The Machine That Cracked a Millennium Problem: Navier-Stokes and the Mathematical Leap of AI

Can artificial intelligence truly be creative, or is it only a machine that remixes what humans already produced? Physicist Brian Greene opens the World Science Festival conversation with exactly this question and recalls how the benchmark kept shifting through history: from the Turing test of conversation to essay writing, from standardized exams to the International Mathematical Olympiad problems solved in the summer of 2025. Every bar was cleared, yet the bar moved further each time.

On the horizon stands a target of a different scale: the seven Millennium Problems, each carrying a one-million-dollar prize announced by the Clay Institute in 2000. A quarter century later only one of the seven has fallen and six remain open. According to the official ClayMath problem list, these seven rank among the deepest open questions in mathematics, and the Institute keeps a dedicated page for each of them.

One of them concerns the Navier-Stokes equations that govern how fluids move. From water in a pipe to air swirling above the earth, from ocean currents to blood inside vessels, these equations produce wonderfully accurate predictions in practice; the engineer designing an aircraft wing and the surgeon developing a heart valve trust the same mathematics.

The secret lies in their nonlinearity : the flow affects the flow itself, interacting with its own motion. In linear systems two solutions can be added to make a new one, but here that pleasant property disappears. The speaker stresses that analysis of linear structures such as the Schrodinger equation is far easier for this reason; real-world fluids offer no such comfort.

The infinite-speed question

Can a flow that starts smoothly develop a singularity in finite time, with velocities growing without bound? Since no real fluid can move infinitely fast, such an outcome would mean the model breaks down. As mathematics writer Kolen explains in a technical survey, blowup scenarios contribute to physics precisely by showing where the equations collapse.

Two families of equations must be kept apart here: the frictionless idealization of the Euler equations and the realistic Navier-Stokes with friction included. Tarek Elgindi's singularity result for Euler sat one notch below full smoothness, and the community only later grasped how critical that gap was; for the speaker this is a classic case of selective scientific memory.

Physical intuition says friction should brake the blowup, yet the non-uniqueness work of Buckmaster and Vlad Vicol treated viscosity as little more than a nuisance. Friction makes blowup harder, but if the nonlinear mechanism is strong enough the equation still runs out of control; everything hinges on which mechanism dominates.

Computers take a seat at the proof table

Computers entering proofs is nothing new: in the four-color theorem and the Hales proof of the Kepler conjecture, pen and paper reduced the problem to finitely many computations that machines finished. Such results were sniffed at by pure mathematicians when first published, yet the method earned legitimacy over time; today's debate reads like that old suspicion returning with far stronger hardware.

The innovation Buckmaster tried with collaborator Levent Alpoge over the past year was to ask not one model but harnessed AI agents . The same question goes to hundreds of agents that coordinate with each other, and even a weaker model can outperform stronger ones through the crowd. DeepMind research teams turned similar large-scale agent infrastructure toward fluid problems and published the outcomes in the autumn of 2025.

The most striking moment of the evening is that proofs generated by AI are written in the worst language any human would want to read, yet they are correct. Buckmaster says Gemini, ChatGPT and Claude could all understand this seemingly nonsensical text while humans would not even recognize it as English; the models behave as if they developed a shared language of their own.

The corporate race

The story did not stay in the laboratory: information leaked from Anthropic to OpenAI spread the rumor that a Millennium Problem had fallen, betting markets opened on social media, and companies went on red alert. In Buckmaster's view the internal models of OpenAI and Anthropic differ little; what makes the difference is compute for running armies of agents, and millions of dollars are being spent on it.

Buckmaster argues the Euler result marks a threshold beyond human intellect, a turning point comparable to Deep Blue beating Kasparov at chess. For him this is no single company's victory but a rupture the whole mathematics community must digest, with consequences reaching far beyond mathematics into all of society.

In its announcement of 8 September 2026, OpenAI stated that an internal model had produced a solution to the Navier-Stokes problem: a flow starting from rest, driven by a smooth force with finite energy throughout, develops a singularity in finite time, in an analytical proof also verified formally in Lean. The company says the system clearly surpasses GPT-6 Astra and settles statements C and D of the official Millennium text.

What intrigues is that the proof's starting point almost exactly matches where Buckmaster's team had stopped, a meeting that happened a day or two after the name was shared. The OpenAI team uses the same mechanism in reverse: the structure Buckmaster used to produce the blowup, they grow to stabilize the singularity. Progress that would span a decade on a human scale appears to have been covered in days by AI.

The unexplained lift

The side branch that surprises physicists is that why planes stay airborne still cannot be fully derived from first principles. As the MIT airfoil notes explain, the standard account is built on pressure difference, camber and angle of attack: viscosity creates the starting vortex, and the vortex induces circulation around the wing. Yet this engineering account is no rigorous derivation from the equations, but a framework patched with extra assumptions.

Meanwhile the relationship between AI and mathematics goes beyond fluids: a study of olympiad-level formal reasoning published in Nature showed models solving competition problems inside environments such as Lean. Buckmaster's story is this line extended into research mathematics, with formal verification as the bridge for trusting proofs the human eye cannot read.

Returning to the fluid question, the picture is now sharp: the equations work flawlessly in engineering, yet under mathematical rigor they either produce singularities or admit countless solutions. The contradiction is no flaw but a boundary map of the model; knowing where the boundary lies means knowing where the equation can be trusted.

The answer to Greene's opening question, in Buckmaster's view, is no longer negative: AI can generate Millennium-scale ideas by compressing months of human labor into days. Still, humans will have the last word; making proofs readable, checking and interpreting them remains mathematicians' work, and this form of collaboration looks set to define scientific practice in the years ahead.