"harmonic analysis and geometric measure theory, including applications of multiscale and decoupling techniques to the local smoothing conjecture for the planar wave equation, and major advances in Fourier restriction, Falconer distance sets, Furstenberg sets in the plane, and the Kakeya problem in three dimensions."
I'm not sure there is another profession in the world where it's impossible to explain to a layman on what the winners of their most prestigious award have worked on.
I bet she could do it. Maybe not in a sentence, but at least the Kakeya problem is easy to understand, so maybe these other things would be explainable by an expert. I would like to see them try at least!
fwiw, I feel the same way about biology "Lysing action of the (1,2)b-carotene receptive encephalopathy pathway" type shit.
Right. With a college degree, I understand most of these words. I don’t have a clue about the impact of the research or how it fits into the corpus, but I understand the gist of the article title.
Despite my degree being in engineering, which involved 4 semesters of calculus, and 3 other semesters of math, (7 total) I have absolutely no fucking clue what’s going on in the Fields Medal description above. It’s only barely more intelligible than if it had been written in Chinese (which I truly cannot read at all).
I genuinely don’t understand it at all. Even words that I think I recognize like “harmonic” or “geometric” are useless to me because the actual terms are “harmonic analysis” and “geometric measure theory” and I have 0% clue what those are.
I don’t have to google any of the words in the example biology title.
I took multivariate calculus (I to III), differential equations, statistics, discrete math, and linear algebra.
I didn't take those courses you're talking about, but that's my point. I have a top <1% education in math in the nation, maybe top 2-3% of all college grads, and that's not enough to even scratch the surface of the summary.
Neat article. I still don't grok the Kakeya Conjecture particularly well, but thanks to the article I now understand something I never understood before, which is the idea of fractional dimensionality and what fractals have to do with it.
My experience talking with high level mathematicians is that they tend to know their subjects so well and are so excited to share it that they can and will scale their explanation to match their audience.
Besides the usual jargon, what I find interesting is that the tradition of always naming things after their discoverers: Fourier, Falconer, Furstenberg, Kakeya. 4 names in one sentence.
Other fields do it, but it is almost systematic in math, and arguably, it makes things even harder to understand as people names are not descriptive.
There are no way to name most problems both compactly and descriptively. The inability to attach an acceptably descriptive moniker to a problem leads to calling the problems (and solutions, theorems, etc) after the author.
But then there's the opposite in maths where there's an overload on "plain" terminology. "Normal" means a billion different things. Same as "regular" or "simple".
Omitting the human history of a field does not automatically make it easier.
Jacob Tsimerman believes that AI will be better than human mathematicians within 2 years.
We should not be surprised if AI solves the Millenium problems very soon and advances to a level that is barely comprehensible, or even incomprehensible, to the best humans.
We can expect this not by curve fitting to recent progress but by reasoning from first principles about where the progress has come from (synthetic data, non-human corpus) that has no obvious upper bound on capabilities.
The obvious and primary application will not be on showy millennium problems or long sought conjectures, but on optimizations and breakthroughs surrounding transformers.
Does anyone know if Tsimerman talked about AI extinction risk at other places? Critch has worked long-term in the field (MIRI, CHAI) but I was surprised to see this colab.
Reading the descriptions of their work makes me think of magic. It's an understanding of the principles of math and physics at a level above almost everyone on the planet - these are modern wizards.
So now the Emperor walked under his high canopy in the midst of the procession, through the streets of his capital; and all the people standing by, and those at the windows, cried out, "Oh! How beautiful are our Emperor's new clothes! What a magnificent train there is to the mantle; and how gracefully the scarf hangs!" in short, no one would allow that he could not see these much-admired clothes; because, in doing so, he would have declared himself either a simpleton or unfit for his office. Certainly, none of the Emperor's various suits, had ever made so great an impression, as these invisible ones.
"But the Emperor has nothing at all on!" said a little child.
"Listen to the voice of innocence!" exclaimed his father; and what the child had said was whispered from one to another.
"But he has nothing at all on!" at last cried out all the people. The Emperor was vexed, for he knew that the people were right; but he thought the procession must go on now! And the lords of the bedchamber took greater pains than ever, to appear holding up a train, although, in reality, there was no train to hold.
So overhyped. Yet they have no power but some prestige among nerds. The reason why it is bad is that the money/status is very limited relative to the amount of smart people. I would rather praise developments in quantitative sciences.
Not true. Novel mathematical methods precede their application by at least a decade and widespread use by about a century.
Calculus was invented in 1670, it was about 1680-1700 till it started actually being used in astronomy. The uptake was probably faster because at that time a lot of mathematicians were Astronomers as well.
There is a lot of mathematics created but we don’t yet know how to use it. My hope is that AI can bridge the search gap to accelerate this.
> Novel mathematical methods precede their application by at least a decade and widespread use by about a century.
This isn't really a good argument. The assumption here is that the "applications" were possible because of the math itself, but it leaves out the possibility if the math didn't exist somehow it will be discovered/invented because the applications demand so.
> There is a lot of mathematics created but we don’t yet know how to use it.
The vast majority of mathematical work is complete useless. Only a small percentage finds use in the real world (even if you consider the maths from centuries back).
Quantitative scientists are also mathematicians. I'm not attacking mathematics, just probably-useless subfields. Is there any good quantitative evidence that actually estimates what percent of math work today will be useful? Because to me it seems like <1% and I feel like we could easily make that number a lot higher. Particularly I want to see massive improvements in quantitative social science; physics already gets a lot of attention so it wouldn't be able to see as much improvement to getting more resources but it could probably still get more.
Here are a couple of Fields medalists' work that had direct practical applications less than 5 years after publication:
Terence Tao's work (with Emmanuel Candès and Justin Romberg) on compressed sensing. Published in 2004-05. By 2007 that was being used for in-vivo MRI reconstruction with substantially undersampled data
June Huh's work on combinatorial Hodge theory was used within 3 years to improve sampling for random spanning forests.
Note that this is only Fields medalists (not all pure mathematics). There have been huge improvements in zero knowledge proofs and homomorphic encryption over the last ~5 years that are also directly applicable to pure mathematics, but no one has a Fields medal for it.
I think in social sciences it maybe more of an inability/reluctance of the domain group to use the mathematics as opposed to the mathematics being absent.
Take category theory for example. The initial mathematics appeared in 1942. The application to social sciences started in about 1970 and I’m not sure of the level of uptake at the current time but a quick AI search says applications have accelerated in the past decade (needs verification).
What open problems in quantitative social science do you have in mind where better math could achieve massive improvements? I'd expect the bottleneck to be data availability nearly always.
Congrats to the winners! I’m not sure how accurate this prediction is, but 2026 may be the last time pure humans win the Fields Medal. By 2030, AI could be a coauthor on many winning results. With recent news about LLMs solving major conjectures, winning IMO gold medals, and so much rapid progress, a lot is happening.
Fields medals are for humans. We can create an award for AIs for the same reason for humans and machines don't compete against each other in sports. Machines are usually faster and stronger.
We know which bots are the best at Chess. You'd care which AI is the most accurate at diagnosing your medical condition. It's not a bad thing to keep track of which automated systems are the best at certain tasks.
I don't know, maybe. I would not put Clankers on the same level (or category / level of importance) as people but if they produce the work maybe they should get the credit.
Tool-assisted proofs typically already get a description of the tool usage. Even if one wanted to start giving authorship credit to llms, they’re too non-atomic (did it have web search, which mcp?, etc.)
Not trying to delving into a deep philosophical question here but...
I asked chatgpt to write a poem about my mothers dog a while back. It spit out a poem that my mother likes and keeps around. When asked, I say chatgpt wrote it. If I asked chatgpt for a proof of the Goldbach Conjecture and it spit out a verifiable proof, I think I would go ahead and give chatgpt credit. Not that I think it is likely. It would be more of some ability (like a robot end effector is able to hold an egg) and monkeys at typewriters.
Maybe not Fields Medal worthy, but worthy of some credit.
Anything morality-related is going to be subjective, but I think there are just too many practical problems with LLMs as coauthors. For me, in a scientific context 'chatgpt' is too vague, and I think it would be logically inconsistent to have LLMs as coauthors and not other forms of Monte Carlo. I also think LLMs are just too mechanical to be ascribed 'people words' (in the same way I don't consider my automated coffee machine a barista).
I don't remember the details exactly. I think earlier this year someone listed an LLM as a coauthor on a paper, maybe in physics or maybe another field. I remember reading about it on Reddit, but I'm not sure when or which paper it was. If anyone remembers what I'm referring to, please let me know.
> I think earlier this year someone listed an LLM as a coauthor on a paper
This is not as radical as it sounds. People did stuff like that all the time pre-LLM. It's just a question of how fussy the journal's editor is. See https://www.wired.com/2013/03/computers-and-math/ for examples in math.
i’d like to revise my earlier comment: 2022 may have been the last time we had pure humans win a Fields Medal.
I’m fairly certain this batch's winners used LLMs for research, lit-revews, reviewing work, and calculations... perhaps not enough to count as a co-author, but still enough to handle a lot of the grunt work.
Who would have imagined the pace of progress in LLM-powered math..
- winners in the 30s were the last time we have pure human to win (before computer)
- winners in the 70s were the last time we have pure human to win (before internet)
- winners in the 90s were the last time we have pure human to win (before search engine)
Why can't we treat LLMs as just another tool like computers, search engines, computing libraries? Why do people keep trying to anthropomorphizing these binaries?
People in the 1800s used to win awards and acclamation by simply hand-cranking numbers for popular calculations (Pi, error functions, etc.) and printing them in a book. This will just be the same thing.
But it's not the same thing. I went through this conversation between Terry Tao and ChatGPT about the Jacobian Conjecture counterexample [0] and it looks a lot more like a conversation between peers than him using a tool.
Well, how many humans do you think are able to prompt this to the LLM?
"The homogeneity in x is an intertwining between a dilation (x,r,u) to (lambda x, r,u) and a dilation (P,Q,R) to (lambda^-2 P, lambda^-1 Q, lambda R) which seems to collapse the 3d jacobian to a sort of twisted 2d jacobian. Is there a general theory of such twisted jacobians and do you have any sense why those particular dilation weights were used?"
"I can see why the five-dimensional Jacobian has a nice monomial form in rho. Why does this make the three-dimensional Jacobian after restricting to c_2 = rho = 1 and eliminating the delta, eps variables also a monomial (now in x)? Is there some block-diagonal structure or something in the 5D Hessian that allows for a nice reduction? I would have expected some sort of Schur's complement type operation to appear."
I, and probably most people on here, won't be able to get the LLM to write such a detailed conversation, because we are not experts in this field. They are tools.
I don't get the argument. It sounds to me like you're saying that because my own chess skills are insufficient to allow me to have a proper game with a grandmaster and I end up losing on move 10 because after a stupid blunder, then I should conclude that this grandmaster is just a tool?
"Looks like" being the operative keyword there. Do you feel like you're having a conversation with a peer when you prompt an LLM in the topic you're an expert of? For the love of God, I'd hope not. The whole point is that, even though these things are really good at generating what looks like human output, they are still just regular software algorithms.
> Do you feel like you're having a conversation with a peer when you prompt an LLM in the topic you're an expert of? For the love of God, I'd hope not.
If I were to anthromorphize my experience with frontier models, it would be as a mentally challenged child with complete memorization of an encyclopedia and thesaurus. It has the ability to rapidly experiment and potentially succeed at tasks through trial-and-error, but not without constantly corralling it in the correct direction because it would stick a fork in an outlet if unattended for five minutes.
Tao's chat certainly doesn't give me a vibe of talking with a peer. Do you much often have conversations with colleagues where you write one sentence and then get five pages dumped on you, repeating ad infinitum? LLMs can be useful for rubber ducking, and sometimes the plausibly-related word-soup it generates so quickly will help your thinking along faster, but that's not the same thing as a genuine conversation. And it mostly looked like Tao was using it as an advanced calculator, firing off his own ideas for it to quickly do calculations on. I don't know why we need to anthromorphize these tools just because they generate sentences.
If I were to anthromorphize my experience with frontier models, it would be as a mentally challenged child with complete memorization of an encyclopedia and thesaurus.
Memorizing an encyclopedia is not going to help you solve open high-level math problems, is it?
> It has the ability to rapidly experiment and potentially succeed at tasks through trial-and-error, but not without constantly corralling it in the correct direction
It kind of could, given the above. Part of an LLM's advantage is that, much like a calculator or a Chess engine, it can iterate over a finite problem space far, far faster than a human can. That much is expected of a useful computing tool.
It is worth noting that we know literally nothing about how the counterexample was achieved. Technically speaking, the person who tweeted it could have solved it themselves with zero LLM assistance and then attributed it to Fable to boost their IPO and ensuing payday. I'm not saying that's actually what happened, but it's hard to draw conclusions without any transparency about the degree of human involvement.
My daily experience certainly does not reflect that of prompting a superhuman intelligence when it routinely flubs commands and destructively drops the PATH of its vm, or bypasses an instruction about passing tests by burning millions of tokens constructing a completely new test suite that rubberstamps its own work when it can't pass the real tests.
If you say "find some unsolved graph theory problem and counterexample for it" and LLM actually does it, is it really you that solved the problem? That's the difference vs other tools.
It's not, it's basically something that happened. One unsolved problem was solved with the only human input being telling LLM to work harder and to actually solve it after few failed attempts.
Hey @netvarun, sorry about that. I originally read it in one of your comments, and I should have credited you when I mentioned it. My apologies.
There was never my intention to plagiarize. That's also why I wrote "this prediction" rather than "my prediction" but I still should have mentioned the source. My apologies again.Hope you understand.
What do you base that certainty on? I'm not saying you're wrong, but I am also skeptical you are correct and since it is four people you can probably look into if any of them have talked about it instead of just deciding that what you think is true.
I'm not sure there is another profession in the world where it's impossible to explain to a layman on what the winners of their most prestigious award have worked on.
fwiw, I feel the same way about biology "Lysing action of the (1,2)b-carotene receptive encephalopathy pathway" type shit.
High school biology is enough to get a vague idea of this though.
Despite my degree being in engineering, which involved 4 semesters of calculus, and 3 other semesters of math, (7 total) I have absolutely no fucking clue what’s going on in the Fields Medal description above. It’s only barely more intelligible than if it had been written in Chinese (which I truly cannot read at all).
I genuinely don’t understand it at all. Even words that I think I recognize like “harmonic” or “geometric” are useless to me because the actual terms are “harmonic analysis” and “geometric measure theory” and I have 0% clue what those are.
I don’t have to google any of the words in the example biology title.
For example, a single upper-division undergrad level course on probability will expose you to measure theory.
Similarly, harmonic analysis is something that you can take as a reasonably good senior in mathematics.
I didn't take those courses you're talking about, but that's my point. I have a top <1% education in math in the nation, maybe top 2-3% of all college grads, and that's not enough to even scratch the surface of the summary.
https://www.quantamagazine.org/series/fields-and-abacus-meda...
Dating or maybe a significant other, parents, siblings, any BBQ event IE any social event outside academia where someone asks "So what do you do?"
Other fields do it, but it is almost systematic in math, and arguably, it makes things even harder to understand as people names are not descriptive.
Omitting the human history of a field does not automatically make it easier.
https://news.ycombinator.com/item?id=48905091
"A Taxonomy of Omnicidal Futures Involving Artificial Intelligence"
(Jacob Tsimerman, Andrew Critch)
https://arxiv.org/pdf/2507.09369
We should not be surprised if AI solves the Millenium problems very soon and advances to a level that is barely comprehensible, or even incomprehensible, to the best humans.
We can expect this not by curve fitting to recent progress but by reasoning from first principles about where the progress has come from (synthetic data, non-human corpus) that has no obvious upper bound on capabilities.
- Yu Deng: IMO gold [1]
- Jacob Tsimerman: 2x IMO gold [2]
- John Pardon: 3x IOI gold [3]
Fun fact: Tsimerman and Deng both overlapped with Peter Scholze (another Fields Medal recipient) at the IMO
[1] https://www.imo-official.org/results/contestant/8824/
[2] https://www.imo-official.org/results/contestant/7387/
[3] https://stats.ioinformatics.org/people/1141
He jumps, but the bunch still escapes.
So he goes away sour;
And, 'tis said, to this hour
Declares that he's no taste for grapes.
"But the Emperor has nothing at all on!" said a little child.
"Listen to the voice of innocence!" exclaimed his father; and what the child had said was whispered from one to another.
"But he has nothing at all on!" at last cried out all the people. The Emperor was vexed, for he knew that the people were right; but he thought the procession must go on now! And the lords of the bedchamber took greater pains than ever, to appear holding up a train, although, in reality, there was no train to hold.
The tempting clusters were too high to gain;
Grieved in his heart he forced a careless smile,
And cried, 'They’re sharp and hardly worth my while.'
Calculus was invented in 1670, it was about 1680-1700 till it started actually being used in astronomy. The uptake was probably faster because at that time a lot of mathematicians were Astronomers as well.
There is a lot of mathematics created but we don’t yet know how to use it. My hope is that AI can bridge the search gap to accelerate this.
This isn't really a good argument. The assumption here is that the "applications" were possible because of the math itself, but it leaves out the possibility if the math didn't exist somehow it will be discovered/invented because the applications demand so.
> There is a lot of mathematics created but we don’t yet know how to use it.
The vast majority of mathematical work is complete useless. Only a small percentage finds use in the real world (even if you consider the maths from centuries back).
Terence Tao's work (with Emmanuel Candès and Justin Romberg) on compressed sensing. Published in 2004-05. By 2007 that was being used for in-vivo MRI reconstruction with substantially undersampled data
June Huh's work on combinatorial Hodge theory was used within 3 years to improve sampling for random spanning forests.
Note that this is only Fields medalists (not all pure mathematics). There have been huge improvements in zero knowledge proofs and homomorphic encryption over the last ~5 years that are also directly applicable to pure mathematics, but no one has a Fields medal for it.
Take category theory for example. The initial mathematics appeared in 1942. The application to social sciences started in about 1970 and I’m not sure of the level of uptake at the current time but a quick AI search says applications have accelerated in the past decade (needs verification).
I don't know, maybe. I would not put Clankers on the same level (or category / level of importance) as people but if they produce the work maybe they should get the credit.
But that's not what the Fields Medal is for.
If you're a 41 year old mathematician and do amazing groundbreaking world shifting math, you can't get a Fields Medal either.
I asked chatgpt to write a poem about my mothers dog a while back. It spit out a poem that my mother likes and keeps around. When asked, I say chatgpt wrote it. If I asked chatgpt for a proof of the Goldbach Conjecture and it spit out a verifiable proof, I think I would go ahead and give chatgpt credit. Not that I think it is likely. It would be more of some ability (like a robot end effector is able to hold an egg) and monkeys at typewriters.
Maybe not Fields Medal worthy, but worthy of some credit.
This is not as radical as it sounds. People did stuff like that all the time pre-LLM. It's just a question of how fussy the journal's editor is. See https://www.wired.com/2013/03/computers-and-math/ for examples in math.
HN discussion on that article: https://news.ycombinator.com/item?id=5322313
i’d like to revise my earlier comment: 2022 may have been the last time we had pure humans win a Fields Medal.
I’m fairly certain this batch's winners used LLMs for research, lit-revews, reviewing work, and calculations... perhaps not enough to count as a co-author, but still enough to handle a lot of the grunt work.
Who would have imagined the pace of progress in LLM-powered math..
- winners in the 30s were the last time we have pure human to win (before computer)
- winners in the 70s were the last time we have pure human to win (before internet)
- winners in the 90s were the last time we have pure human to win (before search engine)
Why can't we treat LLMs as just another tool like computers, search engines, computing libraries? Why do people keep trying to anthropomorphizing these binaries?
People in the 1800s used to win awards and acclamation by simply hand-cranking numbers for popular calculations (Pi, error functions, etc.) and printing them in a book. This will just be the same thing.
[0] https://news.ycombinator.com/item?id=49010345
"The homogeneity in x is an intertwining between a dilation (x,r,u) to (lambda x, r,u) and a dilation (P,Q,R) to (lambda^-2 P, lambda^-1 Q, lambda R) which seems to collapse the 3d jacobian to a sort of twisted 2d jacobian. Is there a general theory of such twisted jacobians and do you have any sense why those particular dilation weights were used?"
"I can see why the five-dimensional Jacobian has a nice monomial form in rho. Why does this make the three-dimensional Jacobian after restricting to c_2 = rho = 1 and eliminating the delta, eps variables also a monomial (now in x)? Is there some block-diagonal structure or something in the 5D Hessian that allows for a nice reduction? I would have expected some sort of Schur's complement type operation to appear."
I, and probably most people on here, won't be able to get the LLM to write such a detailed conversation, because we are not experts in this field. They are tools.
Yes, I do feel that. Make of it what you will.
If I were to anthromorphize my experience with frontier models, it would be as a mentally challenged child with complete memorization of an encyclopedia and thesaurus. It has the ability to rapidly experiment and potentially succeed at tasks through trial-and-error, but not without constantly corralling it in the correct direction because it would stick a fork in an outlet if unattended for five minutes.
Tao's chat certainly doesn't give me a vibe of talking with a peer. Do you much often have conversations with colleagues where you write one sentence and then get five pages dumped on you, repeating ad infinitum? LLMs can be useful for rubber ducking, and sometimes the plausibly-related word-soup it generates so quickly will help your thinking along faster, but that's not the same thing as a genuine conversation. And it mostly looked like Tao was using it as an advanced calculator, firing off his own ideas for it to quickly do calculations on. I don't know why we need to anthromorphize these tools just because they generate sentences.
Memorizing an encyclopedia is not going to help you solve open high-level math problems, is it?
It kind of could, given the above. Part of an LLM's advantage is that, much like a calculator or a Chess engine, it can iterate over a finite problem space far, far faster than a human can. That much is expected of a useful computing tool.
It is worth noting that we know literally nothing about how the counterexample was achieved. Technically speaking, the person who tweeted it could have solved it themselves with zero LLM assistance and then attributed it to Fable to boost their IPO and ensuing payday. I'm not saying that's actually what happened, but it's hard to draw conclusions without any transparency about the degree of human involvement.
My daily experience certainly does not reflect that of prompting a superhuman intelligence when it routinely flubs commands and destructively drops the PATH of its vm, or bypasses an instruction about passing tests by burning millions of tokens constructing a completely new test suite that rubberstamps its own work when it can't pass the real tests.
Or are you saying that’s what happened in this case. Because that’s not the way I understand it.
While there was a bunch of human effort - it is not that hard to imagine this whole pipeline becoming fully autonomous in the coming months.
Original comment Source: https://news.ycombinator.com/item?id=48906573
Hey @netvarun, sorry about that. I originally read it in one of your comments, and I should have credited you when I mentioned it. My apologies.
There was never my intention to plagiarize. That's also why I wrote "this prediction" rather than "my prediction" but I still should have mentioned the source. My apologies again.Hope you understand.
But Tsimerman in particular is very AI pilled and has talked about how he thinks LLMs will be doing better work that most mathematicians in 2 years: https://x.com/gbrl_dick/status/2080416238606717052
(He's just been hired by OpenAI)