@DavidHiggs o3 seems an update towards yes, yet the market has gone down almost 10% since last year.

@DavidHiggs Um o3 is definitely not an update towards yes. It's essentially zero progress. LLMs are not that good at math compared to the best cutting-edge AI. If this happens, which is unlikely, it will come from something like AlphaProof.

@nathanwei it’s an update towards reasoning RL working, the current paradigm (transformers) isn’t hitting a wall, funding will continue, etc.

And it’s an update towards agentic tool use being tractable in the near term, so LLMs don’t have to be explicit superhuman mathematicians by themselves. They can just use the future versions of tools like AlphaProof (in addition to Python, Wolfram, etc.) intelligently with their superhuman reasoning*.

*using reasoning in a slightly narrow sense here, short of full AGI, but still quite general

Related: https://manifold.markets/Thomas42/will-any-of-the-remaining-clay-mill?r=VGhvbWFzNDI

All seven questions and my comments

1. Birch and Swinnerton-Dyer conjecture (1960s --): Mostly number theory, rational points on elliptic curves. Quick literature search did not turn up any relevance to elliptic curve cryptography.

2. Hodge conjecture (1930s --): algebraic topology. Not sure who is interested other than algebraic topologists and geometers (two of the most abstract fields of modern mathematics...).

3. Navier–Stokes existence and smoothness (1850? -- ): It has strong connection to classical mechanics. Presumably physical intuitions can help. Terence Tao has constructed (2016) a singular solution for averaged NS equation that allows liquid computer that constructs a smaller and faster copy of itself, and so on, allowing finite-time blowup. He believes it's possible for this to work with NS itself. See https://www.nature.com/articles/s42254-019-0068-9

4. P versus NP problem (1971--): Extremely difficult. Implications across all computer science, mathematics, and possibly philosophy. See Why Philosophers Should Care About Computational Complexity.

5. Poincaré conjecture (1904 -- 2003): Solved by the Ricci curvature flow (PDE on manifolds).

6. Riemann hypothesis (1859--): Strong connections with number theory and complex analysis. Mathematicians typically say this is the most difficult one.

7. Yang–Mills existence and mass gap (1960s --): A problem in axiomatic Quantum Field Theory. I have no intuition for how difficult it is, but considering how many mathematical physicists there are, probably it's very difficult?

There have been 500 years of cumulative problem time, and only 1 solution, so the baseline rate is 1/500 per year. This gives a probability of 6/500 * 7 = 9% of any solution before 2030.

@YuxiLiu your reasoning made sense if collective maths knowledge didn't progress in the last 500 years & if AI didn't exist or wasn't developing exponentially with applications into scientific research. Following your reasoning, you may also bet that there will be no new iphone next year just because it's been a 350k years since the first homo sapiens and we got only a dozen models by now. I can't say whether this market is correctly priced, but applying linear reasoning to predict future research is simply wrong

@SimoneRomeo I do think a similar method for getting a base rate might be helpful, if you could identify a rough decade when mathematics research reached its current modernized form (something to do with peer review, highly cited papers starting to commonly crop up, maybe something to do with baseline societal levels of numeracy?). Then apply a modifier to pre-ML verses post-ML era research, and you can get a probability out for the scenario where current AI progress isn't especially impactful.

Reasoning models of course do have a substantial chance of being impactful over the next few years, so then you'd estimate the probability they are, plus how impactful they are (maybe something to do with expected productivity increase of mathematics research as a multiplier over base rates?). And then you remember to treat this whole line of reasoning as only 1 model, if perhaps a strongly weighted one.

Running through it real quickly I got ~12-13% chance in the AI pessimistic scenario (1 year has been used up). Then if I think there's ~75% chance AI is impactful, and say ~triples mathematics research productivity over the next 6 years (it's not doing much now, so that's actually decently optimistic), the adjusted probability is ~34%. And then remember this is only 1 line of reasoning, which I would assume biases results to be more pessimistic than I should be.

So, in order to vote yes, you've got to think reasoning model LMM type systems, or their immediate successors, are fairly likely to "solve" math or something else highly impactful and discontinuous. By 2030. Which I do find plausible, but maybe not likely enough to actually vote yes in this market at 53%.

All seven questions and my comments

1. Birch and Swinnerton-Dyer conjecture (1960s --): Mostly number theory, rational points on elliptic curves. Quick literature search did not turn up any relevance to elliptic curve cryptography.

2. Hodge conjecture (1930s --): algebraic topology. Not sure who is interested other than algebraic topologists and geometers (two of the most abstract fields of modern mathematics...).

3. Navier–Stokes existence and smoothness (1850? -- ): It has strong connection to classical mechanics. Presumably physical intuitions can help. Terence Tao has constructed (2016) a singular solution for averaged NS equation that allows liquid computer that constructs a smaller and faster copy of itself, and so on, allowing finite-time blowup. He believes it's possible for this to work with NS itself. See https://www.nature.com/articles/s42254-019-0068-9

4. P versus NP problem (1971--): Extremely difficult. Implications across all computer science, mathematics, and possibly philosophy. See Why Philosophers Should Care About Computational Complexity.

5. Poincaré conjecture (1904 -- 2003): Solved by the Ricci curvature flow (PDE on manifolds).

6. Riemann hypothesis (1859--): Strong connections with number theory and complex analysis. Mathematicians typically say this is the most difficult one.

7. Yang–Mills existence and mass gap (1960s --): A problem in axiomatic Quantum Field Theory. I have no intuition for how difficult it is, but considering how many mathematical physicists there are, probably it's very difficult?

There have been 500 years of cumulative problem time, and only 1 solution, so the baseline rate is 1/500 per year. This gives a probability of 6/500 * 7 = 9% of any solution before 2030.

I made the same market two months ago, and it's currently out of step with this one, so I'm posting it here for anyone who wants to make a dutch book:

You've got to be either the highest tier of A.I doomer or insane to trade yes on this position.

@Phan I dunno, Navier-Stokes looks like it might fall this decade.

With the help of the exponentialy growing field of AI. I would expect at least one of the Clay Mathematics Institute Millennium Problems to verify and accept a solution by 2030.