026). "AI solves 79-year-old mathematical mystery". Scientific American.

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References