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Notes -
A Big Day in the Culture War
Apologies for any incoherency or grammatical errors; today's events led me to down more than my usual share of booze.
Earlier today, big things happened in math. No, not Claude providing a sub quadratic 3SUM. OpenAI released hundreds of notable math results, in a GitHub repo.
Fun results:
Hilbert's tenth problem over (\mathbb{Q}) is undecidable
The quasi-Riemann hypothesis
The rational Hodge conjecture holds for every CM abelian variety
Integer multiplication can be done in sub-log-linear time. Oh, there's also a sub-log linear DFT.
Thoughts and observations:
Mathematicians are big mad. View the relevant subreddit on our progenitor site. Most there are probably, at best, adjuncts at community colleges desperately coping with the downward trajectory of already marginal careers, but it's fair to say that the writing is on the wall for mathematicians. There's probably a double digit number of grad students staring into a glass of whiskey tonight and thinking of hanging themselves.
My immediate question was about whether this closer to the current peak of performance, or just a lazy demonstration of OpenAI's power. So, I took one of the particularly interesting preprints to me (memory and precision in Gaussian models) and tested whether it's at the edge of capabilities or not. 30 minutes of back and forth with Astra (itself behind OAI's internal model) resulted in a significantly stronger result, on multiple dimensions. While I finished my first bottle, I spent a fair amount of time convincing myself of the result; I was convinced the strengthened results were plausible. Write this off as AI psychosis if you want, but try it yourself; I'm genuinely curious for what you get. (The back and forth, here, was entirely me saying "you can do it!", "keep at it, I believe in you", and "you've got this, finish it!")
Probably the most important line in OAI's announcement post is "The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking." This isn't a case of OAI spending millions for a marketing bump. My bet is a kind of Pareto distribution: most took minutes, not hours, with a long tail around Riemann-level results pulling up the average significantly.
Notably, ML related results are nearly entirely absent from this batch of proofs; maybe a half dozen touch on it, distantly. Some problem indices are skipped in overview.md. Conspiratorially, my inclination was to think they filtered them out for competitive advantage. I can't find any evidence of that in the GitHub repo or any of the preprints (equally plausible: deduping), so maybe they judiciously decided not to point their mathematical ballista at ML. ML also doesn't have a meaningful bank of rigorous conjectures, so given the conjecture sources, maybe they're not yet digging into ML math. But color me skeptical.
Does math matter? Is it something to advance civilization and technology, or an artistic pasttime for humans to create logical beauty? Likely both, today, but this is an almost nuclear detonation against the latter.
Sort of true. (I am a ML professor.)
COLT is the main conference for ML theory, and to ML researchers is considered more prestigious than ICML/NeurIPS/etc. It has a much lower impact factor, however, because it is truly a math venue about proving statistical theorems and does not accept any applied/experimental work. It has always had a track for papers presenting open problems. You can find the latest year's CFP at: https://learningtheory.org/colt2026/openproblems.html#cfp
I expect most of these problems to be "much easier" than standard math open problems and that these AI systems could prove most of them. The reason is that they don't receive the same amount of attention as "traditional" math problems. The number of researchers who can meaningfully even understand any one of these problems averages <100. That's similar to many math problems, but the difference here is that the ML researchers do not actually spend time thinking about the ML open problems because they are spending most of their time thinking about realworld ML applications and chasing $$$. Traditional mathematicians don't have either of these distractions.
There are a handful of classes of open problems where a resolution could meaningfully improve user experience with LLMs somehow. For example, there are open problems about improving the sample efficiency of reinforcement learning, automatic hyperparameter search, and A/B testing; and all of these are subproblems that the major labs have to implement to train their models. The actual math is abstract enough, however, that I don't see a lab keeping a result as a trade secret if they do resolve any of these problems.
Guilty. Though I do think about sample efficiency, OOD performance, and methods for encoding expert knowledge as inductive biases, which I think would fall under COLT questions. I just have a practical real-world use case in mind...
My question to you is how many of these are represented as formal mathematical formulations that are solvable without experimental testing? It's one thing to write a tight math proof that can be checked in a Lean Solver, quite another to prove that your new formulation of GNNs functions broadly under distribution shifts in multiple domains.
You will find more greek letters than English ones in a COLT open problem, and never anything empirical. Here's a representative example I just googled: http://proceedings.mlr.press/v125/van-erven20a/van-erven20a.pdf
When googling for the above, I also stumbled on a 3mo old repo that has some AI resolutions to some COLT open problems: https://github.com/Pengbinghui/pipeline-math. The problems they resolve are:
Of these, I've personally spent some time thinking about shuffled SGD and robust conditional probability estimation. The SGD problem is vaguely related to LLM training in a "that's interesting" kind of way, but not something that has any real world impact. The robust estimation result I could see being actually implemented in the backend of FAANG companies to improve performance of various algorithmic feeds and help them make a few more million/year, but no impact on LLM training.
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