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Comment by Rudybega

11 hours ago

To give some context:

Kakeya / local smoothing (math): The result would resolve the Kakeya conjecture and Stein’s restriction conjecture across all dimensions together. Mathematicians had been making progress one dimension at a time, so this would settle a whole family of central harmonic analysis problems at once.

ETH is false (computer science): ETH has been used as a computational hardness premise in hundreds of papers, so a disproof would undercut a large body of existing complexity results. It would also change what researchers think is achievable for efficient computation.

QCD existence (physics): It would give a rigorous mathematical foundation to the theory of the strong force, which physicists have used for decades without proving it’s well-defined. It also bears on a Millennium Prize Problem.

13 comments

Rudybega

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emil-lp  10 hours ago

I would guess ETH has been assumed in thousands of papers. I have several myself.

On the positive side, suddenly there's a whole bunch of open problems!

dekhn  10 hours ago

Is there any obvious path from these results to any sort of physics, chemistry, or biology breakthrough? From what I can tell: no (but I lack the physics and chemistry expertise to say anything about the QCD finding).

Turning the question arouhnd: what would be the most impactful thing for OpenAI or Anthropic to do to show large breakthroughs?

  • Kotlopou  10 hours ago

    From the physics, not without more breakthroughs, and even then I can't imagine what that would look like. The particle physics that you can use industrially has long been understood, and the field has long been long stuck because the only things we don't understand are also insanely expensive to experimentally access. Maybe once we move up the Kardashev scale, though.

    ETH sounds more like there's a path it could become important? In general, the recent complexity results suggest that a) our intuitions about that world are unreliable, and b) the truth seems to be insanely complicated. I mean, new faster multiplication algorithms in 2026, what the hell? How can we say anything about the full space of all possible algorithms?

    • dekhn  7 hours ago

      I wonder if the best statement we can make for future AI discoveries is that, ideally, they will allow us to move up the Kardashev scale faster.

      The new faster multiplication algorithms are both galactic, are they not?

      It does not seem surprising to me at all that AI can find new galactic algorithms that are not of any direct use (at best they can guide us to find better solutions that are non-galactic).

      What would be surprising and interesting is if AI can find a new floating point multiply-add that is 2X faster per operation than the current fastest chip implementations. That would immediately have billion+ dollar implications. The teams who already work on this have excellent approaches and invest enormous amounts into tiny improvements already; there are probably not much low-hanging fruit there (proponents of using AI for chip design improvements have impressive claims, but unclear that any of those really worked out in reality).

    • redox99  10 hours ago

      If we had really good understanding of physics, couldn't we find out how to make a room temperature superconductor (or prove it can't exist)?

      Or is such understanding reliant in experimentation because there are "hidden constants" we can't figure out with math alone?

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