A plume of chimney smoke curls up elegantly into the still evening air. Eddies and whirlpools spin in a stream gurgling over its rocky bed. These familiar forms of fluid flow are among the hardest phenomena to describe scientifically.
It’s not that we don’t understand the physics: the equations governing flow, the Navier-Stokes equations, were written down in the 19th century. They’re basically just Isaac Newton’s laws of motion applied to little parcels of fluid all pushing and deforming one another as they move along. The problem is that the equations are, except in a few cases, impossible to solve with pen and paper – or as mathematicians say, analytically.
Predicting fluid flows is important for all manner of practical problems, from aerodynamics to the turmoil of superhot plasma in nuclear fusion reactors. The equations can be solved “numerically” by computers, but the way they behave mathematically is not fully understood.
It hasn’t even been known if they always have a solution at all – which is to say, one that makes physical sense, with the flow remaining smooth rather than doing something mathematically crazy.
That question was made one of the seven Millennium Prize Problems posed by the private non-profit Clay Mathematics Institute in Denver in 2000. The institute offered prizes of $1m for the first solutions to any of these problems.
Now an answer has been claimed – but not from a living mathematician. Rather, the company OpenAI (which operates ChatGPT) has said that one of its AI models, called GPT-6 Astra, has found an analytical disproof of the smoothness postulate. In other words, there can be situations where, after all, the Navier-Stokes equations blow up, giving the fluid an impossibly infinite speed.
The disproof has not yet been checked and verified by mathematicians. But they are taking it seriously, and if it’s correct, this is arguably the most impactful AI-based solution of a mathematical problem so far. Coming after a slew of other AI-derived solutions to longstanding maths puzzles, this development is sure to leave mathematicians questioning all the more what kind of future they have.
Maths is particularly well suited – or you might say vulnerable – to AI encroachment, because, like chess, it follows well-defined rules, and each of the steps towards a solution or proof can be rigorously checked for correctness. Advances in maths have often come from finding that an obscure discovery or theorem in one subfield can be usefully applied in another – a task for which AI, able to scour the entire digital literature of the subject in an instant, has the upper hand.
Although there is seemingly no rule against AI being used to solve a Millennium Problem, OpenAI has said it won’t be claiming the $1m prize. You have to wonder if that’s partly a gambit to douse the controversy that has erupted around the work.
The OpenAI team says it has been working on the problem using GPT-6 Astra since late August, but stepped up efforts when, on September 1, they heard a rumour that others had cracked it too: specifically, mathematicians Levent Alpöge working at rival company Anthropic, and Tristan Buckmaster at New York University.
As it turned out, the duo instead seemed to have a solution of a closely related problem, which the OpenAI model also solved. OpenAI says it offered the pair a joint announcement of the results. But Buckmaster has alleged that the OpenAI team may have used his work for their own solution, and that they tried to edge Alpöge, from a rival company, out of the story.
The OpenAI team says that it hadn’t seen that work until it was made public. But since Buckmaster and Alpöge themselves used AI models, including Anthropic’s Claude and OpenAI’s Astra, in their own work, the company can’t rule out the possibility that “data derived from their usage of our products helped improve our models.” If so, who’s to blame?
