szegő assistant professor of mathematics @Stanford
Based in United States
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Regular reminder:
"Compression is all you need"
arxiv.org/abs/2603.20396
“An accumulation of facts is no more a science than a heap of stones is a house.” - Poincaré
LLMs are now proving results that have resisted mathematicians for decades. Finding interesting theorems without human guidance is a new bottleneck. We show that we can teach an LLM to do it!
TL;DR:
→ a quantitative notion of interestingness
→ 4.3× higher interestingness
→ a self-expanding discovery loop
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Indeed.
Also, I'm seeing some confusion on the timeline about why Anthropic built a new biolab in SF.
For the past year people have been chanting:
"Where are the tangible positive benefits of AI?"
"You said you were going to cure cancer!"
Well, the biolab was created for scaling up precisely these biological & medical impacts
This whole fiasco is reminiscent of the early days of Erdos math. Less than a year ago.
The first discoveries were “well that was already in the literature, it was just search”, then came “ok you discovered something new but it turns out not a lot of human effort had gone into solving that problem”, next “ok you solved something everyone in the field has tried at some point and failed but it wasn’t an innovative solution”, then “ok you found an innovative solution to a long standing problem but it wasn’t the most difficult problem”, then “ok you solved the Millennium Challenge problem but it wasn’t useful”
Bio is much more practical than pure math, so until AI does something useful that affects people’s lives in the near term, the experts are likely to not recognize progress.
The period of denial will last for much much longer.
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"mathematical archaeology" for the proof that 𝜁(5) is irrational
The much-awaited Calegari take orz
He tries to figure out where Astra's ideas came from in the literature. He focuses on some (uncited-by-Astra) work of Prévost as well as the more obvious connection with Zudilin's applications of Hankel determinants.
galoisrepresentations.org/20…
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Mathematics engineering
I highly recommned the essay on antedisciplinary science by Sean Eddy written 21 years ago.
journals.plos.org/ploscompbi…
It seems to me that we are currently experiencing the birth of a new discipline: mathematics engineering.
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You make a correct observation here, but I would actually have to disagree on the conclusion:
Move 37 was indeed a step that works, which experts would have considered unpromising.
However, the reason why it was considered unpromising was it *represented* a larger coherent strategy that had never been discovered before: namely, to deeply preserve optionality as opposed to naive stone maximizing.
That is, if you only need to win by 1 stone, you don't always have to greedily take the stone in front of you, and execute on disciplined multi-step plans that appear opaque on first glance.
(As a very crude analogy: in WWII when the British cracked the enigma code, they could have greedily saved many boats immediately. However, this would wake up the Germans to their discovery, which would prompt a different German code, and hence invalidate the British's work. Instead, they calculated how many boats to save that would still appear statistically random, and go undetected.)
Moreover, this global non-greedy strategy was the main driver for success in the 4 wins (not just game 2, move 37).
And it is this exact global strategy that made "Move 37" into a synecdoche for AI insight, with such cultural resonance in the public eye.
I would feel more comfortable with your definiton of a 'move 37' as "taking a step experts had ruled out as unpromising, and the step works.".
"construct original method or definition with reinvents the problem or entire field" just sounds dramatic and quite subjective.
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Even now with a solution to the Millenium prize problem, I'm uncertain whether Navier-Stokes would qualify a "Move 37" moment in AI for math.
I'd want to wait until the PDE experts have fully digested it, and hear what they have to say about e.g. the level of ingenuity or originality displayed in the proof.
It is inherently delicate to evaluate, but as of the current moment, AI proofs I'm aware of seem to lie inside the "convex hull" of existing mathematical concepts and techniques.
I am also open to evidence to the contrary:
let's see what the 100 new solutions contain.
For a Move 37 impact in math, one would have to e.g. construct an original method or definition which reinvents the problem (or an entire field...)
So far AI results have found (very clever) combinations of pre-existing techniques (to amazing effect, to be clear)
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Zeta(7) anyone?
WOW!! Zeta(5) is irrational! Here's a Lean formalization: github.com/mo271/zeta5 Amazing what we'll learn (with AI help)
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"The point is that he fell in love with an unassuming superman from the future"
About 40 years ago Siemion Fajtlowicz created Graffiti, a conjecture-generating graph theory computer program.
This essay by Fajtlowicz discusses what Erdős thought of this project (and gives some idea what Erdős may have thought of AI).
thomasbloom.org/Fajtlowicz.p…
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Mathematics, unlike chess or go, is universal.
Too few understand the implications of this.
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Interesting development
We’re working with an independent advisory group of mathematicians to help OpenAI responsibly share advances in AI and mathematics.
The group will advise on how we assess and communicate new mathematical results, uphold academic and professional standards, and build tools that support mathematical research and learning.
Through this work, we want mathematicians to be at the center of shaping how AI supports mathematical understanding and how its benefits reach the wider community.
openai.com/index/advisory-gr…
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Hilbert quite literally used to dream & pray for times like these
It is a privilege and an honor to bear witness
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Jev short for Jevons paradox
Super fast & cheap models will in turn increase overall demand
After co-inventing ChatGPT, I kept asking myself: why have superhuman chat models not led to AGI?
I’ve spent the last 2 years in stealth building a new way to train models (RLCD), and a new type of frontier AI model that we are releasing today: Jev
• 20-200x faster
• 40-400x cheaper (w/ output tokens free)
• Frontier composable intelligence optimized for decisions
AFAICT the shortest path to AI-based economic revolution
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Jevons paradox for mathematics
If I did not believe in the intrinsic and underrated usefulness of mathematics, I would be somewhat sad about what AI is doing to it.
I think Jevons paradox will strike again: much cheaper and deeper mathematics will become even more of the fundamental infrastructure of the future than it is today.
Math will no longer be centered on conjectures posed by humans. It will be continually postulated, solved, processed, and utilized by AI systems as they work on the hard scientific, engineering, and computing problems of the future.
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Jared Duker Lichtman retweeted
"However, there is now the tendency to overreact, going from zero to 100 without nuance."
My PhD advisor, James Maynard (Fields medal, 2022) says he once dismissed predictions about AI as science fiction, but its rapid advances have proved him “consistently wrong”.
I want to commend him for this admission:
"I would have completely dismissed you as a fantasist...I've heard all these predictions that just sound like sci-fi and I've been consistently dismissive and skeptical of all of them."
"But I've been consistently wrong, and the people who sound from a sci-fi novel have been making much more accurate predictions."
It is not easy to admit one's worldview is wrong. So first and foremost, opening up in such a public way must be commended.
Personally, I've come to the realization that my general discomfort over the past several years--as a PhD student at Oxford and now in the math department at Stanford--has arisen from being unable to discuss the trends I've seen with my colleagues, without being dismissed as "sci-fi".
Unlike most in the math community, I have been tracking AI & mathematics daily for the past 7 years, seeing general capabilities increase at a predictable rate (c.f @A_G_I_Joe straight lines on log-log plots). As such, I was excited but not surprised when Navier-Stokes (and possibly more Millenium prize problems) have been solved.
Now, more in the math community are waking up. It is a very unnerving feeling to see your deeply-held, yet embarrassing convictions to suddenly become accepted as mainstream. I feel more disoriented than vindicated, to be honest.
However, there is now the tendency to overreact, going from zero to 100 without nuance. Instead of swinging from one side of the pendulum to the other, one needs only continue to follow trendlines that have been holding up over several orders of magnitude (an empirical fact of first-rate historical importance, but not well understood scientifically).
I would suggest paying close attention to the voices who have been consistently correct in their predictions, rather than the newly converted.
Indeed, these trends suggest great transformation across sectors in the coming months and years. However, this is not cause for undo panic, but rather careful navigation and reflection.
In general, my advice is to keep an open mind on this adventure we are collectively on.
What a time to be alive.
piped.video/watch?v=kMB8WeXr…
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Jared Duker Lichtman retweeted
>In general, my advice is to keep an open mind on this adventure we are collectively on.
Great advice for many people across many different fields!
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My PhD advisor, James Maynard (Fields medal, 2022) says he once dismissed predictions about AI as science fiction, but its rapid advances have proved him “consistently wrong”.
I want to commend him for this admission:
"I would have completely dismissed you as a fantasist...I've heard all these predictions that just sound like sci-fi and I've been consistently dismissive and skeptical of all of them."
"But I've been consistently wrong, and the people who sound from a sci-fi novel have been making much more accurate predictions."
It is not easy to admit one's worldview is wrong. So first and foremost, opening up in such a public way must be commended.
Personally, I've come to the realization that my general discomfort over the past several years--as a PhD student at Oxford and now in the math department at Stanford--has arisen from being unable to discuss the trends I've seen with my colleagues, without being dismissed as "sci-fi".
Unlike most in the math community, I have been tracking AI & mathematics daily for the past 7 years, seeing general capabilities increase at a predictable rate (c.f @A_G_I_Joe straight lines on log-log plots). As such, I was excited but not surprised when Navier-Stokes (and possibly more Millenium prize problems) have been solved.
Now, more in the math community are waking up. It is a very unnerving feeling to see your deeply-held, yet embarrassing convictions to suddenly become accepted as mainstream. I feel more disoriented than vindicated, to be honest.
However, there is now the tendency to overreact, going from zero to 100 without nuance. Instead of swinging from one side of the pendulum to the other, one needs only continue to follow trendlines that have been holding up over several orders of magnitude (an empirical fact of first-rate historical importance, but not well understood scientifically).
I would suggest paying close attention to the voices who have been consistently correct in their predictions, rather than the newly converted.
Indeed, these trends suggest great transformation across sectors in the coming months and years. However, this is not cause for undo panic, but rather careful navigation and reflection.
In general, my advice is to keep an open mind on this adventure we are collectively on.
What a time to be alive.
piped.video/watch?v=kMB8WeXr…
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"A final reason that I didn’t sign the letter is that I wasn’t really sure what it was demanding that isn’t happening already."
"I think the best we can do is recognise the changes that are coming and try to work out the least unsatisfactory way of dealing with them."
I've written a blog post responding to the letter about maths and AI signed by 25 Fields medallists. As with the Leiden Declaration, I didn't sign it, but I agree with much of it and welcome its existence.
gowers.wordpress.com/2026/09…
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