Comment by qnleigh

14 hours ago

It's notable that LLMs have now made substantial progress on four of the seven Millennium Prize problems, resolving one of them: Hodge, Birch-Swinnerton-Dyer, Riemann, and Navier-Stokes, which they resolved. No sign of P vs. NP or Yang-Mills existence and mass gap as far as I can tell, which is interesting.

Talking to some friends in physics this evening, most of the physics-related results that we could recognize were very mathematical, proving things rigorously where the physics community already had strong expectation. For instance, for a certain model of magnetism (the spin-1 Heisenberg chain), it was strongly expected that there is a finite energy gap between the ground state and the first excited state, but proving this rigorously was quite challenging. So while these are major results in mathematical physics, they probably don't rise to the level of a Millennium problem for the field.

It's interesting to think what a comparable breakthrough in physics might look like, since physics tends to favor things like conceptual understanding and applications over mathematical rigor. Maybe a new quantum algorithm, understanding of high-temperature superconductivity, a precise description of M theory...

21 comments

qnleigh

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sebzim4500  10 hours ago

No one really knows a viable approach towards P vs NP so we can't say for sure, but LLMs have created plenty of significant complexity theory results so I wouldn't say there's no progress.

andy_ppp  12 hours ago

I've read that another mathematicians work potentially has been incorporated into the training data with the work done on the Navier-Stokes equations so we should likely asterisk this one. Still it's mad these systems are this good that mathematicians are now using them to see further and probably to check their own work and understanding.

  • nl  10 hours ago

    You are being downvoted for this because OpenAI subsequently checked and clarified than none of the relevant conversations were in the training data in anyway for the Navier-Stokes result.

    • andriy_koval  6 hours ago

      > because OpenAI subsequently checked and clarified than none of the relevant conversations were in the training data in anyway for the Navier-Stokes result.

      could you give link? Because I remember they said they couldn't verify:

      "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."

      1 reply →

    • pvab3  2 hours ago

      for 2 months prior. Not any of the relevant conversations. For a cutoff date a couple months before the announcement. They said they had been working on that problem for a year or more

    • andy_ppp  7 hours ago

      Thanks, it's hard to stay up to date. However, we are just meant to believe that the mathematicians were going about the proof independently in the exact same way as the machines did it. It seems like a very odd coincidence to me.