AI Has Solved One of Math’s $1 Million Millennium Prize Problems

On the morning of Tuesday, September 8, mathematicians at OpenAI announced that a group of 10,000 autonomous AI agents under their direction, running on an advanced model not available to the public, had found a “singularity” in the Navier-Stokes equations in three dimensions — thus resolving one of the six remaining Millennium Prize Problems posed in 2000 by the Clay Mathematics Institute, each of which carries a $1 million prize. Their result has been formally checked in the programming language Lean, giving mathematicians confidence that it is indeed correct. If the result holds up to further scrutiny, it is, by a significant margin, the most important mathematical proof to have been arrived at by an artificial-intelligence model to date, possibly marking a fundamental turning point in how mathematicians tackle difficult problems.

This particular difficult problem deals with differential equations, which express relationships between changing quantities. They are arguably the single most important mathematical tool for explaining the world around us. As a rule, they are easy to write down and hard to solve.

The Navier-Stokes equations are differential equations that use Newton’s second law of motion to describe how fluids, from ocean currents to air flows, behave. They were first written down in the mid-19th century, and have been central to the study of fluid mechanics ever since. But one basic question about the equations has persisted: Are their solutions always well-behaved?

Or can their solutions evolve over time so that some infinitesimally small part of the fluid begins to flow infinitely quickly, creating a so-called singularity? The OpenAI announcement of this long-sought singularity came 12 hours after an announcement from Tristan Buckmaster at New York University that he, together with Levent Alpöge at Anthropic, had resolved several closely related problems with help from a variety of AI models, including those of OpenAI. Both AI-enabled teams relied heavily on work by Diego Córdoba of the Institute for Mathematical Sciences in Madrid and Luis Martínez-Zoroa of CUNEF University, researchers who had developed a strategy to attack the problem that radically departed from the methods most mathematicians were using.

The heroes of the story, he said, are Córdoba and Martínez-Zoroa. As Buckmaster wrote in a statement announcing his results, “Let me make plain what I have said to colleagues in private: in view of this body of work, I believe Luis Martínez-Zoroa deserves a Fields Medal.” The Navier-Stokes equations rely on the assumption that you can zoom in on a fluid, considering endlessly smaller amounts of it. The real world is not like this: Fluids are ultimately made of molecules and atoms.

They are not perfectly smooth. This means that the mathematical results about the formation of singularities don’t have any immediate practical consequences. However, those results are important because it’s surprising that such singularities are possible even in an idealized sense.