Abstract:
Recently, the three-dimensional Ghys conjecture, a world-class problem that has troubled the mathematics community for more than 30 years, was finally solved. This is another case of AI overcoming world-class mathematical problems, but this time it was not an overseas model that solved the problem, but the cooperation of a Chinese team and a large domestic model. According to the Shanghai Artificial Intelligence Laboratory, the team of Xu Disheng, a visiting scholar at the laboratory and a professor at the University of the Greater Bay Area, completed the three-dimensional Ghys conjecture proof through human-machine collaboration based on the Shusheng Science multi-modal large model S2. The team of Professor Dong Bin and Academician Liu Ruochuan of Peking University used the SHEAF framework developed by them to complete formal verification of this proof.
This means that Chinese scientists have used a large Chinese model to overcome a world-class mathematical problem for the first time. But the question that follows is, after AI begins to enter research-level mathematics, how will the research methods of mathematicians be rewritten?
Prior to this, OpenAI had announced that its models had overcome a number of world-class mathematical problems, which caused great repercussions in the industry. The mathematics community has mixed attitudes about this, and the debate on whether AI will change or even harm mathematical research continues to heat up.
Zhao Haiteng, a young researcher at the Shanghai Artificial Intelligence Laboratory, told China Business News that AI has always been a tool to help people increase the upper limit of intellectual production activities they can perform per unit time, and humans are the key players in controlling the value and direction of research. In the end, this collaboration will overall increase the speed of human progress in mathematical research.
How large models can overcome mathematical problems
The Ghys conjecture cracked this time was proposed by French mathematician Étienne Ghys in 1995. It is one of the important issues in the field of complex dynamic systems. Its core is: whether the existence of chaotic dynamic system behavior can determine geometric classification. If this conjecture is true, the problem of chaotic dynamic trajectories can be "reduced" into a concise and regular standard algebraic model.
In the interview, Xu Disheng believed that, like many mathematical problems, cracking the three-dimensional Ghys conjecture does not mean that it will have potential application value in the next one or two months. Instead, we have thought about this problem and invented new methods and new tools, which will provide new help for subsequent mathematical research and even scientific research.
Xu Di has been deeply engaged in the research of three-dimensional Ghys conjecture for a long time. He has collaborated with his mentor and Fields Medal winner Professor Artur Avila to study the three-dimensional case, but no effective proof ideas have been formed. The Shanghai Artificial Intelligence Laboratory has been exploring the shift of model capabilities from competition level to more difficult research-level mathematics, using hard-core mathematical problems to force iteration of model capabilities.
With the same goals of both parties, in early 2026, Xu Disheng’s team and the Shanghai Artificial Intelligence Laboratory launched model training cooperation research. Xu Disheng mentioned that in the early stage, he participated in the Shusheng-S2 model’s suggestions on some selection of training corpus, benchmark (baseline) and other target directions.
The turning point came at the end of July. "When the model was roughly upgraded to the point where we felt we had a chance to make something, we began to test some different mathematical problems on a large scale, hoping to see changes in the depth and breadth of the thinking it produced. I found that the system began to become stronger." So Xu Disheng also fed the Ghys conjecture to the AI, and he found that some key feedback was very different from before.
Xu Disheng invited student Zhang Jiesong to join the team to further tackle the problem. Zhang Jiesong currently works at the Royal Institute of Technology in Sweden. Ghys guessed that this was an issue that he had been paying attention to since he was studying for his doctorate, but he had been stuck on a few key points.
In the Shanghai Artificial Intelligence Laboratory, Zhang Jiesong tried to ask the model about the stuck key points. "Compared to humans, the characteristic of models is that they have a broader accumulation of cross-domain knowledge. AI has output a lot of ideas and ideas from differential geometry research that I am not familiar with."
Zhang Jiesong mentioned that for the first round of reasoning of the three-dimensional Ghys conjecture, Shusheng-S2 produced a 400-page derivation report. Some of them are very enlightening, and some ideas are unreasonable. At this time, people need to do further screening and prompting.
From the detailed report provided by Scholar-S2, the team seized some key breakthroughs and continued to provide feedback to the model and calibrate the direction of exploration. Scholar-S2 has completed 7 batches of continuous deductions, expanded the problem graph from 15 nodes to 389, and output a total of a thousand pages of derivation. After several rounds of screening and prompting, they slowly developed methods to overcome key difficulties, and finally pieced together a complete chain of proofs to crack the three-dimensional Ghys conjecture.
Shanghai Artificial Intelligence Laboratory believes that the significance of this research is to verify a new scientific research paradigm. Traditional mathematical research relies on the long exploration of individual geniuses, but this research combines the keen intuition of experts with the solution loop of large models, forming a link of systematic exploration, expert judgment, achievement accumulation, and feedback correction. This paradigm can be replicated and extended to more basic research fields where the search space is huge and experts are scarce, to achieve more original innovative breakthroughs.
AI rewrites mathematical research
After AI has demonstrated its ability to solve mathematical problems, a more realistic question is: How will it change the way mathematicians conduct research?
Zhang Jiesong told reporters that the model of human and AI collaboratively solving mathematical problems is a better exploration. "In the pre-AI era, asking questions, answering, verifying, understanding, and applying proved to be the most difficult path. Now with the help of AI, what is more important for people is how to ask questions, how to understand, and how to use it deductively."
For example, he said that if you ask an ancient person "what to eat", you may be asking how they fill their stomachs, but if you ask a modern person "what to eat", you may be more concerned about issues such as taste and nutrition. A question will have different meanings under different technical conditions and living environments. This may also be a change in future mathematical research.
In Zhang Jiesong's view, mathematical problems are more like a benchmark. By solving this problem, the understanding of some mathematical objects of interest can be improved. But solving the problem is not the ultimate goal, what is more important is further application and deduction.
Zhao Haiteng, a young researcher at the Shanghai Artificial Intelligence Laboratory, believes that in AI for Math, AI is always positioned as a tool. He believes that AI will not compete with or replace human researchers. In the current cooperation model between human mathematicians and AI, humans are the key player in controlling the direction of research progress, and AI only accelerates this progress.
"A person can only devote a limited amount of time to thinking about a problem every day in a year, but AI can be completed in one day by expanding the computing power and reasoning scale. This can free up more people's time to explore more meaningful and higher-value problems." Zhao Haiteng said.
If this collaborative approach continues to mature, the range of problems that mathematicians can explore may expand, and research directions that have been shelved in the past due to high time costs may also regain opportunities.
The three-dimensional Ghys conjecture is not an isolated case of AI entering cutting-edge mathematical research. In the past period, the mathematical research results released by OpenAI have triggered continued discussions about the boundaries of AI capabilities.
In September, OpenAI announced that its AI model took 88 hours to prove the existence and smoothness of the Navier-Stokes equation, causing shock in the mathematics community. Subsequently, on October 6, OpenAI released 722 mathematical manuscripts generated by internal cutting-edge models at once, saying that its model can solve hundreds of mathematical problems, including long-standing problems such as the quasi-Riemann hypothesis and the four-dimensional Kakeya problem.
French mathematician and Fields Medal winner Cedric Villani once commented that the changes brought about by AI may be one of the most fundamental changes in the field of mathematics in at least four centuries, or even a thousand years.
But not all mathematicians are optimistic about the change. On September 11, 25 Fields Medal winners including Terence Tao jointly issued a statement, pointing out that AI companies' practice of using solving mathematical problems as a benchmark for ability may harm mathematical science and its community.
They worry that as AI companies continue to pursue the number and speed of solving famous problems, mathematicians may not have enough time to verify results, identify new methods, clarify the contributions of existing research, and truly integrate new results into the mathematical knowledge system. The impact of achievement attribution, citation standards, and AI-generated results on academic exchanges and talent cultivation also needs to be taken seriously.
The statement acknowledges that AI has the potential to accelerate mathematical research, but its ultimate impact depends on how technology developers and the mathematical community use and manage this technology.
At a public event on October 11, Qiu Chengtong, chair professor of Tsinghua University, used the term "tsunami" to describe the impact of AI on mathematics research. He said that the recent progress of AI has far exceeded expectations and advocated active participation in this change. "The better AI does, the more helpful it will be to us. We need to surf the entire wave instead of hiding to avoid it."
From ancient times to the present, the development of basic science has often been driven by a few people, and AI is a comprehensive tool that aggregates the thinking of mankind from history to today through powerful computing power. However, Qiu Chengtong also reminded that the rational use of AI may bring epoch-making knowledge, achievements and impact; if it is not handled properly, it may also cause serious consequences.
After AI begins to produce proofs, what humans need to answer is how to turn these proofs into truly mathematical knowledge.
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