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.
We are doing fine so far with just Transformers, we have just crossed Trillion parameter models, we don't know if 100s of Trillion param model wont be better.
The breakthroughs would be mostly in optimizations for computing tokens, not as much in expanding parameter count (if anything we want to shrink that).
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.
May all world-class mathematicians take a look at the AI alignment problem such that humanity can have a better chance of passing through, and may some of them decide to not focus just on sexy parts like coming up with ways to kill us all, and may those who do so anyway at least come up with more interesting or plausible stories than the ones presented in this paper.
I think this says more that being very good at theoretical math does not at all translate into intelligence or subject matter experience about how humans will realistically handle potential armed conflict and potential risks of mass casualties, at the political/nation-state level.
Ed Zitron should write his newsletter with LaTeX and publish them as PDFs in arxiv. Seems to make the techbros automatically take it seriously. Maybe that's what it takes to pop the bubble.
Whether or not Ed is right, he doesn't take any criticism, but peer review is a fundamental tenet of actual science. Vs being a blowhard on the Internet and getting paid for it.
> For comfort, Deng turned to manga. “I remember him at Courant. He was always holding a manga,” said Jalal Shatah, a professor there who formerly chaired the math department. Deng particularly liked stories about deep friendship and romance and found himself drawn to a genre known as yuri, which focuses on those kinds of relationships between women — stories that “feel nice and warm and beautiful,” as he put it. They helped him be kinder to himself, “more comfortable with life.”
----
Note that while the description "anime lesbian" might sound weird to western audiences, within asian nerd subcultures it's relatively a common thingy.
"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.
There's been some new videos uploaded with her and the rest of the prize winners (and the winners of other prizes, like the Gauss Medal): https://www.youtube.com/watch?v=mTPyjR3qmTA But after watching them I still don't think they're great for understanding (I'm still mostly confused) but the human-interest aspects of the videos are something.
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.
if you took an engineering course that covered fourier transform, z transform, or laplace transform, then you have seen an introduction to harmonic analysis on the circle and the real number line.
typically when I see mentions of harmonic analysis together with differential geometry I start thinking of fourier transform on more fancy shapes (on the sphere, for example, the replacements for complex exponentials are called spherical harmonics)
my PhD was EE in signal processing and I took some extra math classes at the graduate level, and continue to read some of this for fun and profit.
I still cant understand most of the terminology in the fields medal citations. that probably requires a more thorough couple of years of graduate education in mathematics.
math is by far the furthest of the sciences in terms of depth of human discovery.
Harmonic analysis is "just" working with harmonic functions which you have enough background for: f(x,y) is harmonic if f_xx+f_yy=0 (f_xx=2nd partial derivative w/r/t/ x, and so on). Higher dimensions work the same: f_{x1x1} + f_{x2x2} + f_{x3x3}+.... = 0.
Every engineering student in the US takes those courses. I did too. That doesn’t make us top 2-3% of college grads.
The stuff you mentioned is first year business for most math undergraduates. You should not expect to know most things in senior level math after taking just first-year math.
It’s like expecting to know computational complexity theory, or distributed systems theory, after taking first year CS.
I used to do something similar (at a lower level), and my solution was to lie, just lie. A big part was about https://en.wikipedia.org/wiki/Wavelet_transform so my description was something like:
I work in something related to Image Compression, so when the computer has to download the image from Internet it's smaller and use less data. Anyway, I study the mathematical part, not the programming part.
The idea is that images usually have big plain parts like the sky or the wall of a house, so you use big blobs of "ink" to paint them. For the border you use smaller blobs of "ink". And very close to the border you use smaller and smaller blobs of "ink". In this method, all the blobs of "ink" has the same shape, the only difference is the size. Also, the plain parts are not perfectly plain, so you use some small blobs of "ink" there.
In a typical image, you need very few blobs of "ink" if you pick the shape of the blobs of "ink" correctly. So you can only send the position and size of the blobs of "ink", that is much smaller than sending all the information of the image. The hard part is choosing a shape of the blobs of "ink" to make this conversion automatically and very fast, without asking the computer to do something smart to select the positions.
If the listener has more technical background:
The blobs of "ink" have white "ink" in some parts and black "ink" in other parts. This correspond to positive and negative values and actually all the blobs of "ink" are an orthonormal base so the calculation is only a orthonormal base change, that is super easy and fast. There is no smart selection of the position of the blobs of "ink" positions, just a boring orthonormal base change.
If the listener has even more technical background:
Something something Fourier Transform.
I don't want to count how many lies that description has. Also, all the parts in this description were done by other persons perhaps 10 year before me. I think I only once compressed an image, just for fun, and got a tiny compression because it was a toy method (¿Haar base?).
At least you are aware of the lies. I'm reminded of Feynman's refusal to lie about the nature of magnets (https://www.youtube.com/watch?v=Q1lL-hXO27Q): "You'd soon ask me about the nature of the [rubber] bands." Even in his digression about "why" questions, in giving a very high level reason for why ice is slippery he adds "they say", since I suspect he knew at the time the usual explanation had problems, and only recently do we have better explanations (https://www.quantamagazine.org/why-is-ice-slippery-a-new-hyp...). But at the same time, I've thought, it can be fun to try and come up with nice sounding lies, or analogies, or oversimplifications, especially for some types of people who won't leave you alone about something until they think they've understood something (even if a lie), or who want something repeatable to tell their friends about so they can brag about you. Still I think honesty is a better policy, even if it's not as satisfying, and if departed from at least making sure both sides know there are convenient lies and oversimplifications. As a student I would be (and have been) very put out if the goal is to understand something on a technical level and the teacher starts with lies.
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.
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.
His work was incredibly APPLIED. Which is precisely why he is remembered and lauded.
Can anyone here not involved in theoretical math name 5 Fields Medalists? Take out Terry Tao or Andrew Wiles or Gigori Perelman and can you name any Pure Mathematicians from the last century at all?
You see that https:// part of the URL in your browser? The 's' started out as pure mathematical research into elliptic curve cryptography based on algebraic geometry and number theory, a solution looking for a problem if there ever was one. When the problem eventually came along, the solution was obvious thanks to pure math.
I wouldn't be the least bit surprised if work done on the behavior of vectors in high-dimensional space by Hong and her colleagues someday becomes relevant to practical neural-net R&D. You never know what you might need in the future, but it will suck if engineers have to stop and figure this stuff out from first principles when they do need it.
von Neumann did a lot of theoretical work, e.g. https://www.academia.edu/download/38046446/Ohta_15_von_Neuma... where he was hanging out intellectually with McCullough and Pitts. He not only wouldn't be surprised at the ANN renaissance under way now, he'd ask us what took so long. He even cited M&P in his EDVAC report. Way ahead of his time, envisioning how hardware that wouldn't be practical for another 80 years might work.
> pure mathematical research into elliptic curve cryptography based on algebraic geometry and number theory, a solution looking for a problem if there ever was one.
This is ahistorical. Elliptic curves had been around for a long time, but elliptic curve cryptography was applied from its birth in 1985. Also the reasons elliptic curves had been studied for so long (Mordell-Weil Theorem, Falting's Theorem, Sato-Tate, etc) didn't really have much to do with the reason it was proposed, which was that its group probably wasn't vulnerable to the same kinds of attacks as DH.
> When the problem eventually came along, the solution was obvious thanks to pure math.
Also ahistorical. ECC wasn't proposed until almost a decade after DHKE (possibly only that early because it was hot at the time thanks to Lenstra (https://www.math.uwaterloo.ca/~ajmeneze/publications/ecc.pdf)) and it took decades to get in common use. Doesn't seem all that obvious.
You're misunderstanding my comment. The OP was amazed by people who can manipulate symbols on paper. Von Neumann would have been considered an incredibly POTENT wizard. He actually impacted the world.
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.
> Published in 2004-05. By 2007 that was being used for in-vivo MRI reconstruction with substantially undersampled data
I have argued exact claim before and from what I recall, Tao maybe contributed 1% to MRI improvements in the last 20 to 30 years. You are way over exaggerating the importance of what he did.
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.
IMU: We've awarded Fields medals to these outstanding mathematicians.
Hackernews: Next time it'll all be LLMs. AI's going to kill us all, though, one of the mathematicians said so! Maths is useless anyway, what a bunch of nerds. Ooh, one of them likes lesbian fanfic.
Oddly. specific. Sorry, I don't get the reference, could you please tell me what/who does that refer to (which fanfic and person, and why said person and fact is relevant to this question)
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.
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.
Because the 4 prompts were actually still prompting for the original prompt (ugh.) in that the computer did not come up with a "complete unconditional counterexample" and the human expertise is required to discern that.
There was an article on Rudi Mandl who pestered Einstein into working on gravitational lensing.
This happened because Rudi was persistent and Einstein was kind enough.
In future, citizen scientists have a chance to work on their ideas using AI, eventhough they don't have the deep domain skills. Of course, an expert would still need to review it as usual. But it is a useful tool to democratize science further.
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.
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.
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.
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.
Because in chess you can be sure that no AI was used (and even there, there are controversies, see the recent Niemann/Carlsen scandal). Since math is not usually done in public, under competition rules, no one will know ever again whether some result was really obtained by a human.
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).
"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.
https://www.quantamagazine.org/yu-deng-wins-the-fields-medal...
> For comfort, Deng turned to manga. “I remember him at Courant. He was always holding a manga,” said Jalal Shatah, a professor there who formerly chaired the math department. Deng particularly liked stories about deep friendship and romance and found himself drawn to a genre known as yuri, which focuses on those kinds of relationships between women — stories that “feel nice and warm and beautiful,” as he put it. They helped him be kinder to himself, “more comfortable with life.”
----
Note that while the description "anime lesbian" might sound weird to western audiences, within asian nerd subcultures it's relatively a common thingy.
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.
typically when I see mentions of harmonic analysis together with differential geometry I start thinking of fourier transform on more fancy shapes (on the sphere, for example, the replacements for complex exponentials are called spherical harmonics)
my PhD was EE in signal processing and I took some extra math classes at the graduate level, and continue to read some of this for fun and profit.
I still cant understand most of the terminology in the fields medal citations. that probably requires a more thorough couple of years of graduate education in mathematics.
math is by far the furthest of the sciences in terms of depth of human discovery.
The stuff you mentioned is first year business for most math undergraduates. You should not expect to know most things in senior level math after taking just first-year math.
It’s like expecting to know computational complexity theory, or distributed systems theory, after taking first year CS.
I work in something related to Image Compression, so when the computer has to download the image from Internet it's smaller and use less data. Anyway, I study the mathematical part, not the programming part.
The idea is that images usually have big plain parts like the sky or the wall of a house, so you use big blobs of "ink" to paint them. For the border you use smaller blobs of "ink". And very close to the border you use smaller and smaller blobs of "ink". In this method, all the blobs of "ink" has the same shape, the only difference is the size. Also, the plain parts are not perfectly plain, so you use some small blobs of "ink" there.
In a typical image, you need very few blobs of "ink" if you pick the shape of the blobs of "ink" correctly. So you can only send the position and size of the blobs of "ink", that is much smaller than sending all the information of the image. The hard part is choosing a shape of the blobs of "ink" to make this conversion automatically and very fast, without asking the computer to do something smart to select the positions.
If the listener has more technical background:
The blobs of "ink" have white "ink" in some parts and black "ink" in other parts. This correspond to positive and negative values and actually all the blobs of "ink" are an orthonormal base so the calculation is only a orthonormal base change, that is super easy and fast. There is no smart selection of the position of the blobs of "ink" positions, just a boring orthonormal base change.
If the listener has even more technical background:
Something something Fourier Transform.
I don't want to count how many lies that description has. Also, all the parts in this description were done by other persons perhaps 10 year before me. I think I only once compressed an image, just for fun, and got a tiny compression because it was a toy method (¿Haar base?).
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?"
Yes we do have plenty of practice adjusting our explanations to the audience.
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.
Same with their ability to summarize and explain things
And their ability to create mathematical objects that are similar, but weird in a funky way, to the actual real objects.
- 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.
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.'
"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.
Can anyone here not involved in theoretical math name 5 Fields Medalists? Take out Terry Tao or Andrew Wiles or Gigori Perelman and can you name any Pure Mathematicians from the last century at all?
I wouldn't be the least bit surprised if work done on the behavior of vectors in high-dimensional space by Hong and her colleagues someday becomes relevant to practical neural-net R&D. You never know what you might need in the future, but it will suck if engineers have to stop and figure this stuff out from first principles when they do need it.
von Neumann did a lot of theoretical work, e.g. https://www.academia.edu/download/38046446/Ohta_15_von_Neuma... where he was hanging out intellectually with McCullough and Pitts. He not only wouldn't be surprised at the ANN renaissance under way now, he'd ask us what took so long. He even cited M&P in his EDVAC report. Way ahead of his time, envisioning how hardware that wouldn't be practical for another 80 years might work.
This is ahistorical. Elliptic curves had been around for a long time, but elliptic curve cryptography was applied from its birth in 1985. Also the reasons elliptic curves had been studied for so long (Mordell-Weil Theorem, Falting's Theorem, Sato-Tate, etc) didn't really have much to do with the reason it was proposed, which was that its group probably wasn't vulnerable to the same kinds of attacks as DH.
> When the problem eventually came along, the solution was obvious thanks to pure math.
Also ahistorical. ECC wasn't proposed until almost a decade after DHKE (possibly only that early because it was hot at the time thanks to Lenstra (https://www.math.uwaterloo.ca/~ajmeneze/publications/ecc.pdf)) and it took decades to get in common use. Doesn't seem all that obvious.
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.
I have argued exact claim before and from what I recall, Tao maybe contributed 1% to MRI improvements in the last 20 to 30 years. You are way over exaggerating the importance of what he did.
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).
https://news.ycombinator.com/item?id=48905091
Hackernews: Next time it'll all be LLMs. AI's going to kill us all, though, one of the mathematicians said so! Maths is useless anyway, what a bunch of nerds. Ooh, one of them likes lesbian fanfic.
Oddly. specific. Sorry, I don't get the reference, could you please tell me what/who does that refer to (which fanfic and person, and why said person and fact is relevant to this question)
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.
https://chatgpt.com/share/6a60b2eb-0b64-83ee-9c76-7931ca1de0...
This happened because Rudi was persistent and Einstein was kind enough.
In future, citizen scientists have a chance to work on their ideas using AI, eventhough they don't have the deep domain skills. Of course, an expert would still need to review it as usual. But it is a useful tool to democratize science further.
https://www.sciencenews.org/blog/context/amateur-who-helped-...
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.
https://chatgpt.com/share/6a60b2eb-0b64-83ee-9c76-7931ca1de0...
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)
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
They might not be officials, they not might represent the official organizations, but those things do happen.
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.