He’s done it again: mathematician Levent Alpöge, at Anthropic, has solved yet another open math problem using artificial intelligence. He had already tackled the Jacobian conjecture and found two new elliptic curves. Now add to these the 1947 Hopf problem. It asks whether a six-dimensional sphere can be completely described using complex numbers, which include the (seemingly impossible) square root of –1. According to Alpöge, the answer is yes, though the result remains unverified.
Some geometric objects, such as a regular sphere (a two-dimensional surface that lives in three-dimensional space), have a “complex structure.” This means that there’s a simple way to assign a different complex number to every point on the sphere. The advantage of this is that many more mathematical tools are available for complex numbers than the usual “real” numbers, which makes some calculations easier. The complex structure on the sphere has had far-reaching uses, such as connecting complicated, ugly equations that live on its surface to simple, polynomial equations.
In a 1947 paper, German mathematician Heinz Hopf posed the question: What about higher-dimensional spheres? He showed that, no, high-dimensional spheres don’t have a complex structure. But he couldn’t prove it for one case: the 6D sphere.
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Ever since, numerous experts have grappled with this scenario. They’ve proposed evidence both for and against the hypothesis that a complex structure exists for the 6D surface of a sphere. But no one has managed to prove their case.
Now AI may have solved the problem. Alpöge used an internal version of Anthropic’s large language model, Claude, to do what humans could not. On August 23 Alpöge posted a 100-plus-page document with a purported proof that the 6D sphere does, in fact, have a complex structure. For the proof, the AI apparently generated a complex 3D structure and demonstrated that it exactly represents the 6D surface of a sphere.
Unfortunately, the full document was nigh-indecipherable—LLMs tend to explain their proofs in unhelpful ways, spending inordinate time on minor details while glancing past important steps.
Mathematician Ilka Agricola at the University of Marburg in Germany says this lack of transparency is typical of tech company announcements: “You don’t know how many prompts were needed to arrive at the result, how much human fine-tuning was required, and so on.” Then, he adds, the mathematical community must verify the claim.
And they got right to work. Within a few days, they’d reconstructed the proof. “There seems to be an emerging consensus that the construction is plausible,” says Duke University mathematician Robert L. Bryant. On August 27 Philip Engel of the University of Illinois Chicago posted a more pedagogical explanation of the LLM’s process on his website. Around the same time, OpenAI researcher Boris Alexeev managed to formalize the proof, verifying its correctness step-by-step using the programming language Lean.
“These examples of ‘new geometry’ have become very rare and precious,” Engel says. “There is a new geometric object in the world that we now know about and can explore.”
Meanwhile, Alpöge has continued to explore other mathematical mysteries. In separate work with mathematician Tristan Buckmaster, he made significant progress toward the Navier-Stokes problem, which made headlines recently when OpenAI claimed to have found a solution.
An earlier version of this article originally appeared in Spektrum der Wissenschaft and was updated with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
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