Two human mathematicians have made, with a “great deal of help” from AI, three important steps towards solving one of the world’s most famous outstanding mathematical problems. Other researchers say the approach could lead to a full solution in time and that it may already be enough to land the pair a $1m Millennium Prize. The Navier-Stokes equations have been used to model the flow of fluids for two centuries – whether that is used to design more efficient and stable aircraft wings, modelling blood flow through arteries or building space rockets – despite their habit of occasionally breaking and outputting nonsense in certain scenarios.
Solving this problem – whether the equations completely tally with the real world, or if their modelled smoothness and turbulence can deviate from it – is one of the six remaining Millennium Problems, the thorny mathematical puzzles published by the Clay Mathematics Institute that will net the solver a $1 million prize. Now, Tristan Buckmaster at New York University has announced groundbreaking results that edge us towards that larger solution via a document uploaded to his website, developed with Levent Alpöge at Harvard University and AI company Anthropic. I made a free AI chatbot solve a decade-long maths problem in 13 minutes The pair claim three results: two of which were published along with Lean formalisation – a process of converting mathematical theories into computer code that allows it to be rigorously checked for logical flaws and errors – and one which isn’t yet published as the pair await a finished formalisation.
The findings relate to close cousins of Navier-Stokes, the Boussinesq approximation and the Euler equations, but aren’t yet generalised to the wider Navier-Stokes problem. In the papers, they describe how they have taken previous work by Diego Córdoba and Luis Martínez-Zoroa to a conclusion with a “great deal of help from LLMs”, including large language models from Anthropic and OpenAI. In his announcement, Buckmaster said he sees the results as being less important than the fact that AI is rapidly becoming a powerful amplifier of human mathematical effort and leading to faster progress.
Neither Buckmaster nor Alpöge immediately responded to New Scientist‘s request for comment. Fermat’s last theorem formalised by AI agents in just 11 days The results can be thought of as “stepping stones” to a full solution of Navier-Stokes, says David Silvester at the University of Manchester, UK. Silvester says that one of the papers, which focuses on the Euler equations, shows that spontaneous “blow-ups” or turbulence can appear.
So it’s not like you’re trying to prove something. You don’t know whether you’re trying to prove it or trying to find a counterexample,” says Silvester. Silvester says that expanding this result to Navier-Stokes won’t be trivial, but that the current work alone may be enough for the pair to claim the Millennium Prize.
Terence Tao at the University of California, Los Angeles, wrote in a social media post that he believes the work takes us very close to a full Navier-Stokes solution. She believes that running the same technique on Navier-Stokes won’t be enough, and that there will be more to the solution than that, but she is optimistic. Despite the fame and longstanding insolubility of Navier-Stokes, a working result is unlikely to deliver any practical benefit, says Silvester.
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