[Disclaimer: The author is a Mathematics and Statistics Department Academic Intern.]
On Sept. 8, OpenAI claimed their software had generated a solution for the Navier-Stokes equations, one of seven Millennium Prize Problems established by the Clay Mathematics Institute (CMI) in May 2000.
“These problems have been selected by the Clay Mathematical Institute because they stand out as great, unresolved problems of the 20th century,” said mathematician Andrew Wiles during his introduction to the CMI Millenium Meeting in 2000. “They are problems which we believe must one day be resolved…problems which we believe will open up whole new possibilities of mathematics, possibilities which we cannot at this present moment even imagine.” CMI’s Board of Directors pledged for $1 million to be awarded to the first correct solution to each problem.
OpenAI published their proof for their solution to the Navier-Stokes equations, which describes the flow of liquids and gases in a three-dimensional space, on Sept. 8. OpenAI shared that, after hearing rumors that two Millenium Prize Problems had been solved, the company decided to focus a new internal model—which had exceeded expectations of mathematical performance—on the remaining unsolved problems.
After utilizing Codex, OpenAI’s coding tool, to synthesize work from various agent groups, a group of nearly 10,000 AI agents generated a potential solution to the equations in approximately 88 hours. The solution was then verified in Lean, a programming language used to process mathematical proofs and theorems.
Professor and Chair of Mathematics and Statistics Jan Cameron emphasized the importance of the peer-review process. “First of all, it’s not a solution,” Cameron stated in an interview with The Miscellany News. “It’s a claimed solution, and this is really important because the mathematics community needs to be able to decide whether or not it is correct.”
While the proof was generated by OpenAI, the work was built atop years of human research. Mathematicians Tristan Buckmaster, a Professor of Mathematics at New York University, and Levent Alpöge, a researcher at the AI company Anthropic, contributed to what became the rumors that motivated OpenAI’s effort. Additionally, Buckmaster and Alpöge credit the work of Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa for the original findings behind their own results.
Vassar alumna Naima Nader ’23, a current third-year Ph.D. student in mathematics at the University of Illinois Chicago cited a recent panel she attended when explaining to the Miscellany how large language model (LLM) softwares are constantly changing based on what users have fed them. “If you use AI and it gives you an idea that you haven’t seen before, usually that’s just a different human’s idea,” Nader said. “It’s on you to not plagiarize and it’s on you to push AI, to be like, ‘Where is this idea coming from?’”
Buckmaster alleged that OpenAI’s proof used the data from his work in Codex. The company responded in a public statement: “We (the researchers and the agents) did not see any of their work through any means until they released it publicly… While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”
Juliet Whidden ’23, a Vassar alumna and current fourth-year Ph.D. student in mathematics at the Georgia Institute of Technology, noted how this potential usage of intellectual property could impact the mathematical community. “Everyone seems willing to share what they are working on, whether to gain ideas or to just spread knowledge to others,” Whidden wrote by email. “However, it seems that this is exactly how Open AI learned that Navier-Stokes was close to being solved and is what prompted them to solve it.”
This tension between Buckmaster and OpenAI led to a greater discussion on how the interaction between intellectual property and LLMs will affect the notion of authorship in the mathematical research community. Vassar Professor of Mathematics Benjamin Lotto told the Miscellany, “Publications have authors. And so the whole reward structure of academia is now feeling a little wobbly, because if producing a proof is the goal, AI can do that for you.”
Additionally, Professor of Mathematics Adam Lowrance asserted that OpenAI’s effort to solve this problem may have been motivated more by a desire to promote its models’ capabilities than to provide valuable contributions to the mathematical field. “At large, mathematics is thought of as being a particularly challenging and difficult field to have successes in,” said Lowrance. “And so [these companies] are incentivized to produce proofs of famous problems so that they can have press releases that say, ‘Look how far our models have come.’”
“You’re not taking differential equations so you can actually solve them. It’s not just about that,” Cai Gordon-Bruno ’27, a fellow mathematics and physics student, said to the Miscellany. “It’s about understanding the tools you use to solve them and how they actually work mathematically. Because this gives you more context, it opens up your mind to new ideas, to new ways of thinking about maths, which is inevitably what results in new progress.”
Vassar’s mathematics faculty echoed this sentiment. “The whole point of [the Millenium Prize Problems] was to help generate a lot of research and turn one question into two,” Assistant Professor of Mathematics Trevor Hyde stated. “The point of setting these problems up on a pedestal was to help direct mathematics in a positive direction.
The mathematicians in the Vassar community explained that mathematical and scientific research still contains a level of subjectivity, as it is dependent on questions and interests produced by humans. “I feel like math is very heavily defined by what people find interesting,” said Taj Holliday ’27, a mathematics and computer science student. “There are so many things you could prove that are just completely useless, so I think it’s a very human-defined field in that sense, even if it feels like it’s a very objective thing, but it really is people deciding where it goes.”
Vassar Assistant Professor of Statistics Daryl DeFord additionally noted that much scientific research is dependent on the context surrounding its practical applications. “One of the reasons [LLMs have] worked out really well for pure math is you have something like Lean that gets to verify you’ve got the answer right,” DeFord explained. “And for applied problems, modeling problems, statistics data problems, there often isn’t a [unique] right answer. But there’s a lot of room for knowledge about the context to inform the analysis.”
OpenAI shared that they do not intend to claim the $1 million Millenium Prize. The company reflected on the finding’s longer-term implications, writing, “This milestone represents substantial work by mathematicians and AI researchers. However, this is not a culmination, but rather a snapshot in time of progress on AI development.” On Sept. 21, OpenAI launched an advisory group of mathematicians to supervise the verification and publication of any new mathematical results from the internal model that generated the Navier-Stokes proof.
Reflecting on AI’s place in scientific research, mathematics and physics student Josh Garraway ’27 said, “I think as we grow to understand AI and its capabilities better, we stay in control. And I think as long as we understand as people what AI should and maybe can do as well as what people are needed for, like diverse perspectives, like sharing our thoughts and ideas with each other, like collaboration… That piece is not going anywhere.”