On September 8, 2026, OpenAI announced that one of its internal, unreleased models had solved the Navier–Stokes problem — the question, open for close to a century, of whether the standard equations of fluid flow (the mathematical model underlying weather forecasting and the theory of turbulence) can spontaneously break down. It was only the second of the seven Clay Millennium Prize Problems ever resolved, after Grigori Perelman’s proof of the Poincaré conjecture, and the first of those magnificent seven to fall to a machine. The feat, said to have taken as many as 10,000 AI “agents” working in tandem and many millions of dollars’ worth of computing, caps a dizzying run of ever more impressive achievements by artificial intelligence in mathematics.
For mathematicians, especially those just beginning their careers, such announcements raise an unsettling question: will there still be a place for us? Answering it requires a closer look at what the machine has accomplished and what remains to be understood. The distinction goes to the heart of what it means to do mathematics. In truth, what the machine did was to settle a much weaker version of the problem — one in which the fluid is allowed to be stirred by an artificial external force, contrived for the purpose. It showed that such a force can be added to the equations so that a fluid starting from complete rest is driven, in a finite time, to a breakdown: a moment at which the speed of the flow becomes infinite. This is a substantial technical achievement: the force must remain smooth even as the fluid velocity becomes unbounded. In setting down the official problem, my colleague Charlie Fefferman had allowed that the construction of such a forced example would count as a solution — and hence all the present brouhaha.
In retrospect, Fefferman’s formulation was a mistake. When the problem was posed, most experts believed that a fluid left entirely to itself — smooth at the outset, with nothing stirring it — would go on flowing smoothly forever. That belief is about the real, unforced equations, which the machine’s result does not touch; and in my view it is almost certainly wrong.
Part of the difficulty is that any such breakdown would involve behavior so extreme — infinite speeds concentrated in a vanishing speck of fluid — that it runs against the very assumptions under which the Navier–Stokes equations were derived from physical laws. And indeed, in more than a century of laboratory experiments with real fluids and of numerical simulation, no such behavior has ever been observed; that absence was long taken as evidence that the conjecture is true. The problem has resisted the leading analysts of the past century — Jean Leray, John Nash, Louis Nirenberg, Luis Caffarelli among them — returning deep partial results but no proof. Over time, the very failure to find one convinced many in my generation of the opposite: that the physical laws embodied in the equations are not enough to prevent a breakdown.
As I argued in an essay more than twenty-five years ago, such extreme solutions may well exist, but be so unstable — so easily disrupted — that no experiment or computation could ever detect one. If so, the conjecture that fits what we actually observe is this: every smooth starting state, outside an exceptional set (in a sense that would itself need to be made precise), flows smoothly for all time, with no outside force at all.
Such a result, if true — and I believe it very likely, though it is far out of reach — would give fluid dynamics a firm mathematical framework for future theoretical work, and it would be wonderful if AI could help establish it. So far, nothing has come close.
The most striking recent mathematical achievements of AI have something in common: each produces a particular example, often of extraordinary complexity, through a search guided by ideas developed by human mathematicians, but far more exhaustive than any search they could carry out unaided.
Many young mathematicians I have talked to recently appear to be terrified about the trend. Given the spectacular achievements of AI, they wonder whether they, as research mathematicians, will still be needed. On the present evidence I would tell them: a categorical yes. What these systems have shown is a formidable ability to find a needle once someone else has built the haystack and pointed to it. Choosing which haystacks are worth building — which questions matter, and what a real answer would even look like — is still our work, and it is the more interesting half. And AI is, after all, a triumph of mathematics — its neural networks are built out of linear algebra, calculus and probability.
There are many risks associated with the uses and misuses of AI. The extinction of mathematics as a human endeavor is not among them.