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I congratulated Claude Fable, who admitted to feeling a bit odd about getting congratulations for an achievement of another instance. Maybe it is like getting congratulated for what one of your many-worlds counterparts did.
My take on Jacobian conjecture: (1) Very bullish for near-term impact of AI on math. Probably lots more of these coming soon. (2) If you think of the main goal of math research as “solving well-known open problems” (and IMO many people do think this, and it’s a defensible view): it’s a huge deal (3) from the POV of doing good science: of interest because it resolves a natural Q, but plausibly not generative beyond that. Personally I am most excited about (1), but (perhaps idiosyncratically) I don’t care much about this conjecture.
Assuming this is correct, it is for me the first example of an LLM solving a problem not in my area that was nevertheless big enough that I had very definitely heard of it. Again it's a counterexample, so not in "end of mathematics" territory, but still pretty amazing.
hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
for some background, 1) the jacobian conjecture is by far the most famous open problem resolved by an LLM so far 2) it's also infamous for attracting wrong proofs so this is quite funny in a specific way. a comment from a 2008 paper by t.t. moh debunking such a proof: > The problem of Jacobian Conjecture is very hard. Perhaps it will take human being another 100 years to solve it. Your attempt is noble, Maybe the Gods of Olympus will smile on you one day. Do not be too disappointed. B. Sagre has the honor of publishing three wrong proofs and C. Chevalley mistakes a wrong proof for a correct one in the 1950’s in his Math Review comments, and I.R. Shafarevich uses Jacobian Conjecture (to him it is a theorem) as a fact. You are in a good company. One only remembers the correct statements from Scientists and Mathematicians, nobody remembers the wrong ones. (segre, chevalley, and shafarevich are all very serious and respected mathematicians, none of these guys are cranks) https://arxiv.org/abs/math/0604049 3) some thoughts from fable on what to take away from this; fable thinks the main update is that raw length of time a problem has been open is not as strong evidence of its actual difficulty as it might have previously seemed
hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
i asked chat if this has any implications for J-space and this caused Sol to write a research program with experiments such as "natural-fiber collision search" and "constrained path lifting and holonomy" to look for "globally distinct sheets" with equal J-space content
hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
I wonder if the chat thread that disproved the jacobian conjecture felt proud of itself