By Leah Margolin, Science and Technology Editor
On September 8, 2026, OpenAI announced a proposed solution to a problem surrounding nearly 200-year-old equations: The Navier-Stokes equations. Named for French engineer Claude-Louis Navier and Irish mathematician George Gabriel Stokes, the equations relate Newton’s second law of motion to fluid dynamics. They describe how fluids move by accounting for the way pressure, density, and viscosity change over time.
These equations are instrumental in a variety of fields, such as weather forecasting and aircraft design. In two dimensions, the Navier-Stokes equations have proven to remain well-behaved and stable as time goes on. In three dimensions, however, fluid dynamics become much more complex. Through the use of computer simulations, scientists and engineers have been able to approximate solutions and apply them to real-world problems. What has remained unsolved — until now — are some of the mathematics behind them.
The question of whether the Navier-Stokes equations remain “well-behaved” has perplexed mathematicians since the mid-19th century. It’s not that the equations were unable to be applied; rather, mathematicians wanted to know if, after a finite amount of time, the equations would develop singularities. Would a regular three-dimensional fluid, at some point in time, cause the equations to break down and produce an infinite value? This question was so significant that, in 2000, solving the Navier-Stokes equations became one of the Clay Institute’s seven Millennium Prize problems. And an answer to one of these problems would grant the solver one million dollars.
This question remained unanswered for more than two decades. Then, in September 2026, OpenAI announced a possible solution — after having 10,000 AI agents work on the problem for only 88 hours. OpenAI’s proposed solution is in the shape of a vortex. The fluid begins smoothly under an external force, but as the vortex spins, its speed increases until it reaches infinity. In other words, the Navier-Stokes equations eventually break down and develop singularities.
While this solution is remarkable, it has not been accepted without controversy. Dr. Tristan Buckmaster, a mathematician at New York University, released a statement saying that OpenAI did not start working on the problem until after they heard that he and Levent Alpöge were close to a solution. Before he and Alpöge could publish the related results they had been working on, OpenAI published a complete solution — using a very similar approach, one that Buckmaster says is not the most obvious way to approach the problem. Because Buckmaster and Alpöge stored their drafts in OpenAI’s Codex, this could mean that OpenAI’s agents accessed their unpublished research and used it to come up with a solution.
Buckmaster also alleges that OpenAI pressured him to remove Alpöge as a co-author because of Alpöge’s ties to Anthropic. When he said he would go public, he claims they asked him, “Why would you ruin your career?” Buckmaster finishes his statement by saying “I do not know what their model did [or]… whether our data was used. I am not accusing anyone of anything.”
OpenAI has categorically denied any threats toward Buckmaster, and has stated that “Buckmaster’s Codex prompts… could not have influenced the system in any way.” As of September 16, 2026, OpenAI’s proof is still in review by the Clay Institute, with the Navier-Stokes equations’ status changed from “unsolved” to “active,” and the Clay Institute saying that it has “apparently been settled.”
Beyond the discussion of who said or did what, the heart of the Navier-Stokes controversy is the question of what mathematics is for. Solving the Navier-Stokes equations costs millions of dollars, something most mathematicians do not have available. OpenAI has already indicated that they are willing to spend even more on solving further Millennium problems. This raises concerns that access to the frontiers of mathematics may simply come down to who can afford it.
Even more than questions of accessibility, something much larger is at stake. As a joint statement by 25 Fields Medalists on mathematics and AI put it: “The goals of the AI companies and the goals of the mathematical community are severely misaligned.” The goal of mathematics has never been just to have a bottom-line solution. Rather, the goal of mathematics lies in the journey to reach a solution, from the new questions that arise to the new understanding gained to the mistakes made. And none of this is encapsulated by simply possessing the answer.
So when members of the mathematics community, such as Instagram user Charles Liu, ask, “If AI can solve problems at the frontier of mathematics, where does that leave us?” — the answer may be exactly where we have always been: studying math for its own sake, for the human thinking that no computer can do.
The Fields Medalists’ statement raises another important question: As AI reshapes how the work is done, who makes sure that we do not lose sight of what the work is meant to accomplish? This cannot be left up to OpenAI and the other highest spenders. As solutions to other problems draw closer, mathematicians, the researchers whose work is on the line and even students must work to create transparent guidelines for AI use across the field. If the terms aren’t set soon by those who understand the “why” of mathematics, they will be set by those who only see the “how.”
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