Shing-Tung Yau Paper Credits GPT 6 Astra and Claude Pro in Answering a Curvature Problem He Listed in 1982

The objects at issue date to 1956, when the American mathematician John Milnor constructed a manifold that is topologically identical to the standard seven-sphere but cannot be smoothly deformed into it, a discovery that earned him a Fields Medal. Seven years later, Michel Kervaire and Milnor showed that dimension seven carries exactly 28 smooth structures, written Θ₇ ≅ Z/28Z. One is the ordinary sphere; the other 27 are known as exotic spheres.

The standard sphere has a familiar property: its sectional curvature is positive everywhere, meaning every small patch bends inward from every direction, with no flat or outward-turning points. Gromoll and Meyer made the first step for the exotic cases in 1974, constructing a metric of nonnegative curvature on one exotic sphere, which allows some directions to be flat but none to turn outward. In 2020, Goette, Kerin and Shankar extended this to all 28 spheres, showing each admits a nonnegatively curved metric.

Moving from nonnegative to strictly positive curvature is the step that stalled. In 2008, Petersen and Wilhelm circulated a preprint claiming positive curvature on one exotic sphere, but the proof never passed peer review. Part of the difficulty comes from the differentiable sphere theorem of Brendle and Schoen: if the ratio of largest to smallest curvature at any point and in any direction stays at or below 4, the manifold must be smoothly equivalent to the standard sphere. Positive curvature on an exotic sphere therefore has to be positive without being too uniform, leaving little room for construction.

The new paper's approach cuts each exotic sphere into two halves using the double-disk model of Durán, Püttmann and Rigas, in which each of the 28 spheres is obtained by gluing two seven-dimensional disks along their boundaries, with 28 distinct gluing maps. Rather than building metrics directly on the seven-dimensional disks, the authors work on a ten-dimensional total space of a principal bundle whose fibers are three-dimensional spheres, then push the metric down to seven dimensions by Riemannian submersion. The O'Neill formula guarantees that curvature does not decrease under this projection, so it suffices to make the ten-dimensional metric positively curved. The key technique is to take the radii of the three-sphere fibers extremely small, since thinner fibers bend more sharply and their large curvature can dominate negative contributions from other directions.

The two disks are built asymmetrically. The southern disk uses a curved base with a carefully chosen connection, while the northern disk uses a flat product connection with a warped radial profile. Their boundary data must match: the induced metrics have to agree exactly, and the second fundamental forms on the two sides must sum to something strictly positive in every direction. The paper establishes a common coordinate system on the shared boundary and shows that the derivatives of the fiber radii with respect to the normal direction are opposite on the two sides, so the fiber contributions to the second fundamental form cancel at the seam while the angular contributions add up positively. A gluing theorem of Reiser and Wraith then yields a smooth positively curved metric on the joined sphere. The paper also includes verification code written in SageMath, which the authors rely on because the construction involves a large number of precise curvature inequalities and parameter choices that would be impractical to check by hand.

Yau's public position on AI has shifted over three years. In April 2023, before a lecture at Fudan University, he said AI could not possibly affect the most advanced mathematicians and could only synthesize existing material. In April 2024, at the Zhongguancun Forum, he said AI would benefit mathematics but would not change it. In January, at the World Congress of Chinese Mathematicians, he described AI as changing research efficiency rather than constituting a paradigm shift; in June he said it can solve many nice, cute questions but remains far from a breakthrough that changes mathematics; in July, at an international string theory conference, he argued students should learn to use AI tools early for查找文献 and summarization. The new paper adds a caveat that responsibility for human verification and for writing the paper rests with the authors.

Other leading mathematicians have taken public positions as well. Terence Tao has treated AI as a research assistant for code generation, literature search and formal proof verification, but he joined 25 Fields medalists, including this year's laureate Deng Yu, in a statement warning that AI companies racing to use mathematical problems as benchmarks have drifted seriously away from the goals of the mathematical community; in their view, the point of mathematics is conceptual understanding and insight, with problem solving as a tool. Four days before the paper appeared, Yau made a related argument, saying AI can aggregate existing knowledge and handle routine work, but that people, having obtained the answer they wanted, often stop daring to ask the more important questions.