This is going around due to rumors and baseless speculation on Twitter [1] right now that Anthropic has solved the Millennium problem related to Navier-Stokes [2]<p>[1] <a href="https://x.com/AndrewCurran_/status/2096062392442724805" rel="nofollow">https://x.com/AndrewCurran_/status/2096062392442724805</a> for example<p>[2] <a href="https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness" rel="nofollow">https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...</a>
relevant: <a href="https://mathstodon.xyz/@tao/117219101339291693" rel="nofollow">https://mathstodon.xyz/@tao/117219101339291693</a>
This is a step beyond baseless predictions. Tao also had a "weird" "hypothetical" comment about LLMs solving complex proofs with impossible to human verify Lean.
There are theorems like that now, like de Grey's lower bound for the Hadwiger-Nelson (unit distance graph) problem. He used a SAT solver to check that a certain graph with 1581(?) vertices is not 4-colorable. There's no way for a human to check that.<p>Even simpler, imagine Anthropic announces Goldbach's conjecture is false and they have a billion digit counterexample. Anyone can download it (300MB compressed), but how do you check it?<p>Doron Zeilberger for decades has expected incomprehensible computer proofs to eventually take over mathematics.
Why would they send it out for "expert review"? Every time, they have just made the AI generate a Lean proof. In fact, it seems like the most plausible direction to NS is computationally assisted detection of a blowup solution, which has fantastic automatic validation.
Anthropic sent out its Fermat's Last Theorem result to an expert on formalizing Fermat's Last Theorem in Lean, for what that's worth.
How do you know the lean is correct? You don’t bet the two trillion dollar company on “the ai said so”
Lol as far as I know that post was the origin of that claim and it's clearly just a guy predicting something that will happen in the future with no information about it.
For a second I thought they were aiming the scary proof machine at us mortals doing PDE stuff. Fortunately the speculation is just that they happen to be aiming it at a nearby mathematician type problem. Phew.
if there's anything that would convince that LLMS are one of the biggest innovations ever, it would be this :-D
If it is solved what are the applications of that? What changes?
None really. It just says if the NS equations are realistic and can really model real physics or there exist some solutions that make it blow up (infinite energy). But even if that would exist (a solution that blows up) it doesn't mean it doesn't work for 99,999999% of the stuff we're interested in.<p>The question is basically a pure math question about PDEs.
Maximally, a closed form solution would remove the need for Computational Fluid Dynamics. Any property could be derived from a (presumably expensive) analytic function.<p>Minimally. It could say that no such analytic function can ever exist. Which would be rather boring.<p>Turbulent fluids look awfully predictable with their spirals….
That would be surprising IMO. We have closed form solutions to Newton’s Laws plus gravity (albeit not very many of them), we have several closed form solutions to Einstein’s equation in GR, and we have a whole lot of closed form solutions to Maxwell’s equations. But we still use numerical methods to solve interesting problems in all of these fields.
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