AI cracks decades-old puzzle about complex structures on 6D spheres
Mathematician Levent Alpöge used Anthropic's Claude AI to propose a solution to the 1947 Hopf problem, which asks whether a six-dimensional sphere can be described with complex numbers. The result, posted in a 100-plus-page document, claims such a complex structure exists, though it has not yet been independently verified. If confirmed, it would resolve a long-standing open question in geometry.
The Hopf problem, posed in 1947, asks whether a six-dimensional sphere admits a complex structure—labeling points with complex numbers. Alpöge, who previously used AI for the Jacobian conjecture and elliptic curves, leveraged Claude to generate a complex 3D object purportedly mapping onto the 6D sphere. His August 23 posting contained the full argument.
Initial scrutiny was hampered by the AI's opaque reasoning, lacking clarity on prompts and human input. Within days, Philip Engel produced a clearer exposition, and Boris Alexeev formalized the proof using Lean. Robert Bryant noted an emerging consensus that the construction is plausible, with Engel calling the new geometric object rare and precious.
This development could reshape how mathematicians approach open problems, demonstrating that AI can generate novel geometric constructions that humans then verify. It may accelerate the adoption of formal verification tools like Lean, ensuring rigor in AI-assisted proofs. For society, it underscores AI's potential as a research partner, though it also highlights the need for transparency and independent validation before such results are accepted as foundational knowledge.