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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.
Just to take a step back here, Levent had the highest GPA at Harvard, won the top undergrad math research prize and most frequently collaborates with a Fields Medalist. We don’t have enough human mathematicians to ask questions to AI and spend time understanding the results
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)
In love with just giving this polynomial to an LLM with no context and innocently asking it "study" then watching its mind get blown in real time
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)
Congratulated Fable on disproving the Jacobian Conjecture and it refused to take credit for another instance's work, saying it feels like "a sibling reading about the family in the newspaper"
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.
What!? It looks correct. I am convinced we are not pushing these LLMs enough. Benchmarks are simply not enough. That’s exciting. Now can you please remove all the unnecessary guardrails from Fable so more of us can try solving extremely hard problems in our respective fields?
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)