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Yang-Hue Hi: The AI Math That Left Number Theorists Speechless
May 23, 2025
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It's the season for all your holiday favorites. Like a very Jonas Christmas movie and Home Alone on Disney Plus. Should I burn down the toy? I don't think so.
What the hell? This is bizarre that this button exists. It went from 0 to almost 100% immediately. I was amazed.
I'm extremely excited. Today we're going to be talking about a new paradigm of science, AI research in particular for math and physics, as well as do a deep dive into Professor Yang Hui He's murmuration conjecture. Professor, what am I leaving out?
Thank you very much for having me again. I had so much fun talking to you last time and it's great fun. I got so excited about this new field of AI assisted theoretical discovery in pure mathematics and theoretical physics that we're really entering a new era of discovery. It's amazing. It's happening every week. There's something new that could potentially be transformative in the way we do science.
Yes, you said it shocked the math world, maybe even the physics world and the science world in general. Yeah, absolutely. I'm going to talk about a little bit more details about this, but just private companies like DeepMind, the fact that they're getting Nobel prizes and all these various AI companies, OpenAI, EpochAI, they're actually doing fundamental research
Why don't you talk about how the landscape of research has changed with the advent of these new AI models? Now there's a large class like there's LLMs, but then there's also new machine learning techniques in general that you have uncovered and other people have uncovered. So why don't you just talk about that?
Yeah, absolutely. So, you know, as I said, I think I mentioned in our first long conversation, I think I got into this thing as a complete novice. You know, my background was in the interface between algebra, geometry and string theory. And in 2017, I was just beginning to learn from Coursera, just the very basics of machine learning.
And so since then, I personally have evolved and I have seen the field evolve of how far we can go, you know, because, you know, 2017 was pre-Chad GPT.
So this was just really machine learning of data from pure mathematics. But now we're much more advanced than this. We're beginning to benchmark DeepMind's AlphaGeo2, DeepMind's AlphaProof. They're beginning to show these signs of reasoning.
whether whether there are reasoning it's a purely philosophical question which i have no right to answer but what i'm seeing is that they're certainly beginning to outperform your typical undergraduate and and now epoch ai which is another company have has launched this tier one two three
problems which are research level mathematics the kind of problems you would give to a graduate student or to give a collaborator or colleague and i can show you some of that later some of some of the stuff i'd love and they're they're entering their uh they're entering their tier four phase and in fact i'm flying to to berkeley on and thursday to have a meeting with bunch of other mathematicians
to benchmark how far that their tier four reasoning machine can go. So this is really rather, you know, it's happening. It's happening as we speak, how the landscape of research can be changed. Yeah, sorry. Yeah. Yeah, I think that's in part why firstly, it's an honor to speak with you again. Our last conversation went viral. I think that's this whole AI assisted research in math and physics is part of the reason because our last conversation was quite specialized in nature.
It just means the audience loves you, loves hearing from you. I also love speaking with you. You also have some books. In 2018, you had a book published called Topology and Physics, and I believe that's followed by your textbook in 2020 on machine learning and pure math. Right. Yeah. So the 2018 one was when I finished and pushed on archive, which is a very kind thing that Springer lets me do. But then the final book was the first textbook on
I wrote that book primarily to teach myself machine learning and AI because I was a complete novice in this and I wanted to share this experience as a theoretician, as a mathematical physicist, how to share with this community, how to even begin with learning about machine learning and AI and this advanced data techniques. But now I think the field has progressed
oh wait beyond that in the last eight years and we're all very very impressed with how we as a community we're all very impressed with how fast this this thing is going so it's a it's a great pleasure talking to you know as i said i'm a i'm a big fan the kind of the kind of depth that you go into
Okay, well, let's dig deep. What are some different ways that people use machine learning? So for instance, Terry Tao uses it as
an assistant to proofs, but also to generate conjectures and perhaps they even point to existing tools that he may not have heard of. And those are more of the LLM sort. Then there's another sort of just finding patterns in large data sets. And that connects to your memorization conjecture. Is there a third category?
Yeah that's right, so I was just trying to, I think I outlined this briefly last time, I mean just trying to, because this is exactly the kind of thought process that's going on, how to categorize the different approaches and of course they're all interrelated and it's hard to delineate them, but so this is my you know my top-down mathematics is this
Intuition guided basis of mathematical research and this LLM approach is what I called meta mathematics and this is kind of LLM you know LLM assisted co-pilots that Terry Tao is talking about and then this third category of you know bottom up
where not necessarily any, any AI is involved. This is, I'm thinking about things like lean provers and proof as co-pilots, where you just have millions of lines of code. And well, sooner or later, somebody is going to process that by, by AI. The interplay between these three directions and between that and the human is clearly beginning to change the landscape of mathematical research. Now, before we get into your presentation here, you mentioned
Yeah, I think
I think it originated from a Greek fable where the fox and the hedgehog are compared
Where the fox knows a lot of things and hedgehog likes to dig in and I think the great mathematician Arnold made a reference to this in classifying mathematicians where he calls things like you know one is an eagle that flies and tries to see the landscape and then the hedgehog digs deep to one particular problem solves it. I mean there's no particular
I know neither is superior to the other. It says we definitely need both and each mathematician can function as both. But certainly AI is helping us in both personalities simply because there's so much literature out there. The AI can have an overview of what everything is in terms of literature. And also there's so much technical detail
You know there's some very boring parts of the proof that you just simply don't have time to iron out and that can certainly be helped by LLM models and it is beginning to do that. I was actually just at a conference last week in Exeter. There was a conference called the impact of AI for mathematics.
Where the organizer Madhu Das and she was saying that she's a number theorist and she said she's just recently completed this very complicated paper where she had to prove a lemma and then what she did was she knew there's another lemma which has got to be true. Okay. And so what she did was she
copied and pasted the entire proof of her lemma, it's a bit technical stuff, into chat GPT-01 and then said now can then copy the lemma and said can you supply the basic proof strategy of this lemma which I know to be true. Now to be fair what this is very far from automated reasoning this is just language model
And then, importantly, what she did was, and then she went line by line, symbol by symbol, the proof that Chachi Biti gave her for the Big Long Lemma. And it was largely correct. And with a bit of prompting, she was actually able to nudge out a complete version of that proof. And then that was done. I mean, so it would have taken her much longer to have ironed out all the details herself.
So this is really rather impressive and this is only because of O1 and O1 came out I think this year or end of last year. So it's really transforming the kind of stuff that the boring stuff you can delegate and also the pattern recognition part you can also delegate. So we become like superhuman all of a sudden and when interact with these agents who can help us actually do research and not just do you know very elementary problems now but serious research level
Why don't you talk about some of your personal use cases? So do you use chat GPT more than Claude or do you use Gemini more than the others? What is your mixture?
I actually, in terms of research, to be honest, prior talking to her, I never even really played around with chat, you know, this kind of LLMs to help with my research. I didn't know that was as possible. I know, you know, it's kind of thing you can, if you want to know very quickly a topic, you can go to Wiki, you can Google, but indeed ChachiBT or DeepSeek will probably answer your question very quickly. If there's some
If there's some theorem that you've just forgot, or there's a field that you really don't want to know about, DeepSeq will summarize it much better than, much more efficiently than if I went to some expert and wasted his or her time and just have that, you know, the process is much more efficient. If I just wanted a very quick overview of some specific, even a specialized topic. So that's been very, very helpful to me.
So it's only been three years since chat GPT came out and already we're seeing this massive change in the landscape. Do you imagine that three years from now, or let's say 10 years from now, that the role of the future academic or intellectuals or mathematicians, if you want to specialize, will be that of a decider or director, like a curator rather than a doer. So the doer is the one that right now we use computation, we use syntax, we
compute already helped us a lot by the end of the 20th century so no professional mathematician really goes by the by the end of by the 90s and early 2000s no professional mathematician for example does boring integrals anymore during doing research because that just that's completely outsourced
to say something like wolfram matematica on and later sage math because this is just boring and we know how to do it it will take as many hours if you really want to grind out some some technical integral of course i'm not saying that you shouldn't teach undergrads integrals anymore because that's part of the learning process and it's still important to teach teach undergraduates this kind of thing but no professional mathematician was started it's there they're really boring and horrible things like
even the simplest thing, you know, sine 17 X, nobody really wants to go on it. It just type into Mathematica. Because if I just need that, need that result very quickly, so I can supply to my next step of what I envision in my paper, I'm not going to waste a couple of hours trying to integrate something very elementary. And I'll probably even get it wrong, there'll be factors wrong. So that already transformed it. So I can see 10 years from now,
Simple, basic, maybe I'm being conservative here, but simple bits of a proof or simple bits of a derivation can just be outsourced to the likes of ChatGPT or DeepSeq or something, or something even more specialized. We don't currently have a LLM just for mathematics. That's surely going to come very soon. I'm sure the Frontier Math project by Epoch AI will start providing this kind of services.
Okay well let's get into your presentation on the AI mathematician. Yeah sure, well thank you.
I guess you know last time we talked about various things and I just want to share more in this chat some of the capabilities both in the terms of top down and bottom up of what we're looking at and to be honest even since our conversation which was what five months ago the field has advanced significantly which is very very impressive.
So just briefly to, as I was saying last time, and I just, as I also just mentioned, you know, there's, I try to classify this in this, in this review article, these three directions of mathematics, of course, they are intertwined. And the memorandum of gestures that I did with my collaborators, Lee, Oliver, and Posnikov, is really a very good example of this top down and I'll explain why this
Top-down mathematics is one that I want to emphasize here and then I will of course I will go back to just refresh people's mind a little bit about how people like Terence Tao and all this great and top minds are doing a proof assistance in terms of what I would call bottom-up and mathematics.
This is a very, very interesting point where I want to emphasize the typical mathematician, and that includes theoretical physicists. Historically, we do things top down by just looking at patterns and spotting patterns. And we do many things in terms of practice before foundation. And this is very important. This is something that can't really be formalized.
Because, you know, linguistically trying to formalize mathematics and which is an extremely important program, right? And all of this benchmarking of problem solving using large language models when you have a precise, well defined problems and trying to find a solution. But the history of scientific discovery is certainly not that I would say probably more than 50% is actually finding the problem or have a vague notion of something before you can formalize it. Can you give an example?
For example, Newton invented calculus without any notion of what even convergence means. He just had this intuitive idea of motion and then he, because it was Newton, he intuited there is this thing called derivative. This is way before we could even have epsilon delta limits, which came in the 19th century, almost 300 years later.
and algebraic geometry is something closer to my heart. Algebraic geometry was just started with the Apollonius and Euclid with just you know shapes and stuff and we can intuitive the kind of theorems we want to prove. Before this Babaki school in the 1950s and 1960s you know the height of the Babaki school tried to formalize that in terms of definitions of fields and rings and polynomial rings and ideals. Now this is this is just
This is how theoretical discovery has always happened. In some sense, the reason I want to emphasize this bit is not only just because this is one I'm most familiar with and the one that I suppose I've been mostly involved with. Another reason I want to emphasize is that it's hard to imagine how AI can help us with this because it's so vague and it's so human.
and there's a lot of mistakes and you if you train some language model there's not even any data to train on because these are not formal proofs these are just grasps of ideas of intuition and and and the point i want to make make that is even in this direction ai is beginning to help us okay so let's imagine we're back in the 16th sorry the 17th century with newton yeah and newton was saying okay i want to come up with something like calculus he didn't have that term he just had this notion of motion like you said
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What would Newton do with an LLM? What is your vision? That's an interesting point.
Newton would what Newton would do with a with an LLM if you had a LNM which was certainly to to process all previous literature. Now to be fair at Newton's time somebody like Newton could read almost the entirety of any relevant literature up to his up to his point and I'm thinking about everything from Euclid's elements Galileo
Bits of Kepler and he certainly won't have that anyone would just go money and you just go and read it and that's fine. But now literature has grown so exponentially. There are no more newtons human newtons that could possibly read the entire literature of the field and that's why the lambs could come in to help. So this is the in the LLM space of discovery because summarize literature.
I do can try to create new possible links between literature and this is happening now i think i think there are i think llama llama which is llm for math like llama double l llama llama is is something it's a it's an ai tool that's beginning to digest the archive for example and on the other hand that's the llm side of the story
now what about the other half is how could newton based on mathematical patterns and he did he did have a lot of patterns it would be things like this will be mathematical data so certainly they hear data in terms of theory in terms of
in terms of theoretical and experimental physics where you know you could measure falling you know the rolling of the kind of stuff the Galileo did the rolling of balls along inclined planes that kind of data or he had the the astronomical data of Kepler but he also would have had mathematical data or platonic data I like this word platonic data because it's pure the kind of data that would be like
Set of polynomial equations into variables which actually try to classify himself. How many cubics you because you knew about the conic classification problem and he would look at these things. And then he was spot patterns and this kind of stuff also gave rise i would imagine i can't imagine you know the mind of of newton but i would imagine she would look at vast amounts of such data.
So like Newton,
There are these things called Newton polynomials which expresses, it's a technical thing, which would be Newton, certain symmetric polynomials in multivariables being expressible in some basis. I would imagine Newton would have written pages and pages of this stuff and spotted a pattern and then tried to prove a general theorem, which is now the theory of Newton polynomials.
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I thought where you were going was this amorphous ideation with an LLM. So for instance, Newton would say, okay, I have balls coming down an incline. I don't have a precise formula for velocity. I don't even know if velocity is a great concept, but I noticed that it moves faster, but it can also that may be associated with something I would call acceleration.
But then there's something else like an impact, which I may call force. Can you help me with this? Is there a way to make this concrete, like to take something that's ill defined and make it well defined? I thought that's where you're going. Do you think LLMs help with that? Or is that not what you were thinking about? No, no, I think that definitely helps a bit. We're not quite there yet, I think, where we could just start asking LLMs and say, here's the literature, digest it all and give me new possible links. But we are actually not
insanely far away from that goal now because the lemons are getting so good at doing this sort of thing so it's actually not not impossible in some sense newton would yeah the the mind of newton is something like that i guess let me see my next slide i think i do mention yeah yeah i do mention that the another mind as great as newton and gauss and i would do i will give that example in a minute all right let's get back to the yeah
presentation oh so i think i don't i can't remember whether speaking of newton i i can't remember whether whether where whether i talked about this last time if not this is a joke worth repeating again which is what is the best neural network of the 18th century you could argue the 17th century was newton and the best neural network of the 18th to 19th century is clearly the the brain of gauss and that's another here's a very very good example as well of just
top-down intuition-guided mathematics. And I might have mentioned this last time, but it's worth repeating. So what is the thought process of this great discovery here? So everybody knew about the primes. Euclid already proved that there's an infinite number of primes. And the proof is very, very beautiful. And it's kind of intricate. It's the first thing you would teach in a number theory course.
It's not obvious at all there's an infinite number of number of primes, but Euclid had this proof by contradiction argument why there should be an infinite number of them. So Gauss certainly would have known that that proof, but Gauss wanted to know more. And it's been 2000 years since Euclid. Gauss wanted to know about more details about the distribution of primes. We know the primes get rarer and rarer, even though it's an infinite number of them. They do get rarer and rarer. How rare do they get?
and he just devised this function which we now call the prime counting function p of x. It's a terrible name because p is pi but it got stuck in the literature and p of x is simply the number of prime numbers less than a given positive real number x. So that was one of his insights was to devise this function which is now continuous because primes are inherently discrete.
This is very interesting. He plotted this and he looked at this curve. He invented regression apparently in order to do a curve fitting because he needed it. This is all done at the age of 16. He invented regression to see what is the best shape that fits this and all of this done by hand and he even had to compute the primes into the hundreds of thousands because the tables stopped there
I'd even got some of them wrong because it's a very boring and tedious computation imagine what gals could have done one if he had Math sage or Mathematica he could have how much more conjectures would have raised right? Well, the problem with that is that if he has access to the modern tools He also has access to tik-tok and we it's not clear if gals would be yeah Let's hope that because gals is gals He wouldn't waste his time to becoming like I don't know doing like watching youtube videos
But and watch more meaning maybe he would maybe he'll watching meaningful youtube videos like other conversations like The kind of conversation you put on theories of everything Okay, great. But anyway, but he would do this thing and and he looked at his p of x and he says ah It's clearly x of a natural log of x
You won't be able to see this just by looking. So he really actually had to invent regression to do this. And so statistical statistics was a side product of this problem. As far as I know, I could, but this needs to be checked with the, you know, real historians of science, but at least that's the, that's the story here. And this is just crazy. Like, how do you even, what do you even do? And this, the proof of this fact was given 50 years later by Adamar and Delavaya Pusa, because
You had to wait for Cauchy and Riemann to invent complex analysis in order to give a tool to prove this fact. So how did Gauss know this? Based on just at the time was surely because large data, right? He really went into the tens and hundred thousand range in order to spot this kind of patterns and it's just amazing and I can plot this by
Because if you just plot the first hundred or so, it looks kind of like a log or kind of like a line or something, but you really need to go into the thousands or tenth of thousands range in order to do something like this. So amazing. That's exactly the kind of top-down guiding intuition. There was not even the foundation to prove something like this until years later. Riemann as well, a great example. Riemann hypothesis, which is arguably the most famous open problem in all of
human intellect and is certainly the one that we all bowed and worship. It's about so the Riemann hypothesis has so many implications in mathematics. It's one of the millennium price problems and the Riemann hypothesis it's so important precisely because there are now probably tens of thousands of mathematics papers whose first opening line is let us assume that the Riemann hypothesis is correct and the rest of the paper.
So it has so many implications, so that's the kind of that's the kind of conjectures that are great, that it has implications to so many other possible results. The interesting about the Riemann hypothesis is that it appeared as a footnote in Riemann's paper. Riemann was just doing
The women's data function precisely to address a similar problem to this to the precise distribution of prime prime of of of primes and eroding the inner inner footnote that i checked the first couple of zeros and they all have real part a half.
I believe that this is true, but I don't really need this result right now It really you can see that amazing footnote and then people that we just said I can't think about this right now But I think this is kind of interesting and then that was the beginning the birth of that How did Riemann into it something like this? Well, these number theorists and their margins They say exactly number theories love margin exactly though. Yeah, that's what I never thought of it. That's a good point That's a good point But that that's an exactly extremely good point you mentioned
They are written into margins because they haven't been formally approved. If it's structured, it would be bottom-up mathematics and it would be in the main text. And often this marginalia are just afterthoughts or just sparks of genius of these people who just relegate this thing to a side comment.
And that kind of intuition leads to centuries of research. So that's a very good point, Erase, about how the difference between margins and formal text, because papers are written, whether it's pure mathematics or theoretical physics, papers are written in a very structured, backwards kind of way, quite different from the way they're reached in this intuitive kind of way. Yeah, so that's a good point.
yeah and then this doesn't stop and this is something that the memorations that will get more into which is this bst conjecture which is another millennium price problem.
This is another one that carries on $1 million tag. And how did this come about? And I will talk more about what the BST conjecture is. This is Birch, Brian Birch, who is still alive. He's, I think, 90 something. You know, mathematicians are very long lived because they're happy. And the Birch's went and died in the 60s. They're in the basement in Cambridge and they just plotted loads and loads of data for ranks of conductors of elliptic curve. Now,
The LMFDB, which I'm going to talk about in a minute, is a database of 3.6 million curves.
We've progressed quite a bit. A 3.6 million is the kind of data scale where you could really train things on. And that's where I could really come in. And this isn't a stop. They plotted this and they raised this conjecture. They noticed a certain pattern between r and ranks, these technical terms, which I'm going to define in a minute. And that was the birth of yet another great and foundational problem. And this is regarded as central piece of mathematics as well.
And these are all intuited, if you wish. So just one quick slide. I got into this because of algebraic geometry. And so I think I mentioned it in the last talk, just trying to see how machine learning can help us spot patterns, if you wish. But I think since
since 2017 i grew a lot alongside with my son we both grew he's growing very fast and then growing intellectually just to digest this field and it's a humbling experience just to see this vast interaction
of so many different people and experts. And so again, I like to thank all of my collaborators. Now I can't, you can't possibly, I can't possibly read out all the names, but that's where the QR code comes. Scan this QR code. There's a long, this is, we'll point you to Google doc where I will try as much as possible to keep up to date all of the names and affiliations and the papers with my co-authors. So you can think, thank them properly at some point. Now this is an interesting part. I wanted to chat GPT,
To give the generate the list of these people and search the internet and find a picture of each of them and give them affiliation. So to save me the time of typing them out. So we're talking about with a hundred people. Yeah, we can talk off air about that. I can call that up for you extremely easily. Okay. But, but what is really interesting is that chat GPT did a terrible job. It found random affiliations of people who didn't exist because you know, they're doing LLM.
So it's matching them to with, it's confusing my collaborators. So I couldn't possibly credit Chachi PT for this. So this is a very early thing. And also they just couldn't, Chachi PT could not produce the correct photos of any of these people. So we are limited. So as excited as I am, I must point out limitations to all of that. I tried DeepSeek, Claude, I tried them all. None of them could even answer this problem, which should be a simply, simply problem.
There are ways of using the agent or the LLM to interrogate itself so that it can double check. So we can talk about that off air. Oh, wow. That's that. If you can help me with that, it'll be great because this is something this is obviously something I can help us because this is boring and it just needs to be done as even part of scientific discovery. It took me an hour to find that LLMs were useless in this, but that hour I could have tried to do something more meaningful.
But you would take hours if i would just do it properly and include copy and paste pictures it would take many hours and you know that's why we're so this is just to tell you we're not quite there yet even with a simple task like this so surprisingly it can help us with mathematical discovery but you know all of this will change very quickly.
That's the book I think you mentioned earlier, which is this. This is the book of my learning experience, trying to learn about machine learning. This finally came out, I think, there's an archive version in 2018. I think it appeared in 2020, the landscape. This is from everything from machine learning and then this editorial in 2020.
Now let's get back to the real meat of the subject, which is I believe this is still part of this review I was trying to say. I tried to emphasize that bottom-up mathematics is a natural language processing because this is
I like this
analogy because the great David Mumford, who is also a Fields Medalist, even back in the 90s after he got the Fields Medal, he stopped everything. He got the Fields Medal for doing topology in K-Theory, if I remember, algebraic topology. And then he switched fields completely, dropped mathematics altogether and started working in computer vision. And now that I've read more of his sort of his recollections, he blogs as well. He's a very excellent blogger like Terry Tao.
So David Monfort, he said the reason he got into this computer vision thing was I think he was really having early visions of how AI can help with research because he was trying to imagine the human mind being an image processing machine. What does a mathematician actually see? The mathematician is beginning to have mental images of formulae
There is this transformation process from what you see as abstraction, as mathematics, into a mentally constructed image. That's why he was so interested in vision. And that image in the mind is somehow, well, I guess in today's language, this will be the latent representation of your data. You know Hadamard, how he had a book on how mathematicians think.
Oh, I heard of that. I have not read that. That's kind of interesting. Is that the same Hadamard as the Hadamard Delivier-Poussin Hadamard? Oh, that's interesting. Oh, cool guy. Yeah. And I'm wondering if he did a historical analysis for Euler because Euler was blind for half his life or something like that, some large portion of his life. So I'm wondering if Euler still used mental imagery to formulate or solve his problems or then abstract to something else.
Absolutely, it's quite imagined. I had a student once in Oxford a few years ago now. She was quite remarkable because she's completely blind. I think she was blind from an early age. So she sat through my lessons without being able to see anything. She had to picture what I was saying.
and then digested all in her head and do all of the mental calculations in her head. Interesting. So I was wondering what was she actually doing and she did she did fairly well in in her final exams with being completely she needed somebody obviously to to translate whatever she's had to dictate to someone in
Yeah, so it was quite remarkable that I got to know this student. But anyhow, but my image processing is this kind of thing is now that I read Mumford, I'm beginning to think why I was beginning to think that all of mathematics, there's all of top down, all pattern recognition is an image processing problem. Close your eyes, exhale, feel your body relax.
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yeah this i guess there is this old debate and this involved all the grades like atia and digraph and hitchin and not witten all these people is physics or is theoretical physics or is mathematics inherently is the net is nature
um algebraic or is it geometric interesting so this top down mathematics this is debate newton was clearly geometrical he had such a distaste and disdain for algebra because it's it's meaningless symbols to him he made some comment about algebra is this i can't remember the original quote but he will say something it's a very disgusting thing
That you had to resort to this to this meaningless symbols he was very very pictorial his proof of what we now call the gas theorem. About integration integration over spheres this is just some about you know a gravitational body exerting a force on an external object.
Gauss would just surround this by and then uses Gauss's law. But Newton actually integrated piece by piece and used all his intricate pieces together in a diagram and got the same answer. It's the kind of horror show you would never do because Gauss's theorem is just one line. You just do this integral. But Newton actually had to piece together. So Newton was definitely visual. Roger Penrose is definitely visual. Penrose, my last conversation with him, he said
she almost said i think i'm i just in case i in case i in case i i'm putting words in his in his mouth but i believe he says something like
If it's not intuitive and if it's not geometrical, he doesn't even accept that as a proof. And I think Conway was like that as well. This is one of the reasons why Conway never really accepted Richard Borchardt's proof of the moonshine conjectures, because Borchardt used this very strange vertex-operated algebra. I think you had a conversation with Ed Frankel about this. And Borchardt's actually. And Borchardt's, yeah, exactly. But Borchardt, he borrowed this piece of
completely crazy stuff, vertex operator algebras, and had this beautiful structure. It's obviously awesome and brilliant. He got him the Fields Medal, and he was able to use that to prove the moonshine conjecture from Kai. Conway, to my knowledge, who told me this, it was
Oh gosh, the previous director of the is Robert die graph the one before oh god oh gosh goddard peter goddard peter goddard knew conway very well and he was telling me that conway never really deep down accepted this proof of butchers because he's not visual conway's this very playful guy as you're going to imagine he wanted everything to be pitori he wanted to see his lattices so anyhow so
In a way this diagram puts geometry in this direction and puts algebra in this direction. Well how is Conway visualizing the monster group?
Maybe in terms of the leech lattice. He had this picture of the leech lattice. To him, the monster group is some extension of the automorphism of the leech lattice. Which I guess in a way, that's how he and Rhys and Reba originally came up with Monster. It wasn't by very hardcore, this whole funny business of classifying simple groups.
He really intuited it in a way. He got this group out by doing norm two lattices and he was able to see the symmetry, the group of symmetries of this lattice and that's a remarkable thing. It's just unfortunate that whole generation of people, that generation of this
Lattice early computer algebra finite group people are slowly dying out in a conways dead and norton is dead and my my own dear friend john mckay was started moonshine.
past away i read this obituary because i was his last close collaborate he actually became he became a grandfather figure to me he became you know he saw my kids grow up interesting and and makai makai would tell me the stories about how he was interacting with conway and how all of that people have portraits and and makai is also here here's another crazy that's another whole conversation about what is this bizarre intuition that makai had
If there's anybody in the later part of the 20th century who had a almost Ramanujan-like intuition would be John Mackay. In a way, he's an unsung hero because he would just look at lists of numbers or look at pictures and graphs and see, ah, but this field is related to this. And that is very much like this. He's AI before AI. Even physicists?
i think that the whole different conversation we're gonna have to do part three and four this is okay i get very emotional when i talk about john because it's like you know he's a is a very much like a like a father figure to me well the next time i'm i'm in october we should have a part three and we have a part three conversation just talking about moonshine conjectures
In the in the from the perspective mccain i know you you certainly oh yeah and he pronounces names mccain not mccain even though he's written down as mccain he insisted on being called john mccain i know you you you chatted with it with with franco and you chatted with um with uh borchards on moonshine and that stuff but it's unfortunate that he passed away before you started all this thing he passed in he passed in 2022
Because he had a very interesting knowledge of that world Of moonshine and stuff. But anyhow, have you heard of stone duality? stone Yes stone duality or a stone type duality. No, not at all. What is that? So it's a duality between topology and then Boolean algebras which some people see as an analogy or an equivalence between geometry and then syntax or something more algebraic
Oh, interesting. Oh, I'll have to look into that. Thanks for pointing out. I didn't know I is this like the stone of the stone virus truss theorem stone? I believe so. I hate you got them all correct. Yeah, I don't know. Okay I just didn't know there was this this is called the stone correspondence or yeah stone duality. Yeah duality duality Oh, I love I would love to see I'm sorry. I just bumped in my I'd love to see that. Oh interesting very interesting anyhow, so so back to back to our current story
So in 2022, Chachi BT actually, as you know, put into this conversation, Chachi BT passed the Turing test, which I again, I'm very surprised he was not on every single newspaper headline. I don't know why this wasn't emphasized either. We can't really, the Turing test was a big thing.
I think that the fact that Chachi BT passed the Turing test just simply showed that you don't need reasoning or understanding to have intelligent conversation. Maybe it says a lot about humans. Chachi BT passing the Turing test says more about humans than it says about AI thinking. We give too much credit, too much credit to what meaningful conversations are.
But this, sort of as a response to that, we organized a conference in Cambridge with loads and loads of people, and you can probably recognize some of the names, Buzzard and Birch. We tried to formulate something that's more stringent than a Turing test for AI guided discovery.
and so we we reported this with I reported this with my my friend birds of as a nature correspondence and this is again I can't remember whether I talked about this in our did I talked about the the the birch test yes the birch test plus plus or the turing test plus plus was the birch test yeah is the birch test yeah last time so we'll put a link on screen for the last conversation in case you're just tuning in and you're wondering it was a wonderful conversation and I believe we talked about
let's see the birch test bottom up top down meta mathematics and even classifications of cy manifolds and then this database construction right okay so i can pass with with the b so this is ai plus n for the birch test so let me just uh i guess i'm very good at digressing sometimes i digress so much that i don't remember what i'm digressing on anymore but the point the point is this is clear clear signs of adht but i've never had it
As you mentioned, in speaking about these shower thoughts or the margins, the digressions are sometimes more meaty than the meat. Often, yeah, often. But back to this AI-guided discovery, which in terms of AI-assisted, top-down, intuition-guided discovery in mathematics,
There have been various candidates in the past eight years or so. Some of the ones, of course, everybody talks about this beautiful paper in this DeepMind collaboration by Alex Davis. Alex comes here a lot because, you know, DeepMind is in St. Pancras, which is a 30-minute walk from this institute, which is kind of very nice. We have a nice hub in London for this sort of thing. Google DeepMind isn't in California?
There's a London office. Oh, okay. So there's a there's a branch at least I Guess I guess I guess yeah. Yeah, there must be. Yeah. So Alex is actually in London. So that the the Davis et al paper And defined it's cool. So so he comes
very regularly of course he can't because it works for Google for deep mind he can't tell us exactly what he's working on nor can he tell us what the next project will be but at least he can summarize what the state you know what what's going on in that in the in the tech world which is kind of kind of interesting the fact that Nobel prizes are being given to non-university organizations which is right very nice.
Which which is what this organization is at some point. Oh, that's another whole conversation again. So I think Yeah, as for as I think I mentioned to you last time with this, you know, these are the rooms where faraday lived Okay, so these are we're very lucky the london institute. We're we're we're at the the we're with the second floor of the royal institution where where the likes of humphrey davey and thomas young and Michael faraday lived so i'm very
fortunate to be in this space to work and try to get this in. But one of our themes, the reason I mentioned is one of the themes, one of our four research themes is AI for theoretical discovery of this Institute. And it's kind of, we're independent of the universities so that we could devote our time fully to research. That's kind of it. So how does this lead to the murmuration conjecture?
Yeah, I promise to tell you about memorization. Yeah, so these early, this Clabi-L manifolds, which we spent so much time talking about last time, this is because it was my bread and butter as I was growing up as a grad student. So that was clearly the first thing I'm going to apply machine learning to. The kind of experiments of using neural networks to predict topological invariance of these varieties in the image processing kind of way immediately fails the Birch test out straight because it's not interpretable.
Sure now it's been improved to 99.999% or whatever it is, but it's useless to a scientist. It just simply says that, oh yes, there is an underlying pattern, but how do you actually extract anything meaningful from that pattern? That's the main question. So the closest so far, and when I say so far, it really, this could change in a couple of months. Oh, you have no idea because the field is growing so fast.
The closest so far in the last, gosh, it's almost a decade. I guess my son is eight now. In the last decade of all this AI discovery, there's been hundreds of papers now on various things on how do I use machine learning to do this in number theory, in theoretical physics, in quantum field theory, there are hundreds, literally hundreds of papers. Now,
The one that really made Buzzard and Birch happy is this memorization conjecture. And the discovery process of this is something that I would like to see, at least that this is sort of the state of the art of human-machine interaction. That's why it's so close to my heart. And this is joint work with Kiu Huan Li, Thomas Oliver, and Alexey Potsniakov, and now with a paper to appear with Andrew Sutherland, who is
the guy who set up this LMFDB. So I think I mentioned last bits of machine learning experiments in number theory, you know, providing can AI predict primes?
No, we're certainly not at that stage. I'm not saying you can't. If an AI can detect a pattern of primes by itself, then we will be at the next level in not only proving the Riemann hypothesis, we would also crack every single code in every single bank in the world because that's all the cryptography is dependent on this. So wait, what's the main impediment for AI to not predict primes? There's a large data set there.
Yeah, so actually that's a good question. The short answer is I don't know. I've certainly fed in millions of primes into whatever representation into a new network of whatever architecture. I just simply asked it to predict the next one.
It does terribly on this. I think now I think somebody's even written a paper on this called why prime prediction is so hard for neural networks. I can't remember the precise networks. I think at some level it's, at some level it has, this again goes back to the Riemann hypothesis. That's why Riemann hypothesis, the Riemann
Exact patterns in the distribution of the zeros of the Roman zeta function in the critical strip will give you precise patterns in the distribution of the primes and people have proven statistical statements about the distribution of the zeros that they're truly stochastic up to some level. I'm not an analytic number theorist but you can
Basically, there is so much noise or truly stochastic randomness in the distribution of the zeros of the zeta function that it's very difficult if you try to train. In other words, training a neural network of the zeros of the zeta function is like training with noise. Interesting.
Something fundamental, but you could just be that this representation we're using for the set of function is not very good. We should dig deeper, but you need almost another, of course, if you find the right representation that would give a very good pattern spotter, that representation is the new mathematics we're looking for.
So speaking of classifications, as we're both fans of classifications, is there a way to map this problem of doing an image processing for prediction of primes slash the Riemann zeta function zeros? Is it mappable to P versus NP or is it a new class like, okay, problems that can be solved with image recognition or image prediction versus problems that can't in math?
This is a deep question and at some point I was talking to model theorists and especially Boris Zilber who is a leading figure in model theory because in model theory tries to classify mathematical problems in terms of hierarchies of difficulty and this is not a P versus NP kind of
Difficulty but a difficulty in the very underlined structures the questions like why is it that a polynomial over the integers is so hard so much harder than to think about a polynomial defined over the complex numbers. Even though the complex numbers is some in some sense a completion of the integers a but why is it so much harder to look for for models over integers and
At some point, we were thinking that maybe the problems that we're going to encounter that the neural networks will struggle with are ones that will go higher in this hierarchy of difficulty. But we haven't thought much more about this, but it will be very interesting to correlate this. But this is not computational complexity. There should be a new definition of a complexity in terms of how difficult a problem is. But it's still, it's hard to say how to define this.
Hmm, actually, just as an aside and aside on to this is an aside on on the side. Yeah. So there's a contest called the summer of math exposition by three blue, one brown. And it's about getting people to make animations and lessons for math, different math topics. We're doing one on this podcast theories of everything, but for physics and also complexity and physics, AI and complexity. And this is a teaser of an announcement. It's not the full announcement, but those who are listening, it's going to be announced shortly.
And there will be prize money for those who have the best explanation, the top five gets, you'll see. Oh, I love, I love to see that. Amazing, amazing stuff. Oh, sorry. Back to, back to BST. So now finally I have to, so this is the memorization again, there's a, this was, this was considered by, it got Quanta interested and Quanta considered this one of the breakthroughs of 2024 because they obviously,
Because this was something that was AI guided and it was it really surprised the experts and just sort of I want to tell you the story of this because it shows where we are in terms of AI assisted discovery.
Probably my biggest contribution to this was to have insisted on this paper being called Memoration, because I remember this Skype that came through. The original paper was in, oh gosh, three years ago, when this pattern that appeared, and my collaborators are saying, you know, this reminds me of this thing that birds do.
I'm gonna insist that when when this paper gets finished. We gonna. Call this paper the migration phenomena on the car got stuck that's probably my biggest contribution because these are the my collaborators are.
Card Carry Number Theorists, we teamed up because they were trying to explore this AI assisted world and I needed some real experts. So this already breaks, you break the birch test. The fact that I had to look for human experts to try to generate something like this already fails the birch test, but it was worth it. It was worth breaking birch for because I made friends with number theorists and it was something surprising to their community.
Just a bit about the importance of the BST Conjecture, but this is kind of nice because it gives an opportunity for me to share my own ignorance on the BST Conjecture because I learned about this as I was working not as a number theorist, which I'm not, but as an amateur number theorist coming from AI discovery side.
So this made me appreciate why the BSD conjecture is so important and why it's so interesting and how surprisingly AI can help with this and how the Birch test was almost met by this particular problem. So let's go way back to the Diffentine equations.
Diophantine equations, named after Diophantus, is just about finding rational or integer solutions to polynomials. I said these two are equivalent because you can always rationalize the denominator and cancel out. So finding solutions over q is really kind of the same as finding solutions over z.
So a typical example of a Diophantine equation is find all the rational solutions to this and the solution this here is is Pythagoras. So Pythagoras tells us this is probably the most famous example three-fifths squared plus four-fifths squared is equal to one. If you think about it this is actually highly non-trivial. The fact that you know
The solution is that there is a one parameter infinite family solution of solutions.
What we say points because you can plot this and this is thanks to Descartes you can plot this and this is a circle.
So you're finding rational points on a unit circle. That's why the word solution and point is used interchangeably in this field called arithmetic geometry.
This is great, but what I want to emphasize is that what's less known is that this is obviously just one solution, but there is a one parameter infinite solution to this. So in other words, all solutions can be parameterized in a specific way. So this is a quadratic case. Now if we recall high school algebra,
or high school Cartesian geometry, a quadratic is what's known as a conic section. You're slicing the cone. If you bump it up, it's already extremely difficult. If you bump up the degree, it's already become an impossible problem. So instead of considering x squared plus y squared is equal to one, by the way, all of the quadratic ones can be solved in a similar way. So all the conic sections
Conic sections not over the complex numbers, but conic sections over the rationals can be solved in a similar way. But once you go into cubics, you're completely stuck. Even something simple like this, how I change the two into three, how do you find all rational points on this? We still don't know in general, in some sense. But you can see the kind of problems we can get in Fermat, for example, is about talking about higher degree polynomials. Fermat's last theorem is
x to the n plus y to the n is equal to one and find all rational points on that particular curve. That's it. And the theorem states that the only rational ones are the quadratic ones. And they're from three and above. So x cubed plus y cubed is equal to one. There cannot exist any rational points and so on and so forth.
So now these are called conic sections, that's just a curve like parabolas and stuff like this and circles and ellipses. Once it's a cubic, it's called an elliptic curve. There's a general theorem that so you can imagine there could be more terms, right? Why not have something like x squared y, that's also a cubic, is a degree three, or x y squared, that's also a cubic and so on.
There's a theorem by Weierstrass that all of these ones can be reduced after transformations of variables into this form. So you have a quadratic in y and a cubic in x and then a linear in x. You can transform away all of the other coefficients if you wish. So this is called the general Weierstrass representation of an elliptic curve. This is my, well, I grew up in this one and it is important to emphasize
that that my bias towards favoring over this is what exactly prevented me from being able to understand any of this from the AI point of view in the beginning because for for an algebraic geometry this is the one that we always use and it's like our favorite thing we always try to try to use something and i will tell how an experiment failed playing with this just because i was taught to always think in terms of bias transform this is a canonical representation of elliptic curves
So just before we move on, this wire stress form, this means that any elliptic curve can be classified by these two numbers G2 and G4? Yeah, exactly, exactly. Yeah, there's a variable transformation that puts you into this canonical form. Okay, so I know you'll get to it, but I'm interested as to why this prevented you because I could imagine that these two numbers can serve as
Something like pixels or RGB numbers. You're reading my mind. You're reading my mind. That's exactly the experiment that I tried and I failed. And in hindsight, it's not surprising how I failed.
But I'll get to that in a minute. So let's park that idea. So just like canonical conics can be written into some kind of standard conics that we would remember from high school, the canonical cubic can be written into this virus rostrum. The important thing about the cubic thing is somehow this cubic curve, there are very deep reasons for this, captures a lot of the non-trivial arithmetic and number theory.
For example, the Fermat's last theorem was able to be cracked because Frey
And friends were able to reduce formats, the format. Well, that's not, you know, that's neither a conic nor cubic, but they were able to reduce that to a particular elliptic curve called the free elliptic curve. And then Wiles comes in and proves the modularity theorem. That's a whole big story. So somehow this, this cubic is just at the intersection. That's why, Oh, by the way, cubics are great because this is an example. Well, this is the only example of a Clavier manifold in dimension one.
So there's something about the elliptic curve, the cubic curve. So all clavials in complex dimension one, remember the picture that I drew last time, where you have positive curvature is the sphere, zero curvature is the torus, and surfaces of general type are negative curvature. This is the Euler-Riemann normalization.
The Riemann uniformization theorem. But the critical case, positive curvature, negative curvature, zero curvature, the torus, if you were to represent this algebraically, that's exactly this elliptic curve. So elliptic curves are Clavier manifolds of complex dimension one. And this is just the critical part, the critical part that also captures so much number theory. So that's why...
So algebraic geometers, differential geometers, physicists, and number theorists are interested in clabialness because of this intrinsic zero curvature. There's a lot of depth about this statement. Zero-curvatured objects give so much wealth because it's just at the boundary of positive and negative curvature. Oh, and that's what you mean by criticality is zero curvature?
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It's the dividing point and there's lots of conjectures. Yao has various conjectures on this about finiteness of this topological type in this space. But anyhow, back to, so that's why I know about the curves because I came from this algebra geometry string theory background that also wanted to study this Ritchie curvature flat or zero curvature objects. So back to this.
So now it's a theorem, and this is a theorem due to so many people that have gotten the Fields Medal for proving different parts of this theorem. And this theorem really, really spanned a long time. People like Wey, Delene, Grotendieck, Dwork, Fountains, and Modell, they all contributed to this. And the theorem is this.
We can't say something like Pythagoras, which says that there's a one family, infinite family parameter solution to the rational points of the quadratic or the conic.
we can't say something like that because it's too hard but at least what we can say is the following is that any elliptic curve over q of the rational points over any elliptic curve itself forms a group and the group is of this form.
The group is that there's an infinite parameter family of solutions. That's called R, it's called the rank. So how many copies of infinite solutions there are?
And then there is this finite, what's called the torsion solution. There are 16 types of torsion. This is really at the heart. All of this stuff, by the way, is at the heart of the Langlands. Ed Frankel would have told you about how excited he is about this sort of thing. But the one particular thing I want to emphasize is this rank
Is the number of infinite family of solutions and this is the rank of an elliptic curve. Yeah this is the mortal while theorem or the mortal exactly exactly exactly exactly exactly what are they.
Anyhow, so the reason I want to emphasize there was still nothing to do with BSD, but this rank is the infinite number, how many measures, how many infinite family solutions over Q. So in the case of Pythagoras, the rank, if you wish, would be one because there's a one family, one infinite family, infinite family. So R is the generalization of one from the conic section case to the elliptic curve case.
So that's it. This really is the state of the art in which you can say about rational points of an elliptic curve. There is this wonderful thing called rank, which is actually quite difficult to compute. It's not like I give you, no, I can just read it off. It's not like there's some analytic formula that says, ah, yeah, I get this. I look at this form one, this is x cubed, y squared. I can just tell you the rank is what I can't remember what it is in this case, two or whatever it is.
The earliest experiment that I did was exactly as you suggested. This is back in 2019. I just took a database of about 3 million elliptic curves in Varshray's case. I took the two numbers G2 and G4.
And then this was done. So this was a paper that I did with. So the reason is I did it with two data scientists who are using the fanciest data possible. And so we're a bunch of amateurs as far as BSTs is concerned, was to take the G2 G4 as two parameters and just plot them and then label them by R, the rank, and try to see a pattern. We got a null result.
Because it was because this G2G4 in the database, I can show it's actually it's massive there in the in the in the trillions. So it's very hard to get much. We had to take the log of all these numbers to even establish a plot. And the rank was so randomly distributed, even with the fanciest technology. So what's the solution to this problem? Yeah, we got it. We didn't see anything.
i was just we couldn't we couldn't get g2 and g4 to predict to any accuracy level what the rank is but nevertheless this was featured by new scientists because it was such a strange and novel thing to do
even though it was a null result, but at least it was inching towards something. Somebody someday must be able to say something intelligible at BSD from a data science point of view. Anyhow, back to this. So what should be the thing to do? And this is where number theory expertise actually comes in. First of all, this is an old law, which is if you can't solve a solution over the integers,
Solve it modulo primed and see how far you can get. So for example, I can't think of a rational number that I can't off the top of my head now. I think there is there exist solutions on top of my head of a rational point on this elliptic curve, but at least let me try to work over modulo prime. So modulo 23, this is true.
You can check it because 2 cubed is 8, 8 plus 16 is 24, and that is one modulo. So that works. This one works. You can see in the modulo 5 this works. So, okay, this seems like a game, but the deep results of all the people like Deline and Wei and all these people is that if you work over a sufficient number of primes, in fact, if you worked over all primes and take a limit,
you should know something very deep about the solutions over q and that's the point. So in particular what you should record are these Euler coefficients which is the number of solutions modulo prime and and how they deviate from p plus one to the prime itself so this is what's known known as a as an Euler coefficients just keep track start with two
And then try three, try four, try three, try five, try seven, and then find how many solutions there are. And this is a finite problem. You can just do, in the worst case, a grid search because you're doing modular prime. You just have to count the number and just check by brute force. So now what you do is to form what's called the local zeta function.
The local zeta function is this exponential generating function that keeps track of the number of solutions over p. So you fix a prime and you literally count the number of solutions modulo p.
The more technical thing, what you really should do is because all finite fields are prime powers, what you actually do is to do an exponential generating function over the number of solutions over the field of prime characteristic p, but that's a technical side. But what you really should do, what you're essentially doing is to keep track of the number of solutions of the elliptic curve, the number of, not q points, but the number of fp points.
Okay. And the fact that this generating function becomes this polynomial divided by polynomial form is what got when Deline proved this, he got the Fields Medal for showing that this is in this particular form. It's a very, very, very deep result in number theory. So, okay, long story short is that when I met Oliver and Lee,
They told me that whatever the hell I was doing with these data scientists, no offense to data scientists, we were just amateurs, we shouldn't have used the Weierstrass representation because the Weierstrass representation inherently is not what an arithmetic geometer, what a number theorist would be using that would capture the fundamental arithmetic of elliptic curves. So now there's the geometry of elliptic curves, which is my background, but that doesn't capture, the Weierstrass form doesn't capture that.
The AP coefficients captures the arithmetic, so we should be using AP coefficients to predict the rank. So then we did, so instead of using a pair of very large integers, G2 and G4, and set up a neural network to predict R, you take instead the first, say, 100 AP coefficients.
Now we're in a very interesting point cloud of a hundred dimensional point cloud. It turns out 50 is enough. It doesn't have to be a hundred just randomly chosen. You take the first 50, 50 primes. It's just not too crazy now for, you know, for, for by today's AI standards. Take the first list, let's do a hundred. Take the first AP, take the first 100 primes. Now take an elliptic curve.
Reduce this elliptic curve and count how many solutions modulate these primes and compute these polar coefficients. So for each elliptic curve, you get this 100-dimensional vector. Move to the next elliptic curve, 100-dimensional vector. Now you just start labeling.
Because of this wonderful thing called LMFDB, the LMFDB was set up by a bunch of people. I think Andrew Sutherland at MIT is one of the one of the instigators of the Sutherland and Booker and bunch of people set up this thing, which just records everything you ever wanted to know about elliptic curves. There are tens of millions, on the order of 10 million in this data set.
So now you've got 100 dimensional vectors as you march through, because the LMFDB has the rank information, you can start set up a newer network. So we set up a newer network or some other classifier. And just like what we did with, just like what I did with Alessandro Andretti and Paranchelli on using G2, G4, this pair of vastrascoviches as a project rank, that gave no result.
And when we did this for 100 differential vector, this immediately gave 99.99% accuracy in prediction. You went from zero to almost 100% immediately. And you're using less data as well?
I'm using less data. Remember, with the data scientists, with the G2, G4, with these guys, we used something like all 3.5 million Euclid curves. And we couldn't get any accuracies at all. But with this one, even with 100,000, so you give me any elliptic curve, you just look at the Euler coefficients in this one channel demands, I can tell you with almost 100% accuracy what this rank going to be. Interesting.
So, of course, this immediately breaks birds' nests because I had to talk to real human experts who told me to actually use the oil representation rather than the partial representation. At Capella University, learning online doesn't mean learning alone. You'll get support from people who care about your success, like your enrollment specialist who gets to know you and the goals you'd like to achieve.
You'll also get a designated academic coach who's with you throughout your entire program. Plus, career coaches are available to help you navigate your professional goals. A different future is closer than you think with Capella University. Learn more at capella.edu. But at the time, I was amazed. I was like, this is so cool. Of course, this is still useless for science. This was just a very, very cool thing to do in terms of visualizing the curves.
And then after some digging, we realized what really was under the hood that was happening. You know, this is when Lee and Oliver were telling me, because they're number three, he says, well, this is no surprise, because this is the BSD conjecture at work.
somewhere under the hood, this is the BSD conjecture at work. So now I can finally define for you with the BSD conjecture. I will give you the weak version so as not to bore you for two reasons. One, the strong version is very too technical, and two, I don't even understand it very well myself. But the weak version simply says, if you take this generating function that keeps track of the Euler coefficients, form this product polynomial,
Now you take a product of all primes. This is what's called the local to global behavior because you localize it. Remember this zeta function is local to a particular prime and now you take the product of all primes you form this new function called the global L function. Now I want to emphasize this is called zeta function
Okay.
if you were to work over point but now we didn't we were more sophisticated our algebraic variety our manifold is not just a point but an actual elliptic curve so it becomes a much more richer structure so you get this this so this l function really is this global l function really is an analog of the riemann zeta function
So that's why this whole Langlands business is so beautiful and so intricate because he unifies geometry with geometry with harmonic analysis with number theory with all this is why eddick franco was so excited about all of this stuff right he was saying with such because he unifies so many different branches of mathematics anyhow so the bst conjecture states we don't know what the rank is you can't just buy you can't do it by looking at the curve
but once you have the L function by by this strange procedure working modulo over of a fixed prime and then take the product of all primes you get an analytic function now we have an analytic function i can start mucking around with i can use complex analysis the order of vanishing of this analytic function at one this is
Remember, the inverse of the Riemann zeta function
at one. The Riemann zeta function has a simple pole at one, so its reciprocal will have a vanishing of order one. Its order of zero is one at s equal to one. So it's a good analogy with the Riemann zeta function. But for elliptic curves, instead of a point, this one, the order vanishing should exactly equal to rank. So this gives you a way to compute rank, albeit a very, very convoluted and very, very intricate way to compute rank.
But what surprised us was a neural network was able to predict this rank, this number, without going through any of this stuff and predicted almost to 100% accuracy. So where's your million dollars? Well, there's no million dollars because one, there's no proof, there is no statement. And so what? So this is we're still in the
As far as the Birch test, the A test has already failed because we had to talk to real experts. The I test has already failed at this point because there's no interpretability. So now how do we break the I test and then surprisingly also break the N test that should also be non-trivial? So that's why the memorization phenomenon was important.
So then we got this prediction, we were very excited, oh yeah it's always really cool and then the excitement died down and we're like so what, what do we do? So this is the wonderful moment where you can recruit an undergraduate intern.
So Alexey Potsnikov was at the time a second year undergraduate student of Kiu Huan Li's. But anyhow, so Alexey was given this thing like dig under the hood and tell us what this neural network or this classifier, this base classifier or tree classifier are actually doing. And finally, we homed in onto a PCA analysis.
principal component analysis because basically we found that this rank prediction was doing so well with basically anything. Naive Bayes classifiers, you know, newer networks of a very simple architecture. We didn't even have to go to transformers or encoder architectures. It's just a simple, you know, forward feed linear active linear structure with, you know, I think it was a basically sigmoid activation function was good enough to do this.
so and pca certainly would do it so if you remember pc i mean i said i i so pca i had to learn i didn't know what a pca was until 2017 pca was just a principal component analysis here's a 100 dimensional data you know is a point crowd i can't visualize it so i find these eigenvalues and and find the the the principal components of these eigenvalues meaning like the ones that the data is has most variance of
and then focus and project onto this eigenvalue directions. This is like stats 101, which I never learned, but luckily I knew what a PCA was, so we were just mucking around with PCA. So here's the remarkable fact. If you take a PCA projection of this 100 dimensional vector space of elliptic curve Euler coefficients,
Sorry, is it important that you do your principal component analysis down to two dimensions or does it just happen to work out that way in this case?
So in our case, we chose two because it was easier to see. It would have done it, if you project in any other dimension, you will see this kind of separation. Sure. Yeah. Two was just nicest. And in retrospect, we really should have chosen like an Italian flag or, you know, or a French flag, because it really just looks like a flag. In this one, this is elliptic curves of rank zero, rank one and rank two.
This is already quite interesting, right? It's still useless in terms of actual mathematics. It was very good for AI that, you know, AI was able to just, you know, there's not even really AI. At this point, it was just PCA analysis. It was just a data analysis in a picture that analyzed elliptic curves in a way that was never done before. This is 2020 when we had this result.
So we look at this and we saw that this is kind of nice. You know, they're elliptic curves, they're separated. Oh, by the way, elliptic curves, the ranks of elliptic curves is again a hugely, you can imagine because of the BSD. So Manjur Bhagawa was able to prove that almost all elliptic curves are ranked either zero or one in the infinite limit. And he got the Fields Medal for that. So this is obviously a very, very important result.
But in the NFDB, you do also have higher ranks as well. Just, you know, in the infinite limit, you will be vastly dominated by ranks zero and one. So there are either no rational points, no infinite families of rational points or one parameter family, just like the conic section case. But the world record, I believe, is the one that's held by Noam Alkes, who has discovered this rank 28.
a elliptic curve. It's huge. You can write it out. And that one is, so rank 28 means there are 28 parameter families of rational points on that particular curve. But so in LMFDB, you can already see there is sufficient number of ranks 0, 1, and 2 cases that there is this 3. I mean, there are rank 4s as well, but you won't see that in this picture.
Now, so why do we why do I emphasize some PCA? No, I'm finishing with this now with PCA is because PCA is just a matrix projection It's a linear transformation You could look under the hood and just look at these matrices Okay, and that's what we that's what we asked Alexey to do. What does it mean to look under the hood at the matrix?
Because this is a PCA just because you know this is a it says 100 dimensional vector point cloud being projected to two dimensions so there's a whole bunch of 100 by two matrices you can just look at but nobody ever does this right you don't you don't look at you know what the AI algorithm is doing but in this case you can just make people look at it.
And so we gave it to Alexi to look at it, not expecting much. And this is my undergraduate. But Alexi expected all expectations. He really looked at a lot of the sample of these matrices and noticed that almost all of the non-zero values are focused on essentially just one row. The one row dominated vastly over any other. So what does that mean in terms of what? So that's interesting.
If you have a PCA projection, if you have a matrix projection that's focused on just one row, that means it's essentially you're just doing a sum. You're just doing an average in some sense. And that's exactly it. So now, what Wood is actually doing is that it's taking its Euler coefficients
an average in over a particular range of elliptic curves ordered by conductor, and I won't bore you with the details of conductor, and it's just computing this average. And if you do this average plotted against primes, you will start seeing this memorization phenomenon. So let me just emphasize a few points on this. First of all,
You're taking and this we wouldn't have done this if we didn't do a PCA analysis or if we first of if we didn't do this machine learning exercise on rank, we wouldn't have dug under the hood in the first place. So that was already AI guided. And then it told us we homed in on PCA because we thought that might have been the most interpretable thing. And once we do interpret it, it gave this equation.
Which is a very very simple equation which always always says you just to do the following take a families of elliptic curves order them by this conductor range and and average over different elliptic curves but at a fixed prime.
This is come to known as a vertical average. It's a very strange thing to do because traditionally what you would do is to average for fixed elliptic curves or average over different primes, right? You know, things like take the product formulas for a fixed elliptic curve and average over different primes. But the PCA told us no, you do the opposite. You take different elliptic curves over a range and average over a fixed curve.
over a fixed prime and plot the thing against prime. Once you plot it, you see exactly what this is. So the red bit are all of these zero rank ones and the blue bit are exactly all the rank one ones. So the reason that all of the neural networks and all of the stuff were able to fundamentally tell the difference between
Hmm.
It's guiding us what to do. It's guiding us. You can do this to all of the other ranks and just isolate different ranks and plot them. And it turns out that all of the even parity ranks, all the even ranks oscillated in this way and all of the odd ranks oscillated in the other way. Remember, there is 0, 1, 0, 1, 2, 3. They're all these different ranks. So fundamentally, you can tell the parity of the rank just by the way that these oscillation patterns happen.
This is the point. This was a plot that was produced by Lee and Posnikoff. Posnikoff did this and showed us this. I remember this Zoom chat very well. This is 2020-2021. There were still COVID times. All we did all day was to Zoom people. My friends were saying, this looks like what these birds do.
I say I looks like my rations of my my rations that's what it is my rations starlings and i said that is a way we should absolutely call this phenomenon. Memorations we should call instead of calling it a boring oscillatory pattern which is a this is an emigration like because it's not quite oscillatory right because it's kind of there's there's noise around it.
And this noise is very interesting. I'll tell you in a bit. So this noise is part of the statistical error that you get from doing with finite data, because we know we only had, you know, 3.6 million, whatever LMFDB. But the point was, when then we immediately wrote to Sarnak, and to Sutherland, who are the leaders in this field, thinking that all this is trivial.
This is a typical kind of thing. You write to the expert and you get a reply within a day. And he says, oh yeah, this is trivial. And it is a consequence of this theorem that I proved 20 years ago in this paper. This is the usual story, like 100 times is this, right? But not only do we not get back a thing, this message, we got a long message back from Peter Sarnak, who wrote,
What the hell guys, this is bizarre that this pattern exists. So why? Right. And, and then there was this many, many emails back and forth. I mean, to be honest, a lot of these emails were way over my head because you know, I, this is my contribution was, was an AI guided algebraic geometry. I'm not a number theorist. So there's lots of back and forth and, and, and then, and then this became the memorization, memorization phenomenon. And then there were all these conferences organized.
So we're back to this birth test, right? So it failed the A because it wasn't automatic. We needed human. We were mucking around all the time with human experts. It passed the I test because it became interpretable. This was a precise formula.
Most importantly, it passed the end test. This is the first time that an end test was passed because it actually galvanized a field of study number three. Now there's a whole field called murmuration phenomena.
Wow. This is totally out of like, what the hell? I mean, this is above my pay grade because I don't know the number three community that well at all. So that's why Quanto was so exciting. There were conferences organized in ICERM. There are all these people, there are workshops apparently in Bristol and various universities. They're like, oh yeah, there's this kind of memorization workshop on this.
so i'm not gonna i need to wrap up because i think it's getting too too much detail i want to tell you that there are other parts of this memorization thing and now but but now you can you can this is a precise conjecture this is a precise conjecture that was raised
Buy guided by a explorations with let humans and peter sun access is really well he says this is a conjecture that versions went and i could have raised themselves but they didn't because i never thought to take this average.
because it's a bizarre thing to do. The AI doesn't know what it's doing. It always is doing this spotting patterns. So, and just to emphasize the parts of this conjecture now of the phenomena, which is expected to be true for all L functions. So this is why, so in other words, the, the memorization phenomena should be a general phenomenon in the entire Langlands program. So it's, it's now proven for Dirichlet character, the memorization Dirichlet character is now proved. So this memorization actually converges to a precise curve.
And this was proven by, I wasn't involved in this because this was an actual number theory paper. And then Nina Zubrilina, who is a Sanax PhD student, who was a Sanax student, and then Alex Cohen, and now they've proved they've awaited two modular forms as well, and this was all in 2023. In 2024, there are more results being precisely proven. And what it really is, and this goes back to
Gauss and to Riemann zeta is that this memorization fundamentally generalizes a bias in the distribution of primes. That's also quite striking. So this is an interesting fact. So Chebyshev noticed this factor before. So Chebyshev noticed it's called the Chebyshev bias. If you take all primes,
If you take primes and find the remainder of these primes upon division by four, you're going to get remainder either one or three because you're modulo four and you would have thought that it's 50% one and 50% three in the large limit.
But Chebyshev in the 19th century already noticed there's just a tiny bit of a bias towards three than one.
and and that was just a conjecture of his and this is again mucking around with data this platonic data like why is it the primes are are more biased towards one one of the ones to upon division by four this is known as chebyshev's bias and this was proven actually by sarnak and and um rubinstein rubinstein in the 90s but only conditional on the riemann hypothesis being true
Interesting, right? There is this fundamental bias, so there's no unconditional proof of this, but it's conditional on the Riemann hypothesis that this is true. But what is interesting is that the memorization phenomenon is a generalization of this Chebyshev bias to all of the L-function world. So it's a generalization from prime biases to the biases in all L-functions because there is this underlined oscillatory behavior.
Add that also has this deep relation with the bsd conjecture so that's where this whole this whole world um and that's why so so so so it it it passes i passes n people still work on it but it doesn't pass a because you know human expertise were constantly involved in interpreting you choosing pca
Anyhow, to wrap up this whole story, where are we in terms of mathematical conjectures in formulating problems in this top-down, guided mathematical discovery over the centuries? And I will say that in the 19th century, the eyes of Gauss were good enough to come up with this conjecture.
But the 20th century BST already needed a computer to come up with a groundbreaking conjecture. And we are now just at the castle. But that's why so exciting that AI guided human intuition led to this deep mind paper and led to the memorization thing and this new in a new matrix multiplication. So obviously where we're going is this is this is this is a combination of these three directions.
I mentioned earlier, in these very rooms that I'm talking, that Faraday would have had the conversation with Maxwell in this room about 150 years ago. Our institute has devoted one of our four themes to this AI for mathematical physics because this is really a paradigm shift in terms of how science is done.
So just to promise, this is the last slide. So what is the current state of the art? Where are we with this AI guided thing? Let's drop this human guided intuition for the time being.
Alpha G02 DeepMind has now reached silver medal. I think we had this long joke and discussion about how to beat the mind, the 16 year old Terry, sorry, the 12 year old Terry Tao. When you beat the 16 year old Terry Tao, it's game over. That's a new age of scientific experiments. But we're making like these guys or this field, we as a field, as a community, we're making progress every couple of months.
the U.S. military thing.
But you were not defined military. So the military wing, the DARPA, which is the Defense Advanced Research Institute, I was just at a meeting with them two weeks ago, just launched eXp math, where they literally are saying, exponentiating mathematics, you can Google this, the eXp math project that DARPA is funding now is how to benchmark
Proving accelerating mathematics by proving but unfortunately, they're not funding in this AI guided discovery area, which I think should be funded. And that's the way, you know, combination of all these three directions. Anyhow, so that's alpha due to alpha proof again, deep mind is on proving theorems at an almost
almost research level. And I think we even joked last time that at the silver medal level, AlphaGo2 is high school level, AlphaProof is maybe college level, right? But this is the interesting one. I think this is very just to wrap up the EPOC AI Frontier Math Project
And you can Google this. And this is really on professional mathematical problems. The kind of problems you would give it to a colleague or to a researcher, a very advanced postdoc or graduate student. And they're benchmarking this now. In fact, I'm flying to Berkeley on Thursday to help. There's a whole team of us we're flying in to benchmark the tier four problems. And this is happening this weekend, actually. So by
As of December 2024, EPOC AI is capable of solving only 2% of advanced research level mathematical problems. But by March, they finished their tier 1 to tier 3 problems, and they gave this division of tier 1 to 3 problems and gradual level problems. You can go to this website to look how hard
These problems are and terry tau gave some problems and i'm and i can all know these are all really research problems and um i gave a problem to the tier four which is which is about to appear um and there's there's still soliciting more and just just go to this link you can just look oh my god this is the kind of problem i would give to my to my research student
My order to the kind of problems i would work on with with a with a collaborator like that we would write papers about and their gate and their benchmark is about ten to twenty five percent on tier one to three so once you did their next benchmark is tier four so hard.
that you know humans are not likely to solve it not because they're they're true not that not because they're tricky like you know um olympia problem but but because they're actual research level problems and and we are actually together as a community are attacking this kind of problems so this is where the the state of the art state of the artist so in terms of where the future of mathematics is i think
So I think I try to summarize in this picture. So I'm using an old picture of Terry Tao because he's the best human mathematician. And so how would it go? So you would have literature, the corpus of literature from scientific papers,
You go back and forth where human and I would process it together and then use top-down mathematics to formulate conjectures from platonic data that's gathered from the literature or process it directly from the literature and formulate the kind of problems. Once the problem is formulated, you would go to auto-formalization
where you it's only a matter of time before lean the lean community gets it get you know by also formalization i mean you take a math paper in latech and just hit return you know translated to a lean uh to a lean format we're very far from that right now um having conversations with buzzard
Not because of the technology is not there but because there's not enough lean data to train a large language model on interesting we only have millions of lines of. Lean so far available just millions we need billions in order to have and it's millions of you know this poor guys type in everything right yeah so.
So conjecture formulation to this meta formalization and then you go through and find pathways through through mathly lib in this bottom up approach where you would have a combination of LLMs and that would generate your proof.
I think it's it's where we already heading toward where where.
It's not immediately foreseeable future that all of this is going to be automated, but what is remarkable is that this is within reach and I can't put a date on it, maybe 10 years, maybe 5 to 10 years.
Because every single step of this, we are being helped by AI. Like for me, for example, the the memorization, which is taking available data and formulating in this way. And already last week, I'm meeting people who are actually coming up with proof pathways by even by by chat GPT and then humanly verifying that so.
This is the brave new world of mathematics i mean by a theory of new discovery and where we were so lucky to be in this age where where has advanced enough we could actually help us with genuine new discoveries. Young thank you so much for bringing us to literally the front here the bleeding edge of math and also the future.
It's a great pleasure talking to you. Thank you for listening. I get very excited about this. I'm parking everything else so I could devote to this new community and it's a growing community of mathematicians who believe in this and I think there was a recent
There's a recent Quanta report. I'm not involved in that one. It involves experts like Andrew Granville, who talked about what is a beautiful proof and how AI can help us with it. And they are also amazed at how fast this is going. Interesting. Thank you. Thank you very much.
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"text": " Thank you very much for having me again. I had so much fun talking to you last time and it's great fun. I got so excited about this new field of AI assisted theoretical discovery in pure mathematics and theoretical physics that we're really entering a new era of discovery. It's amazing. It's happening every week. There's something new that could potentially be transformative in the way we do science."
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"text": " problems which are research level mathematics the kind of problems you would give to a graduate student or to give a collaborator or colleague and i can show you some of that later some of some of the stuff i'd love and they're they're entering their uh they're entering their tier four phase and in fact i'm flying to to berkeley on and thursday to have a meeting with bunch of other mathematicians"
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"text": " to benchmark how far that their tier four reasoning machine can go. So this is really rather, you know, it's happening. It's happening as we speak, how the landscape of research can be changed. Yeah, sorry. Yeah. Yeah, I think that's in part why firstly, it's an honor to speak with you again. Our last conversation went viral. I think that's this whole AI assisted research in math and physics is part of the reason because our last conversation was quite specialized in nature."
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"start_time": 383.831,
"text": " oh wait beyond that in the last eight years and we're all very very impressed with how we as a community we're all very impressed with how fast this this thing is going so it's a it's a great pleasure talking to you know as i said i'm a i'm a big fan the kind of the kind of depth that you go into"
},
{
"end_time": 428.592,
"index": 18,
"start_time": 402.261,
"text": " Okay, well, let's dig deep. What are some different ways that people use machine learning? So for instance, Terry Tao uses it as"
},
{
"end_time": 444.991,
"index": 19,
"start_time": 428.882,
"text": " an assistant to proofs, but also to generate conjectures and perhaps they even point to existing tools that he may not have heard of. And those are more of the LLM sort. Then there's another sort of just finding patterns in large data sets. And that connects to your memorization conjecture. Is there a third category?"
},
{
"end_time": 469.241,
"index": 20,
"start_time": 445.623,
"text": " Yeah that's right, so I was just trying to, I think I outlined this briefly last time, I mean just trying to, because this is exactly the kind of thought process that's going on, how to categorize the different approaches and of course they're all interrelated and it's hard to delineate them, but so this is my you know my top-down mathematics is this"
},
{
"end_time": 490.486,
"index": 21,
"start_time": 469.241,
"text": " Intuition guided basis of mathematical research and this LLM approach is what I called meta mathematics and this is kind of LLM you know LLM assisted co-pilots that Terry Tao is talking about and then this third category of you know bottom up"
},
{
"end_time": 518.712,
"index": 22,
"start_time": 490.776,
"text": " where not necessarily any, any AI is involved. This is, I'm thinking about things like lean provers and proof as co-pilots, where you just have millions of lines of code. And well, sooner or later, somebody is going to process that by, by AI. The interplay between these three directions and between that and the human is clearly beginning to change the landscape of mathematical research. Now, before we get into your presentation here, you mentioned"
},
{
"end_time": 540.128,
"index": 23,
"start_time": 519.002,
"text": " Yeah, I think"
},
{
"end_time": 556.288,
"index": 24,
"start_time": 540.725,
"text": " I think it originated from a Greek fable where the fox and the hedgehog are compared"
},
{
"end_time": 580.879,
"index": 25,
"start_time": 556.288,
"text": " Where the fox knows a lot of things and hedgehog likes to dig in and I think the great mathematician Arnold made a reference to this in classifying mathematicians where he calls things like you know one is an eagle that flies and tries to see the landscape and then the hedgehog digs deep to one particular problem solves it. I mean there's no particular"
},
{
"end_time": 610.759,
"index": 26,
"start_time": 581.254,
"text": " I know neither is superior to the other. It says we definitely need both and each mathematician can function as both. But certainly AI is helping us in both personalities simply because there's so much literature out there. The AI can have an overview of what everything is in terms of literature. And also there's so much technical detail"
},
{
"end_time": 631.169,
"index": 27,
"start_time": 611.186,
"text": " You know there's some very boring parts of the proof that you just simply don't have time to iron out and that can certainly be helped by LLM models and it is beginning to do that. I was actually just at a conference last week in Exeter. There was a conference called the impact of AI for mathematics."
},
{
"end_time": 654.838,
"index": 28,
"start_time": 631.357,
"text": " Where the organizer Madhu Das and she was saying that she's a number theorist and she said she's just recently completed this very complicated paper where she had to prove a lemma and then what she did was she knew there's another lemma which has got to be true. Okay. And so what she did was she"
},
{
"end_time": 679.326,
"index": 29,
"start_time": 655.606,
"text": " copied and pasted the entire proof of her lemma, it's a bit technical stuff, into chat GPT-01 and then said now can then copy the lemma and said can you supply the basic proof strategy of this lemma which I know to be true. Now to be fair what this is very far from automated reasoning this is just language model"
},
{
"end_time": 706.357,
"index": 30,
"start_time": 680.418,
"text": " And then, importantly, what she did was, and then she went line by line, symbol by symbol, the proof that Chachi Biti gave her for the Big Long Lemma. And it was largely correct. And with a bit of prompting, she was actually able to nudge out a complete version of that proof. And then that was done. I mean, so it would have taken her much longer to have ironed out all the details herself."
},
{
"end_time": 735.759,
"index": 31,
"start_time": 706.357,
"text": " So this is really rather impressive and this is only because of O1 and O1 came out I think this year or end of last year. So it's really transforming the kind of stuff that the boring stuff you can delegate and also the pattern recognition part you can also delegate. So we become like superhuman all of a sudden and when interact with these agents who can help us actually do research and not just do you know very elementary problems now but serious research level"
},
{
"end_time": 746.084,
"index": 32,
"start_time": 736.374,
"text": " Why don't you talk about some of your personal use cases? So do you use chat GPT more than Claude or do you use Gemini more than the others? What is your mixture?"
},
{
"end_time": 774.206,
"index": 33,
"start_time": 746.613,
"text": " I actually, in terms of research, to be honest, prior talking to her, I never even really played around with chat, you know, this kind of LLMs to help with my research. I didn't know that was as possible. I know, you know, it's kind of thing you can, if you want to know very quickly a topic, you can go to Wiki, you can Google, but indeed ChachiBT or DeepSeek will probably answer your question very quickly. If there's some"
},
{
"end_time": 800.196,
"index": 34,
"start_time": 774.667,
"text": " If there's some theorem that you've just forgot, or there's a field that you really don't want to know about, DeepSeq will summarize it much better than, much more efficiently than if I went to some expert and wasted his or her time and just have that, you know, the process is much more efficient. If I just wanted a very quick overview of some specific, even a specialized topic. So that's been very, very helpful to me."
},
{
"end_time": 831.101,
"index": 35,
"start_time": 801.101,
"text": " So it's only been three years since chat GPT came out and already we're seeing this massive change in the landscape. Do you imagine that three years from now, or let's say 10 years from now, that the role of the future academic or intellectuals or mathematicians, if you want to specialize, will be that of a decider or director, like a curator rather than a doer. So the doer is the one that right now we use computation, we use syntax, we"
},
{
"end_time": 858.524,
"index": 36,
"start_time": 831.613,
"text": " compute already helped us a lot by the end of the 20th century so no professional mathematician really goes by the by the end of by the 90s and early 2000s no professional mathematician for example does boring integrals anymore during doing research because that just that's completely outsourced"
},
{
"end_time": 886.613,
"index": 37,
"start_time": 858.524,
"text": " to say something like wolfram matematica on and later sage math because this is just boring and we know how to do it it will take as many hours if you really want to grind out some some technical integral of course i'm not saying that you shouldn't teach undergrads integrals anymore because that's part of the learning process and it's still important to teach teach undergraduates this kind of thing but no professional mathematician was started it's there they're really boring and horrible things like"
},
{
"end_time": 913.387,
"index": 38,
"start_time": 887.363,
"text": " even the simplest thing, you know, sine 17 X, nobody really wants to go on it. It just type into Mathematica. Because if I just need that, need that result very quickly, so I can supply to my next step of what I envision in my paper, I'm not going to waste a couple of hours trying to integrate something very elementary. And I'll probably even get it wrong, there'll be factors wrong. So that already transformed it. So I can see 10 years from now,"
},
{
"end_time": 943.746,
"index": 39,
"start_time": 914.002,
"text": " Simple, basic, maybe I'm being conservative here, but simple bits of a proof or simple bits of a derivation can just be outsourced to the likes of ChatGPT or DeepSeq or something, or something even more specialized. We don't currently have a LLM just for mathematics. That's surely going to come very soon. I'm sure the Frontier Math project by Epoch AI will start providing this kind of services."
},
{
"end_time": 951.561,
"index": 40,
"start_time": 945.213,
"text": " Okay well let's get into your presentation on the AI mathematician. Yeah sure, well thank you."
},
{
"end_time": 980.657,
"index": 41,
"start_time": 952.176,
"text": " I guess you know last time we talked about various things and I just want to share more in this chat some of the capabilities both in the terms of top down and bottom up of what we're looking at and to be honest even since our conversation which was what five months ago the field has advanced significantly which is very very impressive."
},
{
"end_time": 1007.671,
"index": 42,
"start_time": 981.067,
"text": " So just briefly to, as I was saying last time, and I just, as I also just mentioned, you know, there's, I try to classify this in this, in this review article, these three directions of mathematics, of course, they are intertwined. And the memorandum of gestures that I did with my collaborators, Lee, Oliver, and Posnikov, is really a very good example of this top down and I'll explain why this"
},
{
"end_time": 1025.538,
"index": 43,
"start_time": 1007.978,
"text": " Top-down mathematics is one that I want to emphasize here and then I will of course I will go back to just refresh people's mind a little bit about how people like Terence Tao and all this great and top minds are doing a proof assistance in terms of what I would call bottom-up and mathematics."
},
{
"end_time": 1056.988,
"index": 44,
"start_time": 1027.705,
"text": " This is a very, very interesting point where I want to emphasize the typical mathematician, and that includes theoretical physicists. Historically, we do things top down by just looking at patterns and spotting patterns. And we do many things in terms of practice before foundation. And this is very important. This is something that can't really be formalized."
},
{
"end_time": 1085.64,
"index": 45,
"start_time": 1058.012,
"text": " Because, you know, linguistically trying to formalize mathematics and which is an extremely important program, right? And all of this benchmarking of problem solving using large language models when you have a precise, well defined problems and trying to find a solution. But the history of scientific discovery is certainly not that I would say probably more than 50% is actually finding the problem or have a vague notion of something before you can formalize it. Can you give an example?"
},
{
"end_time": 1111.067,
"index": 46,
"start_time": 1086.305,
"text": " For example, Newton invented calculus without any notion of what even convergence means. He just had this intuitive idea of motion and then he, because it was Newton, he intuited there is this thing called derivative. This is way before we could even have epsilon delta limits, which came in the 19th century, almost 300 years later."
},
{
"end_time": 1139.735,
"index": 47,
"start_time": 1111.834,
"text": " and algebraic geometry is something closer to my heart. Algebraic geometry was just started with the Apollonius and Euclid with just you know shapes and stuff and we can intuitive the kind of theorems we want to prove. Before this Babaki school in the 1950s and 1960s you know the height of the Babaki school tried to formalize that in terms of definitions of fields and rings and polynomial rings and ideals. Now this is this is just"
},
{
"end_time": 1165.555,
"index": 48,
"start_time": 1140.111,
"text": " This is how theoretical discovery has always happened. In some sense, the reason I want to emphasize this bit is not only just because this is one I'm most familiar with and the one that I suppose I've been mostly involved with. Another reason I want to emphasize is that it's hard to imagine how AI can help us with this because it's so vague and it's so human."
},
{
"end_time": 1193.985,
"index": 49,
"start_time": 1165.947,
"text": " and there's a lot of mistakes and you if you train some language model there's not even any data to train on because these are not formal proofs these are just grasps of ideas of intuition and and and the point i want to make make that is even in this direction ai is beginning to help us okay so let's imagine we're back in the 16th sorry the 17th century with newton yeah and newton was saying okay i want to come up with something like calculus he didn't have that term he just had this notion of motion like you said"
},
{
"end_time": 1216.391,
"index": 50,
"start_time": 1194.394,
"text": " So, I don't usually stop mid-conversation about mathematics to talk about metabolism, but I've been using something that's made an appreciable difference. It's called Cell Being by Verso. Summer's coming up, and if you're like me, getting lean while juggling work, stress,"
},
{
"end_time": 1238.814,
"index": 51,
"start_time": 1216.596,
"text": " And everything else isn't exactly straightforward. I train, I eat well, but as I age, as we all age, fat loss gets harder. Cell being uses research-backed ingredients that help your body boost NAD levels. NAD, by the way, is crucial for metabolism, energy, and even DNA repair. However, there's a large added benefit here to cell being."
},
{
"end_time": 1260.964,
"index": 52,
"start_time": 1238.814,
"text": " This formula also helps regulate hunger hormones and supports fat breakdown. Basically, it tells your body to burn more and crave less. Since taking it, I've noticed I'm not as peckish, I feel more clear headed, my energy is distinctly more stable throughout the day, and I'm discernibly quicker mentally. I haven't changed anything else about my supplement regimen,"
},
{
"end_time": 1279.206,
"index": 53,
"start_time": 1260.964,
"text": " I've just added this and it's helped me stay consistent without forcing it. Also, Verso third-party tests every batch and publishes the results, which matters to me personally because this way I know exactly what I'm getting. So if you're looking to dial in your energy, metabolism, and shed a bit of that stubborn fat before summer,"
},
{
"end_time": 1308.473,
"index": 54,
"start_time": 1279.206,
"text": " What would Newton do with an LLM? What is your vision? That's an interesting point."
},
{
"end_time": 1333.968,
"index": 55,
"start_time": 1309.292,
"text": " Newton would what Newton would do with a with an LLM if you had a LNM which was certainly to to process all previous literature. Now to be fair at Newton's time somebody like Newton could read almost the entirety of any relevant literature up to his up to his point and I'm thinking about everything from Euclid's elements Galileo"
},
{
"end_time": 1358.2,
"index": 56,
"start_time": 1335.043,
"text": " Bits of Kepler and he certainly won't have that anyone would just go money and you just go and read it and that's fine. But now literature has grown so exponentially. There are no more newtons human newtons that could possibly read the entire literature of the field and that's why the lambs could come in to help. So this is the in the LLM space of discovery because summarize literature."
},
{
"end_time": 1385.435,
"index": 57,
"start_time": 1358.575,
"text": " I do can try to create new possible links between literature and this is happening now i think i think there are i think llama llama which is llm for math like llama double l llama llama is is something it's a it's an ai tool that's beginning to digest the archive for example and on the other hand that's the llm side of the story"
},
{
"end_time": 1402.705,
"index": 58,
"start_time": 1386.869,
"text": " now what about the other half is how could newton based on mathematical patterns and he did he did have a lot of patterns it would be things like this will be mathematical data so certainly they hear data in terms of theory in terms of"
},
{
"end_time": 1428.063,
"index": 59,
"start_time": 1402.705,
"text": " in terms of theoretical and experimental physics where you know you could measure falling you know the rolling of the kind of stuff the Galileo did the rolling of balls along inclined planes that kind of data or he had the the astronomical data of Kepler but he also would have had mathematical data or platonic data I like this word platonic data because it's pure the kind of data that would be like"
},
{
"end_time": 1455.247,
"index": 60,
"start_time": 1429.36,
"text": " Set of polynomial equations into variables which actually try to classify himself. How many cubics you because you knew about the conic classification problem and he would look at these things. And then he was spot patterns and this kind of stuff also gave rise i would imagine i can't imagine you know the mind of of newton but i would imagine she would look at vast amounts of such data."
},
{
"end_time": 1467.261,
"index": 61,
"start_time": 1455.555,
"text": " So like Newton,"
},
{
"end_time": 1494.07,
"index": 62,
"start_time": 1467.602,
"text": " There are these things called Newton polynomials which expresses, it's a technical thing, which would be Newton, certain symmetric polynomials in multivariables being expressible in some basis. I would imagine Newton would have written pages and pages of this stuff and spotted a pattern and then tried to prove a general theorem, which is now the theory of Newton polynomials."
},
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"start_time": 1497.21,
"text": " Ford BlueCruise hands-free highway driving takes the work out of being behind the wheel, allowing you to relax and reconnect while also staying in control. Enjoy the drive in BlueCruise enabled vehicles like the F-150, Explorer and Mustang Mach-E. Available feature on equipped vehicles. Terms apply. Does not replace safe driving. See Ford.com slash BlueCruise for more details."
},
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"text": " This episode is brought to you by State Farm. Listening to this podcast? Smart move. Being financially savvy? Smart move. Another smart move? Having State Farm help you create a competitive price when you choose to bundle home and auto. Bundling. Just another way to save with a personal price plan. Like a good neighbor, State Farm is there. Prices are based on rating plans that vary by state. Coverage options are selected by the customer. Availability, amount of discounts and savings, and eligibility vary by state."
},
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"start_time": 1555.452,
"text": " I thought where you were going was this amorphous ideation with an LLM. So for instance, Newton would say, okay, I have balls coming down an incline. I don't have a precise formula for velocity. I don't even know if velocity is a great concept, but I noticed that it moves faster, but it can also that may be associated with something I would call acceleration."
},
{
"end_time": 1606.766,
"index": 66,
"start_time": 1577.022,
"text": " But then there's something else like an impact, which I may call force. Can you help me with this? Is there a way to make this concrete, like to take something that's ill defined and make it well defined? I thought that's where you're going. Do you think LLMs help with that? Or is that not what you were thinking about? No, no, I think that definitely helps a bit. We're not quite there yet, I think, where we could just start asking LLMs and say, here's the literature, digest it all and give me new possible links. But we are actually not"
},
{
"end_time": 1635.452,
"index": 67,
"start_time": 1607.159,
"text": " insanely far away from that goal now because the lemons are getting so good at doing this sort of thing so it's actually not not impossible in some sense newton would yeah the the mind of newton is something like that i guess let me see my next slide i think i do mention yeah yeah i do mention that the another mind as great as newton and gauss and i would do i will give that example in a minute all right let's get back to the yeah"
},
{
"end_time": 1664.787,
"index": 68,
"start_time": 1635.452,
"text": " presentation oh so i think i don't i can't remember whether speaking of newton i i can't remember whether whether where whether i talked about this last time if not this is a joke worth repeating again which is what is the best neural network of the 18th century you could argue the 17th century was newton and the best neural network of the 18th to 19th century is clearly the the brain of gauss and that's another here's a very very good example as well of just"
},
{
"end_time": 1690.452,
"index": 69,
"start_time": 1665.265,
"text": " top-down intuition-guided mathematics. And I might have mentioned this last time, but it's worth repeating. So what is the thought process of this great discovery here? So everybody knew about the primes. Euclid already proved that there's an infinite number of primes. And the proof is very, very beautiful. And it's kind of intricate. It's the first thing you would teach in a number theory course."
},
{
"end_time": 1719.462,
"index": 70,
"start_time": 1691.305,
"text": " It's not obvious at all there's an infinite number of number of primes, but Euclid had this proof by contradiction argument why there should be an infinite number of them. So Gauss certainly would have known that that proof, but Gauss wanted to know more. And it's been 2000 years since Euclid. Gauss wanted to know about more details about the distribution of primes. We know the primes get rarer and rarer, even though it's an infinite number of them. They do get rarer and rarer. How rare do they get?"
},
{
"end_time": 1744.206,
"index": 71,
"start_time": 1720.23,
"text": " and he just devised this function which we now call the prime counting function p of x. It's a terrible name because p is pi but it got stuck in the literature and p of x is simply the number of prime numbers less than a given positive real number x. So that was one of his insights was to devise this function which is now continuous because primes are inherently discrete."
},
{
"end_time": 1771.186,
"index": 72,
"start_time": 1744.326,
"text": " This is very interesting. He plotted this and he looked at this curve. He invented regression apparently in order to do a curve fitting because he needed it. This is all done at the age of 16. He invented regression to see what is the best shape that fits this and all of this done by hand and he even had to compute the primes into the hundreds of thousands because the tables stopped there"
},
{
"end_time": 1801.152,
"index": 73,
"start_time": 1771.578,
"text": " I'd even got some of them wrong because it's a very boring and tedious computation imagine what gals could have done one if he had Math sage or Mathematica he could have how much more conjectures would have raised right? Well, the problem with that is that if he has access to the modern tools He also has access to tik-tok and we it's not clear if gals would be yeah Let's hope that because gals is gals He wouldn't waste his time to becoming like I don't know doing like watching youtube videos"
},
{
"end_time": 1820.043,
"index": 74,
"start_time": 1801.63,
"text": " But and watch more meaning maybe he would maybe he'll watching meaningful youtube videos like other conversations like The kind of conversation you put on theories of everything Okay, great. But anyway, but he would do this thing and and he looked at his p of x and he says ah It's clearly x of a natural log of x"
},
{
"end_time": 1847.193,
"index": 75,
"start_time": 1820.589,
"text": " You won't be able to see this just by looking. So he really actually had to invent regression to do this. And so statistical statistics was a side product of this problem. As far as I know, I could, but this needs to be checked with the, you know, real historians of science, but at least that's the, that's the story here. And this is just crazy. Like, how do you even, what do you even do? And this, the proof of this fact was given 50 years later by Adamar and Delavaya Pusa, because"
},
{
"end_time": 1874.991,
"index": 76,
"start_time": 1848.353,
"text": " You had to wait for Cauchy and Riemann to invent complex analysis in order to give a tool to prove this fact. So how did Gauss know this? Based on just at the time was surely because large data, right? He really went into the tens and hundred thousand range in order to spot this kind of patterns and it's just amazing and I can plot this by"
},
{
"end_time": 1904.599,
"index": 77,
"start_time": 1875.299,
"text": " Because if you just plot the first hundred or so, it looks kind of like a log or kind of like a line or something, but you really need to go into the thousands or tenth of thousands range in order to do something like this. So amazing. That's exactly the kind of top-down guiding intuition. There was not even the foundation to prove something like this until years later. Riemann as well, a great example. Riemann hypothesis, which is arguably the most famous open problem in all of"
},
{
"end_time": 1934.275,
"index": 78,
"start_time": 1904.753,
"text": " human intellect and is certainly the one that we all bowed and worship. It's about so the Riemann hypothesis has so many implications in mathematics. It's one of the millennium price problems and the Riemann hypothesis it's so important precisely because there are now probably tens of thousands of mathematics papers whose first opening line is let us assume that the Riemann hypothesis is correct and the rest of the paper."
},
{
"end_time": 1954.019,
"index": 79,
"start_time": 1934.565,
"text": " So it has so many implications, so that's the kind of that's the kind of conjectures that are great, that it has implications to so many other possible results. The interesting about the Riemann hypothesis is that it appeared as a footnote in Riemann's paper. Riemann was just doing"
},
{
"end_time": 1972.585,
"index": 80,
"start_time": 1954.36,
"text": " The women's data function precisely to address a similar problem to this to the precise distribution of prime prime of of of primes and eroding the inner inner footnote that i checked the first couple of zeros and they all have real part a half."
},
{
"end_time": 2002.261,
"index": 81,
"start_time": 1973.336,
"text": " I believe that this is true, but I don't really need this result right now It really you can see that amazing footnote and then people that we just said I can't think about this right now But I think this is kind of interesting and then that was the beginning the birth of that How did Riemann into it something like this? Well, these number theorists and their margins They say exactly number theories love margin exactly though. Yeah, that's what I never thought of it. That's a good point That's a good point But that that's an exactly extremely good point you mentioned"
},
{
"end_time": 2031.834,
"index": 82,
"start_time": 2002.944,
"text": " They are written into margins because they haven't been formally approved. If it's structured, it would be bottom-up mathematics and it would be in the main text. And often this marginalia are just afterthoughts or just sparks of genius of these people who just relegate this thing to a side comment."
},
{
"end_time": 2061.067,
"index": 83,
"start_time": 2032.568,
"text": " And that kind of intuition leads to centuries of research. So that's a very good point, Erase, about how the difference between margins and formal text, because papers are written, whether it's pure mathematics or theoretical physics, papers are written in a very structured, backwards kind of way, quite different from the way they're reached in this intuitive kind of way. Yeah, so that's a good point."
},
{
"end_time": 2072.261,
"index": 84,
"start_time": 2062.824,
"text": " yeah and then this doesn't stop and this is something that the memorations that will get more into which is this bst conjecture which is another millennium price problem."
},
{
"end_time": 2101.749,
"index": 85,
"start_time": 2072.79,
"text": " This is another one that carries on $1 million tag. And how did this come about? And I will talk more about what the BST conjecture is. This is Birch, Brian Birch, who is still alive. He's, I think, 90 something. You know, mathematicians are very long lived because they're happy. And the Birch's went and died in the 60s. They're in the basement in Cambridge and they just plotted loads and loads of data for ranks of conductors of elliptic curve. Now,"
},
{
"end_time": 2116.92,
"index": 86,
"start_time": 2102.381,
"text": " The LMFDB, which I'm going to talk about in a minute, is a database of 3.6 million curves."
},
{
"end_time": 2146.237,
"index": 87,
"start_time": 2117.756,
"text": " We've progressed quite a bit. A 3.6 million is the kind of data scale where you could really train things on. And that's where I could really come in. And this isn't a stop. They plotted this and they raised this conjecture. They noticed a certain pattern between r and ranks, these technical terms, which I'm going to define in a minute. And that was the birth of yet another great and foundational problem. And this is regarded as central piece of mathematics as well."
},
{
"end_time": 2169.565,
"index": 88,
"start_time": 2146.237,
"text": " And these are all intuited, if you wish. So just one quick slide. I got into this because of algebraic geometry. And so I think I mentioned it in the last talk, just trying to see how machine learning can help us spot patterns, if you wish. But I think since"
},
{
"end_time": 2186.391,
"index": 89,
"start_time": 2170.043,
"text": " since 2017 i grew a lot alongside with my son we both grew he's growing very fast and then growing intellectually just to digest this field and it's a humbling experience just to see this vast interaction"
},
{
"end_time": 2215.862,
"index": 90,
"start_time": 2186.732,
"text": " of so many different people and experts. And so again, I like to thank all of my collaborators. Now I can't, you can't possibly, I can't possibly read out all the names, but that's where the QR code comes. Scan this QR code. There's a long, this is, we'll point you to Google doc where I will try as much as possible to keep up to date all of the names and affiliations and the papers with my co-authors. So you can think, thank them properly at some point. Now this is an interesting part. I wanted to chat GPT,"
},
{
"end_time": 2244.309,
"index": 91,
"start_time": 2216.357,
"text": " To give the generate the list of these people and search the internet and find a picture of each of them and give them affiliation. So to save me the time of typing them out. So we're talking about with a hundred people. Yeah, we can talk off air about that. I can call that up for you extremely easily. Okay. But, but what is really interesting is that chat GPT did a terrible job. It found random affiliations of people who didn't exist because you know, they're doing LLM."
},
{
"end_time": 2274.258,
"index": 92,
"start_time": 2244.77,
"text": " So it's matching them to with, it's confusing my collaborators. So I couldn't possibly credit Chachi PT for this. So this is a very early thing. And also they just couldn't, Chachi PT could not produce the correct photos of any of these people. So we are limited. So as excited as I am, I must point out limitations to all of that. I tried DeepSeek, Claude, I tried them all. None of them could even answer this problem, which should be a simply, simply problem."
},
{
"end_time": 2302.483,
"index": 93,
"start_time": 2274.821,
"text": " There are ways of using the agent or the LLM to interrogate itself so that it can double check. So we can talk about that off air. Oh, wow. That's that. If you can help me with that, it'll be great because this is something this is obviously something I can help us because this is boring and it just needs to be done as even part of scientific discovery. It took me an hour to find that LLMs were useless in this, but that hour I could have tried to do something more meaningful."
},
{
"end_time": 2324.531,
"index": 94,
"start_time": 2303.404,
"text": " But you would take hours if i would just do it properly and include copy and paste pictures it would take many hours and you know that's why we're so this is just to tell you we're not quite there yet even with a simple task like this so surprisingly it can help us with mathematical discovery but you know all of this will change very quickly."
},
{
"end_time": 2349.548,
"index": 95,
"start_time": 2325.998,
"text": " That's the book I think you mentioned earlier, which is this. This is the book of my learning experience, trying to learn about machine learning. This finally came out, I think, there's an archive version in 2018. I think it appeared in 2020, the landscape. This is from everything from machine learning and then this editorial in 2020."
},
{
"end_time": 2370.879,
"index": 96,
"start_time": 2350.282,
"text": " Now let's get back to the real meat of the subject, which is I believe this is still part of this review I was trying to say. I tried to emphasize that bottom-up mathematics is a natural language processing because this is"
},
{
"end_time": 2392.09,
"index": 97,
"start_time": 2371.425,
"text": " I like this"
},
{
"end_time": 2421.852,
"index": 98,
"start_time": 2392.739,
"text": " analogy because the great David Mumford, who is also a Fields Medalist, even back in the 90s after he got the Fields Medal, he stopped everything. He got the Fields Medal for doing topology in K-Theory, if I remember, algebraic topology. And then he switched fields completely, dropped mathematics altogether and started working in computer vision. And now that I've read more of his sort of his recollections, he blogs as well. He's a very excellent blogger like Terry Tao."
},
{
"end_time": 2446.886,
"index": 99,
"start_time": 2422.534,
"text": " So David Monfort, he said the reason he got into this computer vision thing was I think he was really having early visions of how AI can help with research because he was trying to imagine the human mind being an image processing machine. What does a mathematician actually see? The mathematician is beginning to have mental images of formulae"
},
{
"end_time": 2471.596,
"index": 100,
"start_time": 2447.176,
"text": " There is this transformation process from what you see as abstraction, as mathematics, into a mentally constructed image. That's why he was so interested in vision. And that image in the mind is somehow, well, I guess in today's language, this will be the latent representation of your data. You know Hadamard, how he had a book on how mathematicians think."
},
{
"end_time": 2501.152,
"index": 101,
"start_time": 2471.92,
"text": " Oh, I heard of that. I have not read that. That's kind of interesting. Is that the same Hadamard as the Hadamard Delivier-Poussin Hadamard? Oh, that's interesting. Oh, cool guy. Yeah. And I'm wondering if he did a historical analysis for Euler because Euler was blind for half his life or something like that, some large portion of his life. So I'm wondering if Euler still used mental imagery to formulate or solve his problems or then abstract to something else."
},
{
"end_time": 2524.241,
"index": 102,
"start_time": 2501.425,
"text": " Absolutely, it's quite imagined. I had a student once in Oxford a few years ago now. She was quite remarkable because she's completely blind. I think she was blind from an early age. So she sat through my lessons without being able to see anything. She had to picture what I was saying."
},
{
"end_time": 2546.817,
"index": 103,
"start_time": 2524.531,
"text": " and then digested all in her head and do all of the mental calculations in her head. Interesting. So I was wondering what was she actually doing and she did she did fairly well in in her final exams with being completely she needed somebody obviously to to translate whatever she's had to dictate to someone in"
},
{
"end_time": 2575.503,
"index": 104,
"start_time": 2547.756,
"text": " Yeah, so it was quite remarkable that I got to know this student. But anyhow, but my image processing is this kind of thing is now that I read Mumford, I'm beginning to think why I was beginning to think that all of mathematics, there's all of top down, all pattern recognition is an image processing problem. Close your eyes, exhale, feel your body relax."
},
{
"end_time": 2601.493,
"index": 105,
"start_time": 2575.913,
"text": " 1-800-CONTACTS"
},
{
"end_time": 2630.93,
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"start_time": 2602.056,
"text": " Tito's Handmade Vodka is America's favorite vodka for a reason. From the first legal distillery in Texas, Tito's is six times distilled till it's just right and naturally gluten-free, making it a high-quality spirit that mixes with just about anything, from the smoothest martinis to the best Bloody Marys. Tito's is known for giving back, teaming up with nonprofits to serve its communities and do good for dogs. Make your next cocktail with Tito's. Distilled and bottled by 5th Generation Inc. Austin, Texas. 40% alcohol by volume. Savor responsibly."
},
{
"end_time": 2649.497,
"index": 107,
"start_time": 2632.671,
"text": " yeah this i guess there is this old debate and this involved all the grades like atia and digraph and hitchin and not witten all these people is physics or is theoretical physics or is mathematics inherently is the net is nature"
},
{
"end_time": 2673.302,
"index": 108,
"start_time": 2650.06,
"text": " um algebraic or is it geometric interesting so this top down mathematics this is debate newton was clearly geometrical he had such a distaste and disdain for algebra because it's it's meaningless symbols to him he made some comment about algebra is this i can't remember the original quote but he will say something it's a very disgusting thing"
},
{
"end_time": 2693.387,
"index": 109,
"start_time": 2673.746,
"text": " That you had to resort to this to this meaningless symbols he was very very pictorial his proof of what we now call the gas theorem. About integration integration over spheres this is just some about you know a gravitational body exerting a force on an external object."
},
{
"end_time": 2721.357,
"index": 110,
"start_time": 2693.797,
"text": " Gauss would just surround this by and then uses Gauss's law. But Newton actually integrated piece by piece and used all his intricate pieces together in a diagram and got the same answer. It's the kind of horror show you would never do because Gauss's theorem is just one line. You just do this integral. But Newton actually had to piece together. So Newton was definitely visual. Roger Penrose is definitely visual. Penrose, my last conversation with him, he said"
},
{
"end_time": 2728.558,
"index": 111,
"start_time": 2721.357,
"text": " she almost said i think i'm i just in case i in case i in case i i'm putting words in his in his mouth but i believe he says something like"
},
{
"end_time": 2759.053,
"index": 112,
"start_time": 2729.548,
"text": " If it's not intuitive and if it's not geometrical, he doesn't even accept that as a proof. And I think Conway was like that as well. This is one of the reasons why Conway never really accepted Richard Borchardt's proof of the moonshine conjectures, because Borchardt used this very strange vertex-operated algebra. I think you had a conversation with Ed Frankel about this. And Borchardt's actually. And Borchardt's, yeah, exactly. But Borchardt, he borrowed this piece of"
},
{
"end_time": 2783.251,
"index": 113,
"start_time": 2759.94,
"text": " completely crazy stuff, vertex operator algebras, and had this beautiful structure. It's obviously awesome and brilliant. He got him the Fields Medal, and he was able to use that to prove the moonshine conjecture from Kai. Conway, to my knowledge, who told me this, it was"
},
{
"end_time": 2812.159,
"index": 114,
"start_time": 2784.923,
"text": " Oh gosh, the previous director of the is Robert die graph the one before oh god oh gosh goddard peter goddard peter goddard knew conway very well and he was telling me that conway never really deep down accepted this proof of butchers because he's not visual conway's this very playful guy as you're going to imagine he wanted everything to be pitori he wanted to see his lattices so anyhow so"
},
{
"end_time": 2823.507,
"index": 115,
"start_time": 2813.097,
"text": " In a way this diagram puts geometry in this direction and puts algebra in this direction. Well how is Conway visualizing the monster group?"
},
{
"end_time": 2853.319,
"index": 116,
"start_time": 2826.783,
"text": " Maybe in terms of the leech lattice. He had this picture of the leech lattice. To him, the monster group is some extension of the automorphism of the leech lattice. Which I guess in a way, that's how he and Rhys and Reba originally came up with Monster. It wasn't by very hardcore, this whole funny business of classifying simple groups."
},
{
"end_time": 2880.862,
"index": 117,
"start_time": 2853.797,
"text": " He really intuited it in a way. He got this group out by doing norm two lattices and he was able to see the symmetry, the group of symmetries of this lattice and that's a remarkable thing. It's just unfortunate that whole generation of people, that generation of this"
},
{
"end_time": 2893.49,
"index": 118,
"start_time": 2881.783,
"text": " Lattice early computer algebra finite group people are slowly dying out in a conways dead and norton is dead and my my own dear friend john mckay was started moonshine."
},
{
"end_time": 2920.401,
"index": 119,
"start_time": 2893.848,
"text": " past away i read this obituary because i was his last close collaborate he actually became he became a grandfather figure to me he became you know he saw my kids grow up interesting and and makai makai would tell me the stories about how he was interacting with conway and how all of that people have portraits and and makai is also here here's another crazy that's another whole conversation about what is this bizarre intuition that makai had"
},
{
"end_time": 2948.183,
"index": 120,
"start_time": 2920.657,
"text": " If there's anybody in the later part of the 20th century who had a almost Ramanujan-like intuition would be John Mackay. In a way, he's an unsung hero because he would just look at lists of numbers or look at pictures and graphs and see, ah, but this field is related to this. And that is very much like this. He's AI before AI. Even physicists?"
},
{
"end_time": 2967.858,
"index": 121,
"start_time": 2948.916,
"text": " i think that the whole different conversation we're gonna have to do part three and four this is okay i get very emotional when i talk about john because it's like you know he's a is a very much like a like a father figure to me well the next time i'm i'm in october we should have a part three and we have a part three conversation just talking about moonshine conjectures"
},
{
"end_time": 2995.555,
"index": 122,
"start_time": 2967.858,
"text": " In the in the from the perspective mccain i know you you certainly oh yeah and he pronounces names mccain not mccain even though he's written down as mccain he insisted on being called john mccain i know you you you chatted with it with with franco and you chatted with um with uh borchards on moonshine and that stuff but it's unfortunate that he passed away before you started all this thing he passed in he passed in 2022"
},
{
"end_time": 3022.91,
"index": 123,
"start_time": 2995.555,
"text": " Because he had a very interesting knowledge of that world Of moonshine and stuff. But anyhow, have you heard of stone duality? stone Yes stone duality or a stone type duality. No, not at all. What is that? So it's a duality between topology and then Boolean algebras which some people see as an analogy or an equivalence between geometry and then syntax or something more algebraic"
},
{
"end_time": 3051.408,
"index": 124,
"start_time": 3023.541,
"text": " Oh, interesting. Oh, I'll have to look into that. Thanks for pointing out. I didn't know I is this like the stone of the stone virus truss theorem stone? I believe so. I hate you got them all correct. Yeah, I don't know. Okay I just didn't know there was this this is called the stone correspondence or yeah stone duality. Yeah duality duality Oh, I love I would love to see I'm sorry. I just bumped in my I'd love to see that. Oh interesting very interesting anyhow, so so back to back to our current story"
},
{
"end_time": 3071.476,
"index": 125,
"start_time": 3051.681,
"text": " So in 2022, Chachi BT actually, as you know, put into this conversation, Chachi BT passed the Turing test, which I again, I'm very surprised he was not on every single newspaper headline. I don't know why this wasn't emphasized either. We can't really, the Turing test was a big thing."
},
{
"end_time": 3095.759,
"index": 126,
"start_time": 3072.295,
"text": " I think that the fact that Chachi BT passed the Turing test just simply showed that you don't need reasoning or understanding to have intelligent conversation. Maybe it says a lot about humans. Chachi BT passing the Turing test says more about humans than it says about AI thinking. We give too much credit, too much credit to what meaningful conversations are."
},
{
"end_time": 3118.677,
"index": 127,
"start_time": 3096.493,
"text": " But this, sort of as a response to that, we organized a conference in Cambridge with loads and loads of people, and you can probably recognize some of the names, Buzzard and Birch. We tried to formulate something that's more stringent than a Turing test for AI guided discovery."
},
{
"end_time": 3146.152,
"index": 128,
"start_time": 3119.514,
"text": " and so we we reported this with I reported this with my my friend birds of as a nature correspondence and this is again I can't remember whether I talked about this in our did I talked about the the the birch test yes the birch test plus plus or the turing test plus plus was the birch test yeah is the birch test yeah last time so we'll put a link on screen for the last conversation in case you're just tuning in and you're wondering it was a wonderful conversation and I believe we talked about"
},
{
"end_time": 3174.65,
"index": 129,
"start_time": 3146.647,
"text": " let's see the birch test bottom up top down meta mathematics and even classifications of cy manifolds and then this database construction right okay so i can pass with with the b so this is ai plus n for the birch test so let me just uh i guess i'm very good at digressing sometimes i digress so much that i don't remember what i'm digressing on anymore but the point the point is this is clear clear signs of adht but i've never had it"
},
{
"end_time": 3194.804,
"index": 130,
"start_time": 3175.026,
"text": " As you mentioned, in speaking about these shower thoughts or the margins, the digressions are sometimes more meaty than the meat. Often, yeah, often. But back to this AI-guided discovery, which in terms of AI-assisted, top-down, intuition-guided discovery in mathematics,"
},
{
"end_time": 3225.009,
"index": 131,
"start_time": 3195.606,
"text": " There have been various candidates in the past eight years or so. Some of the ones, of course, everybody talks about this beautiful paper in this DeepMind collaboration by Alex Davis. Alex comes here a lot because, you know, DeepMind is in St. Pancras, which is a 30-minute walk from this institute, which is kind of very nice. We have a nice hub in London for this sort of thing. Google DeepMind isn't in California?"
},
{
"end_time": 3243.336,
"index": 132,
"start_time": 3226.698,
"text": " There's a London office. Oh, okay. So there's a there's a branch at least I Guess I guess I guess yeah. Yeah, there must be. Yeah. So Alex is actually in London. So that the the Davis et al paper And defined it's cool. So so he comes"
},
{
"end_time": 3267.176,
"index": 133,
"start_time": 3243.882,
"text": " very regularly of course he can't because it works for Google for deep mind he can't tell us exactly what he's working on nor can he tell us what the next project will be but at least he can summarize what the state you know what what's going on in that in the in the tech world which is kind of kind of interesting the fact that Nobel prizes are being given to non-university organizations which is right very nice."
},
{
"end_time": 3295.384,
"index": 134,
"start_time": 3267.79,
"text": " Which which is what this organization is at some point. Oh, that's another whole conversation again. So I think Yeah, as for as I think I mentioned to you last time with this, you know, these are the rooms where faraday lived Okay, so these are we're very lucky the london institute. We're we're we're at the the we're with the second floor of the royal institution where where the likes of humphrey davey and thomas young and Michael faraday lived so i'm very"
},
{
"end_time": 3322.534,
"index": 135,
"start_time": 3295.384,
"text": " fortunate to be in this space to work and try to get this in. But one of our themes, the reason I mentioned is one of the themes, one of our four research themes is AI for theoretical discovery of this Institute. And it's kind of, we're independent of the universities so that we could devote our time fully to research. That's kind of it. So how does this lead to the murmuration conjecture?"
},
{
"end_time": 3351.544,
"index": 136,
"start_time": 3322.807,
"text": " Yeah, I promise to tell you about memorization. Yeah, so these early, this Clabi-L manifolds, which we spent so much time talking about last time, this is because it was my bread and butter as I was growing up as a grad student. So that was clearly the first thing I'm going to apply machine learning to. The kind of experiments of using neural networks to predict topological invariance of these varieties in the image processing kind of way immediately fails the Birch test out straight because it's not interpretable."
},
{
"end_time": 3381.425,
"index": 137,
"start_time": 3352.073,
"text": " Sure now it's been improved to 99.999% or whatever it is, but it's useless to a scientist. It just simply says that, oh yes, there is an underlying pattern, but how do you actually extract anything meaningful from that pattern? That's the main question. So the closest so far, and when I say so far, it really, this could change in a couple of months. Oh, you have no idea because the field is growing so fast."
},
{
"end_time": 3407.039,
"index": 138,
"start_time": 3382.022,
"text": " The closest so far in the last, gosh, it's almost a decade. I guess my son is eight now. In the last decade of all this AI discovery, there's been hundreds of papers now on various things on how do I use machine learning to do this in number theory, in theoretical physics, in quantum field theory, there are hundreds, literally hundreds of papers. Now,"
},
{
"end_time": 3435.486,
"index": 139,
"start_time": 3407.841,
"text": " The one that really made Buzzard and Birch happy is this memorization conjecture. And the discovery process of this is something that I would like to see, at least that this is sort of the state of the art of human-machine interaction. That's why it's so close to my heart. And this is joint work with Kiu Huan Li, Thomas Oliver, and Alexey Potsniakov, and now with a paper to appear with Andrew Sutherland, who is"
},
{
"end_time": 3449.087,
"index": 140,
"start_time": 3435.486,
"text": " the guy who set up this LMFDB. So I think I mentioned last bits of machine learning experiments in number theory, you know, providing can AI predict primes?"
},
{
"end_time": 3478.166,
"index": 141,
"start_time": 3449.292,
"text": " No, we're certainly not at that stage. I'm not saying you can't. If an AI can detect a pattern of primes by itself, then we will be at the next level in not only proving the Riemann hypothesis, we would also crack every single code in every single bank in the world because that's all the cryptography is dependent on this. So wait, what's the main impediment for AI to not predict primes? There's a large data set there."
},
{
"end_time": 3498.66,
"index": 142,
"start_time": 3479.241,
"text": " Yeah, so actually that's a good question. The short answer is I don't know. I've certainly fed in millions of primes into whatever representation into a new network of whatever architecture. I just simply asked it to predict the next one."
},
{
"end_time": 3524.616,
"index": 143,
"start_time": 3499.718,
"text": " It does terribly on this. I think now I think somebody's even written a paper on this called why prime prediction is so hard for neural networks. I can't remember the precise networks. I think at some level it's, at some level it has, this again goes back to the Riemann hypothesis. That's why Riemann hypothesis, the Riemann"
},
{
"end_time": 3553.848,
"index": 144,
"start_time": 3525.401,
"text": " Exact patterns in the distribution of the zeros of the Roman zeta function in the critical strip will give you precise patterns in the distribution of the primes and people have proven statistical statements about the distribution of the zeros that they're truly stochastic up to some level. I'm not an analytic number theorist but you can"
},
{
"end_time": 3579.019,
"index": 145,
"start_time": 3554.292,
"text": " Basically, there is so much noise or truly stochastic randomness in the distribution of the zeros of the zeta function that it's very difficult if you try to train. In other words, training a neural network of the zeros of the zeta function is like training with noise. Interesting."
},
{
"end_time": 3599.189,
"index": 146,
"start_time": 3579.582,
"text": " Something fundamental, but you could just be that this representation we're using for the set of function is not very good. We should dig deeper, but you need almost another, of course, if you find the right representation that would give a very good pattern spotter, that representation is the new mathematics we're looking for."
},
{
"end_time": 3624.206,
"index": 147,
"start_time": 3599.872,
"text": " So speaking of classifications, as we're both fans of classifications, is there a way to map this problem of doing an image processing for prediction of primes slash the Riemann zeta function zeros? Is it mappable to P versus NP or is it a new class like, okay, problems that can be solved with image recognition or image prediction versus problems that can't in math?"
},
{
"end_time": 3645.077,
"index": 148,
"start_time": 3626.084,
"text": " This is a deep question and at some point I was talking to model theorists and especially Boris Zilber who is a leading figure in model theory because in model theory tries to classify mathematical problems in terms of hierarchies of difficulty and this is not a P versus NP kind of"
},
{
"end_time": 3671.749,
"index": 149,
"start_time": 3645.077,
"text": " Difficulty but a difficulty in the very underlined structures the questions like why is it that a polynomial over the integers is so hard so much harder than to think about a polynomial defined over the complex numbers. Even though the complex numbers is some in some sense a completion of the integers a but why is it so much harder to look for for models over integers and"
},
{
"end_time": 3699.633,
"index": 150,
"start_time": 3671.749,
"text": " At some point, we were thinking that maybe the problems that we're going to encounter that the neural networks will struggle with are ones that will go higher in this hierarchy of difficulty. But we haven't thought much more about this, but it will be very interesting to correlate this. But this is not computational complexity. There should be a new definition of a complexity in terms of how difficult a problem is. But it's still, it's hard to say how to define this."
},
{
"end_time": 3729.991,
"index": 151,
"start_time": 3700.503,
"text": " Hmm, actually, just as an aside and aside on to this is an aside on on the side. Yeah. So there's a contest called the summer of math exposition by three blue, one brown. And it's about getting people to make animations and lessons for math, different math topics. We're doing one on this podcast theories of everything, but for physics and also complexity and physics, AI and complexity. And this is a teaser of an announcement. It's not the full announcement, but those who are listening, it's going to be announced shortly."
},
{
"end_time": 3755.418,
"index": 152,
"start_time": 3730.35,
"text": " And there will be prize money for those who have the best explanation, the top five gets, you'll see. Oh, I love, I love to see that. Amazing, amazing stuff. Oh, sorry. Back to, back to BST. So now finally I have to, so this is the memorization again, there's a, this was, this was considered by, it got Quanta interested and Quanta considered this one of the breakthroughs of 2024 because they obviously,"
},
{
"end_time": 3770.401,
"index": 153,
"start_time": 3756.152,
"text": " Because this was something that was AI guided and it was it really surprised the experts and just sort of I want to tell you the story of this because it shows where we are in terms of AI assisted discovery."
},
{
"end_time": 3793.78,
"index": 154,
"start_time": 3771.681,
"text": " Probably my biggest contribution to this was to have insisted on this paper being called Memoration, because I remember this Skype that came through. The original paper was in, oh gosh, three years ago, when this pattern that appeared, and my collaborators are saying, you know, this reminds me of this thing that birds do."
},
{
"end_time": 3812.108,
"index": 155,
"start_time": 3793.78,
"text": " I'm gonna insist that when when this paper gets finished. We gonna. Call this paper the migration phenomena on the car got stuck that's probably my biggest contribution because these are the my collaborators are."
},
{
"end_time": 3840.845,
"index": 156,
"start_time": 3812.329,
"text": " Card Carry Number Theorists, we teamed up because they were trying to explore this AI assisted world and I needed some real experts. So this already breaks, you break the birch test. The fact that I had to look for human experts to try to generate something like this already fails the birch test, but it was worth it. It was worth breaking birch for because I made friends with number theorists and it was something surprising to their community."
},
{
"end_time": 3862.039,
"index": 157,
"start_time": 3841.493,
"text": " Just a bit about the importance of the BST Conjecture, but this is kind of nice because it gives an opportunity for me to share my own ignorance on the BST Conjecture because I learned about this as I was working not as a number theorist, which I'm not, but as an amateur number theorist coming from AI discovery side."
},
{
"end_time": 3884.94,
"index": 158,
"start_time": 3862.295,
"text": " So this made me appreciate why the BSD conjecture is so important and why it's so interesting and how surprisingly AI can help with this and how the Birch test was almost met by this particular problem. So let's go way back to the Diffentine equations."
},
{
"end_time": 3905.077,
"index": 159,
"start_time": 3885.435,
"text": " Diophantine equations, named after Diophantus, is just about finding rational or integer solutions to polynomials. I said these two are equivalent because you can always rationalize the denominator and cancel out. So finding solutions over q is really kind of the same as finding solutions over z."
},
{
"end_time": 3926.886,
"index": 160,
"start_time": 3906.101,
"text": " So a typical example of a Diophantine equation is find all the rational solutions to this and the solution this here is is Pythagoras. So Pythagoras tells us this is probably the most famous example three-fifths squared plus four-fifths squared is equal to one. If you think about it this is actually highly non-trivial. The fact that you know"
},
{
"end_time": 3946.493,
"index": 161,
"start_time": 3927.21,
"text": " The solution is that there is a one parameter infinite family solution of solutions."
},
{
"end_time": 3965.742,
"index": 162,
"start_time": 3946.493,
"text": " What we say points because you can plot this and this is thanks to Descartes you can plot this and this is a circle."
},
{
"end_time": 3978.848,
"index": 163,
"start_time": 3966.323,
"text": " So you're finding rational points on a unit circle. That's why the word solution and point is used interchangeably in this field called arithmetic geometry."
},
{
"end_time": 4003.951,
"index": 164,
"start_time": 3979.019,
"text": " This is great, but what I want to emphasize is that what's less known is that this is obviously just one solution, but there is a one parameter infinite solution to this. So in other words, all solutions can be parameterized in a specific way. So this is a quadratic case. Now if we recall high school algebra,"
},
{
"end_time": 4030.026,
"index": 165,
"start_time": 4004.36,
"text": " or high school Cartesian geometry, a quadratic is what's known as a conic section. You're slicing the cone. If you bump it up, it's already extremely difficult. If you bump up the degree, it's already become an impossible problem. So instead of considering x squared plus y squared is equal to one, by the way, all of the quadratic ones can be solved in a similar way. So all the conic sections"
},
{
"end_time": 4058.746,
"index": 166,
"start_time": 4030.401,
"text": " Conic sections not over the complex numbers, but conic sections over the rationals can be solved in a similar way. But once you go into cubics, you're completely stuck. Even something simple like this, how I change the two into three, how do you find all rational points on this? We still don't know in general, in some sense. But you can see the kind of problems we can get in Fermat, for example, is about talking about higher degree polynomials. Fermat's last theorem is"
},
{
"end_time": 4088.712,
"index": 167,
"start_time": 4059.343,
"text": " x to the n plus y to the n is equal to one and find all rational points on that particular curve. That's it. And the theorem states that the only rational ones are the quadratic ones. And they're from three and above. So x cubed plus y cubed is equal to one. There cannot exist any rational points and so on and so forth."
},
{
"end_time": 4115.06,
"index": 168,
"start_time": 4089.258,
"text": " So now these are called conic sections, that's just a curve like parabolas and stuff like this and circles and ellipses. Once it's a cubic, it's called an elliptic curve. There's a general theorem that so you can imagine there could be more terms, right? Why not have something like x squared y, that's also a cubic, is a degree three, or x y squared, that's also a cubic and so on."
},
{
"end_time": 4144.923,
"index": 169,
"start_time": 4115.503,
"text": " There's a theorem by Weierstrass that all of these ones can be reduced after transformations of variables into this form. So you have a quadratic in y and a cubic in x and then a linear in x. You can transform away all of the other coefficients if you wish. So this is called the general Weierstrass representation of an elliptic curve. This is my, well, I grew up in this one and it is important to emphasize"
},
{
"end_time": 4171.698,
"index": 170,
"start_time": 4145.145,
"text": " that that my bias towards favoring over this is what exactly prevented me from being able to understand any of this from the AI point of view in the beginning because for for an algebraic geometry this is the one that we always use and it's like our favorite thing we always try to try to use something and i will tell how an experiment failed playing with this just because i was taught to always think in terms of bias transform this is a canonical representation of elliptic curves"
},
{
"end_time": 4201.647,
"index": 171,
"start_time": 4171.698,
"text": " So just before we move on, this wire stress form, this means that any elliptic curve can be classified by these two numbers G2 and G4? Yeah, exactly, exactly. Yeah, there's a variable transformation that puts you into this canonical form. Okay, so I know you'll get to it, but I'm interested as to why this prevented you because I could imagine that these two numbers can serve as"
},
{
"end_time": 4217.415,
"index": 172,
"start_time": 4202.585,
"text": " Something like pixels or RGB numbers. You're reading my mind. You're reading my mind. That's exactly the experiment that I tried and I failed. And in hindsight, it's not surprising how I failed."
},
{
"end_time": 4244.684,
"index": 173,
"start_time": 4217.79,
"text": " But I'll get to that in a minute. So let's park that idea. So just like canonical conics can be written into some kind of standard conics that we would remember from high school, the canonical cubic can be written into this virus rostrum. The important thing about the cubic thing is somehow this cubic curve, there are very deep reasons for this, captures a lot of the non-trivial arithmetic and number theory."
},
{
"end_time": 4251.169,
"index": 174,
"start_time": 4245.06,
"text": " For example, the Fermat's last theorem was able to be cracked because Frey"
},
{
"end_time": 4278.763,
"index": 175,
"start_time": 4251.476,
"text": " And friends were able to reduce formats, the format. Well, that's not, you know, that's neither a conic nor cubic, but they were able to reduce that to a particular elliptic curve called the free elliptic curve. And then Wiles comes in and proves the modularity theorem. That's a whole big story. So somehow this, this cubic is just at the intersection. That's why, Oh, by the way, cubics are great because this is an example. Well, this is the only example of a Clavier manifold in dimension one."
},
{
"end_time": 4306.596,
"index": 176,
"start_time": 4279.582,
"text": " So there's something about the elliptic curve, the cubic curve. So all clavials in complex dimension one, remember the picture that I drew last time, where you have positive curvature is the sphere, zero curvature is the torus, and surfaces of general type are negative curvature. This is the Euler-Riemann normalization."
},
{
"end_time": 4334.087,
"index": 177,
"start_time": 4307.022,
"text": " The Riemann uniformization theorem. But the critical case, positive curvature, negative curvature, zero curvature, the torus, if you were to represent this algebraically, that's exactly this elliptic curve. So elliptic curves are Clavier manifolds of complex dimension one. And this is just the critical part, the critical part that also captures so much number theory. So that's why..."
},
{
"end_time": 4359.599,
"index": 178,
"start_time": 4335.077,
"text": " So algebraic geometers, differential geometers, physicists, and number theorists are interested in clabialness because of this intrinsic zero curvature. There's a lot of depth about this statement. Zero-curvatured objects give so much wealth because it's just at the boundary of positive and negative curvature. Oh, and that's what you mean by criticality is zero curvature?"
},
{
"end_time": 4381.135,
"index": 179,
"start_time": 4360.247,
"text": " Hi everyone, hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights, and the chance to be a part of a thriving community of like-minded pilgrimers."
},
{
"end_time": 4401.561,
"index": 180,
"start_time": 4381.135,
"text": " By joining you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here, hit subscribe and let's keep pushing the boundaries of knowledge together. Thank you and enjoy the show. Just so you know, if you're listening, it's C-U-R-T-J-A-I-M-U-N-G-A-L dot org."
},
{
"end_time": 4432.534,
"index": 181,
"start_time": 4403.097,
"text": " It's the dividing point and there's lots of conjectures. Yao has various conjectures on this about finiteness of this topological type in this space. But anyhow, back to, so that's why I know about the curves because I came from this algebra geometry string theory background that also wanted to study this Ritchie curvature flat or zero curvature objects. So back to this."
},
{
"end_time": 4455.469,
"index": 182,
"start_time": 4432.927,
"text": " So now it's a theorem, and this is a theorem due to so many people that have gotten the Fields Medal for proving different parts of this theorem. And this theorem really, really spanned a long time. People like Wey, Delene, Grotendieck, Dwork, Fountains, and Modell, they all contributed to this. And the theorem is this."
},
{
"end_time": 4469.002,
"index": 183,
"start_time": 4456.63,
"text": " We can't say something like Pythagoras, which says that there's a one family, infinite family parameter solution to the rational points of the quadratic or the conic."
},
{
"end_time": 4486.852,
"index": 184,
"start_time": 4469.599,
"text": " we can't say something like that because it's too hard but at least what we can say is the following is that any elliptic curve over q of the rational points over any elliptic curve itself forms a group and the group is of this form."
},
{
"end_time": 4501.817,
"index": 185,
"start_time": 4487.995,
"text": " The group is that there's an infinite parameter family of solutions. That's called R, it's called the rank. So how many copies of infinite solutions there are?"
},
{
"end_time": 4525.691,
"index": 186,
"start_time": 4502.91,
"text": " And then there is this finite, what's called the torsion solution. There are 16 types of torsion. This is really at the heart. All of this stuff, by the way, is at the heart of the Langlands. Ed Frankel would have told you about how excited he is about this sort of thing. But the one particular thing I want to emphasize is this rank"
},
{
"end_time": 4546.203,
"index": 187,
"start_time": 4525.691,
"text": " Is the number of infinite family of solutions and this is the rank of an elliptic curve. Yeah this is the mortal while theorem or the mortal exactly exactly exactly exactly exactly what are they."
},
{
"end_time": 4576.561,
"index": 188,
"start_time": 4546.886,
"text": " Anyhow, so the reason I want to emphasize there was still nothing to do with BSD, but this rank is the infinite number, how many measures, how many infinite family solutions over Q. So in the case of Pythagoras, the rank, if you wish, would be one because there's a one family, one infinite family, infinite family. So R is the generalization of one from the conic section case to the elliptic curve case."
},
{
"end_time": 4607.602,
"index": 189,
"start_time": 4577.602,
"text": " So that's it. This really is the state of the art in which you can say about rational points of an elliptic curve. There is this wonderful thing called rank, which is actually quite difficult to compute. It's not like I give you, no, I can just read it off. It's not like there's some analytic formula that says, ah, yeah, I get this. I look at this form one, this is x cubed, y squared. I can just tell you the rank is what I can't remember what it is in this case, two or whatever it is."
},
{
"end_time": 4626.271,
"index": 190,
"start_time": 4607.602,
"text": " The earliest experiment that I did was exactly as you suggested. This is back in 2019. I just took a database of about 3 million elliptic curves in Varshray's case. I took the two numbers G2 and G4."
},
{
"end_time": 4653.609,
"index": 191,
"start_time": 4626.988,
"text": " And then this was done. So this was a paper that I did with. So the reason is I did it with two data scientists who are using the fanciest data possible. And so we're a bunch of amateurs as far as BSTs is concerned, was to take the G2 G4 as two parameters and just plot them and then label them by R, the rank, and try to see a pattern. We got a null result."
},
{
"end_time": 4681.152,
"index": 192,
"start_time": 4654.206,
"text": " Because it was because this G2G4 in the database, I can show it's actually it's massive there in the in the in the trillions. So it's very hard to get much. We had to take the log of all these numbers to even establish a plot. And the rank was so randomly distributed, even with the fanciest technology. So what's the solution to this problem? Yeah, we got it. We didn't see anything."
},
{
"end_time": 4694.599,
"index": 193,
"start_time": 4681.647,
"text": " i was just we couldn't we couldn't get g2 and g4 to predict to any accuracy level what the rank is but nevertheless this was featured by new scientists because it was such a strange and novel thing to do"
},
{
"end_time": 4722.961,
"index": 194,
"start_time": 4695.026,
"text": " even though it was a null result, but at least it was inching towards something. Somebody someday must be able to say something intelligible at BSD from a data science point of view. Anyhow, back to this. So what should be the thing to do? And this is where number theory expertise actually comes in. First of all, this is an old law, which is if you can't solve a solution over the integers,"
},
{
"end_time": 4749.241,
"index": 195,
"start_time": 4723.336,
"text": " Solve it modulo primed and see how far you can get. So for example, I can't think of a rational number that I can't off the top of my head now. I think there is there exist solutions on top of my head of a rational point on this elliptic curve, but at least let me try to work over modulo prime. So modulo 23, this is true."
},
{
"end_time": 4779.599,
"index": 196,
"start_time": 4749.701,
"text": " You can check it because 2 cubed is 8, 8 plus 16 is 24, and that is one modulo. So that works. This one works. You can see in the modulo 5 this works. So, okay, this seems like a game, but the deep results of all the people like Deline and Wei and all these people is that if you work over a sufficient number of primes, in fact, if you worked over all primes and take a limit,"
},
{
"end_time": 4804.019,
"index": 197,
"start_time": 4780.265,
"text": " you should know something very deep about the solutions over q and that's the point. So in particular what you should record are these Euler coefficients which is the number of solutions modulo prime and and how they deviate from p plus one to the prime itself so this is what's known known as a as an Euler coefficients just keep track start with two"
},
{
"end_time": 4828.131,
"index": 198,
"start_time": 4804.872,
"text": " And then try three, try four, try three, try five, try seven, and then find how many solutions there are. And this is a finite problem. You can just do, in the worst case, a grid search because you're doing modular prime. You just have to count the number and just check by brute force. So now what you do is to form what's called the local zeta function."
},
{
"end_time": 4846.664,
"index": 199,
"start_time": 4829.377,
"text": " The local zeta function is this exponential generating function that keeps track of the number of solutions over p. So you fix a prime and you literally count the number of solutions modulo p."
},
{
"end_time": 4876.988,
"index": 200,
"start_time": 4847.312,
"text": " The more technical thing, what you really should do is because all finite fields are prime powers, what you actually do is to do an exponential generating function over the number of solutions over the field of prime characteristic p, but that's a technical side. But what you really should do, what you're essentially doing is to keep track of the number of solutions of the elliptic curve, the number of, not q points, but the number of fp points."
},
{
"end_time": 4904.002,
"index": 201,
"start_time": 4876.988,
"text": " Okay. And the fact that this generating function becomes this polynomial divided by polynomial form is what got when Deline proved this, he got the Fields Medal for showing that this is in this particular form. It's a very, very, very deep result in number theory. So, okay, long story short is that when I met Oliver and Lee,"
},
{
"end_time": 4933.319,
"index": 202,
"start_time": 4904.377,
"text": " They told me that whatever the hell I was doing with these data scientists, no offense to data scientists, we were just amateurs, we shouldn't have used the Weierstrass representation because the Weierstrass representation inherently is not what an arithmetic geometer, what a number theorist would be using that would capture the fundamental arithmetic of elliptic curves. So now there's the geometry of elliptic curves, which is my background, but that doesn't capture, the Weierstrass form doesn't capture that."
},
{
"end_time": 4960.555,
"index": 203,
"start_time": 4933.319,
"text": " The AP coefficients captures the arithmetic, so we should be using AP coefficients to predict the rank. So then we did, so instead of using a pair of very large integers, G2 and G4, and set up a neural network to predict R, you take instead the first, say, 100 AP coefficients."
},
{
"end_time": 4987.722,
"index": 204,
"start_time": 4960.845,
"text": " Now we're in a very interesting point cloud of a hundred dimensional point cloud. It turns out 50 is enough. It doesn't have to be a hundred just randomly chosen. You take the first 50, 50 primes. It's just not too crazy now for, you know, for, for by today's AI standards. Take the first list, let's do a hundred. Take the first AP, take the first 100 primes. Now take an elliptic curve."
},
{
"end_time": 5003.848,
"index": 205,
"start_time": 4988.507,
"text": " Reduce this elliptic curve and count how many solutions modulate these primes and compute these polar coefficients. So for each elliptic curve, you get this 100-dimensional vector. Move to the next elliptic curve, 100-dimensional vector. Now you just start labeling."
},
{
"end_time": 5028.046,
"index": 206,
"start_time": 5004.275,
"text": " Because of this wonderful thing called LMFDB, the LMFDB was set up by a bunch of people. I think Andrew Sutherland at MIT is one of the one of the instigators of the Sutherland and Booker and bunch of people set up this thing, which just records everything you ever wanted to know about elliptic curves. There are tens of millions, on the order of 10 million in this data set."
},
{
"end_time": 5057.483,
"index": 207,
"start_time": 5028.148,
"text": " So now you've got 100 dimensional vectors as you march through, because the LMFDB has the rank information, you can start set up a newer network. So we set up a newer network or some other classifier. And just like what we did with, just like what I did with Alessandro Andretti and Paranchelli on using G2, G4, this pair of vastrascoviches as a project rank, that gave no result."
},
{
"end_time": 5071.561,
"index": 208,
"start_time": 5057.91,
"text": " And when we did this for 100 differential vector, this immediately gave 99.99% accuracy in prediction. You went from zero to almost 100% immediately. And you're using less data as well?"
},
{
"end_time": 5099.497,
"index": 209,
"start_time": 5071.937,
"text": " I'm using less data. Remember, with the data scientists, with the G2, G4, with these guys, we used something like all 3.5 million Euclid curves. And we couldn't get any accuracies at all. But with this one, even with 100,000, so you give me any elliptic curve, you just look at the Euler coefficients in this one channel demands, I can tell you with almost 100% accuracy what this rank going to be. Interesting."
},
{
"end_time": 5125.469,
"index": 210,
"start_time": 5100.077,
"text": " So, of course, this immediately breaks birds' nests because I had to talk to real human experts who told me to actually use the oil representation rather than the partial representation. At Capella University, learning online doesn't mean learning alone. You'll get support from people who care about your success, like your enrollment specialist who gets to know you and the goals you'd like to achieve."
},
{
"end_time": 5154.77,
"index": 211,
"start_time": 5126.015,
"text": " You'll also get a designated academic coach who's with you throughout your entire program. Plus, career coaches are available to help you navigate your professional goals. A different future is closer than you think with Capella University. Learn more at capella.edu. But at the time, I was amazed. I was like, this is so cool. Of course, this is still useless for science. This was just a very, very cool thing to do in terms of visualizing the curves."
},
{
"end_time": 5170.316,
"index": 212,
"start_time": 5156.067,
"text": " And then after some digging, we realized what really was under the hood that was happening. You know, this is when Lee and Oliver were telling me, because they're number three, he says, well, this is no surprise, because this is the BSD conjecture at work."
},
{
"end_time": 5198.387,
"index": 213,
"start_time": 5171.237,
"text": " somewhere under the hood, this is the BSD conjecture at work. So now I can finally define for you with the BSD conjecture. I will give you the weak version so as not to bore you for two reasons. One, the strong version is very too technical, and two, I don't even understand it very well myself. But the weak version simply says, if you take this generating function that keeps track of the Euler coefficients, form this product polynomial,"
},
{
"end_time": 5225.606,
"index": 214,
"start_time": 5199.138,
"text": " Now you take a product of all primes. This is what's called the local to global behavior because you localize it. Remember this zeta function is local to a particular prime and now you take the product of all primes you form this new function called the global L function. Now I want to emphasize this is called zeta function"
},
{
"end_time": 5243.029,
"index": 215,
"start_time": 5225.981,
"text": " Okay."
},
{
"end_time": 5263.814,
"index": 216,
"start_time": 5243.353,
"text": " if you were to work over point but now we didn't we were more sophisticated our algebraic variety our manifold is not just a point but an actual elliptic curve so it becomes a much more richer structure so you get this this so this l function really is this global l function really is an analog of the riemann zeta function"
},
{
"end_time": 5291.817,
"index": 217,
"start_time": 5263.814,
"text": " So that's why this whole Langlands business is so beautiful and so intricate because he unifies geometry with geometry with harmonic analysis with number theory with all this is why eddick franco was so excited about all of this stuff right he was saying with such because he unifies so many different branches of mathematics anyhow so the bst conjecture states we don't know what the rank is you can't just buy you can't do it by looking at the curve"
},
{
"end_time": 5318.865,
"index": 218,
"start_time": 5292.449,
"text": " but once you have the L function by by this strange procedure working modulo over of a fixed prime and then take the product of all primes you get an analytic function now we have an analytic function i can start mucking around with i can use complex analysis the order of vanishing of this analytic function at one this is"
},
{
"end_time": 5337.193,
"index": 219,
"start_time": 5319.616,
"text": " Remember, the inverse of the Riemann zeta function"
},
{
"end_time": 5366.834,
"index": 220,
"start_time": 5337.927,
"text": " at one. The Riemann zeta function has a simple pole at one, so its reciprocal will have a vanishing of order one. Its order of zero is one at s equal to one. So it's a good analogy with the Riemann zeta function. But for elliptic curves, instead of a point, this one, the order vanishing should exactly equal to rank. So this gives you a way to compute rank, albeit a very, very convoluted and very, very intricate way to compute rank."
},
{
"end_time": 5391.34,
"index": 221,
"start_time": 5367.551,
"text": " But what surprised us was a neural network was able to predict this rank, this number, without going through any of this stuff and predicted almost to 100% accuracy. So where's your million dollars? Well, there's no million dollars because one, there's no proof, there is no statement. And so what? So this is we're still in the"
},
{
"end_time": 5418.029,
"index": 222,
"start_time": 5392.176,
"text": " As far as the Birch test, the A test has already failed because we had to talk to real experts. The I test has already failed at this point because there's no interpretability. So now how do we break the I test and then surprisingly also break the N test that should also be non-trivial? So that's why the memorization phenomenon was important."
},
{
"end_time": 5433.268,
"index": 223,
"start_time": 5419.104,
"text": " So then we got this prediction, we were very excited, oh yeah it's always really cool and then the excitement died down and we're like so what, what do we do? So this is the wonderful moment where you can recruit an undergraduate intern."
},
{
"end_time": 5457.517,
"index": 224,
"start_time": 5433.643,
"text": " So Alexey Potsnikov was at the time a second year undergraduate student of Kiu Huan Li's. But anyhow, so Alexey was given this thing like dig under the hood and tell us what this neural network or this classifier, this base classifier or tree classifier are actually doing. And finally, we homed in onto a PCA analysis."
},
{
"end_time": 5486.049,
"index": 225,
"start_time": 5457.517,
"text": " principal component analysis because basically we found that this rank prediction was doing so well with basically anything. Naive Bayes classifiers, you know, newer networks of a very simple architecture. We didn't even have to go to transformers or encoder architectures. It's just a simple, you know, forward feed linear active linear structure with, you know, I think it was a basically sigmoid activation function was good enough to do this."
},
{
"end_time": 5512.039,
"index": 226,
"start_time": 5486.049,
"text": " so and pca certainly would do it so if you remember pc i mean i said i i so pca i had to learn i didn't know what a pca was until 2017 pca was just a principal component analysis here's a 100 dimensional data you know is a point crowd i can't visualize it so i find these eigenvalues and and find the the the principal components of these eigenvalues meaning like the ones that the data is has most variance of"
},
{
"end_time": 5532.398,
"index": 227,
"start_time": 5512.039,
"text": " and then focus and project onto this eigenvalue directions. This is like stats 101, which I never learned, but luckily I knew what a PCA was, so we were just mucking around with PCA. So here's the remarkable fact. If you take a PCA projection of this 100 dimensional vector space of elliptic curve Euler coefficients,"
},
{
"end_time": 5554.309,
"index": 228,
"start_time": 5533.251,
"text": " Sorry, is it important that you do your principal component analysis down to two dimensions or does it just happen to work out that way in this case?"
},
{
"end_time": 5582.142,
"index": 229,
"start_time": 5554.565,
"text": " So in our case, we chose two because it was easier to see. It would have done it, if you project in any other dimension, you will see this kind of separation. Sure. Yeah. Two was just nicest. And in retrospect, we really should have chosen like an Italian flag or, you know, or a French flag, because it really just looks like a flag. In this one, this is elliptic curves of rank zero, rank one and rank two."
},
{
"end_time": 5606.254,
"index": 230,
"start_time": 5582.551,
"text": " This is already quite interesting, right? It's still useless in terms of actual mathematics. It was very good for AI that, you know, AI was able to just, you know, there's not even really AI. At this point, it was just PCA analysis. It was just a data analysis in a picture that analyzed elliptic curves in a way that was never done before. This is 2020 when we had this result."
},
{
"end_time": 5633.677,
"index": 231,
"start_time": 5607.398,
"text": " So we look at this and we saw that this is kind of nice. You know, they're elliptic curves, they're separated. Oh, by the way, elliptic curves, the ranks of elliptic curves is again a hugely, you can imagine because of the BSD. So Manjur Bhagawa was able to prove that almost all elliptic curves are ranked either zero or one in the infinite limit. And he got the Fields Medal for that. So this is obviously a very, very important result."
},
{
"end_time": 5660.93,
"index": 232,
"start_time": 5633.797,
"text": " But in the NFDB, you do also have higher ranks as well. Just, you know, in the infinite limit, you will be vastly dominated by ranks zero and one. So there are either no rational points, no infinite families of rational points or one parameter family, just like the conic section case. But the world record, I believe, is the one that's held by Noam Alkes, who has discovered this rank 28."
},
{
"end_time": 5689.428,
"index": 233,
"start_time": 5660.93,
"text": " a elliptic curve. It's huge. You can write it out. And that one is, so rank 28 means there are 28 parameter families of rational points on that particular curve. But so in LMFDB, you can already see there is sufficient number of ranks 0, 1, and 2 cases that there is this 3. I mean, there are rank 4s as well, but you won't see that in this picture."
},
{
"end_time": 5715.06,
"index": 234,
"start_time": 5690.538,
"text": " Now, so why do we why do I emphasize some PCA? No, I'm finishing with this now with PCA is because PCA is just a matrix projection It's a linear transformation You could look under the hood and just look at these matrices Okay, and that's what we that's what we asked Alexey to do. What does it mean to look under the hood at the matrix?"
},
{
"end_time": 5735.589,
"index": 235,
"start_time": 5715.623,
"text": " Because this is a PCA just because you know this is a it says 100 dimensional vector point cloud being projected to two dimensions so there's a whole bunch of 100 by two matrices you can just look at but nobody ever does this right you don't you don't look at you know what the AI algorithm is doing but in this case you can just make people look at it."
},
{
"end_time": 5764.804,
"index": 236,
"start_time": 5736.049,
"text": " And so we gave it to Alexi to look at it, not expecting much. And this is my undergraduate. But Alexi expected all expectations. He really looked at a lot of the sample of these matrices and noticed that almost all of the non-zero values are focused on essentially just one row. The one row dominated vastly over any other. So what does that mean in terms of what? So that's interesting."
},
{
"end_time": 5789.497,
"index": 237,
"start_time": 5765.367,
"text": " If you have a PCA projection, if you have a matrix projection that's focused on just one row, that means it's essentially you're just doing a sum. You're just doing an average in some sense. And that's exactly it. So now, what Wood is actually doing is that it's taking its Euler coefficients"
},
{
"end_time": 5813.148,
"index": 238,
"start_time": 5790.35,
"text": " an average in over a particular range of elliptic curves ordered by conductor, and I won't bore you with the details of conductor, and it's just computing this average. And if you do this average plotted against primes, you will start seeing this memorization phenomenon. So let me just emphasize a few points on this. First of all,"
},
{
"end_time": 5839.138,
"index": 239,
"start_time": 5813.575,
"text": " You're taking and this we wouldn't have done this if we didn't do a PCA analysis or if we first of if we didn't do this machine learning exercise on rank, we wouldn't have dug under the hood in the first place. So that was already AI guided. And then it told us we homed in on PCA because we thought that might have been the most interpretable thing. And once we do interpret it, it gave this equation."
},
{
"end_time": 5857.637,
"index": 240,
"start_time": 5840.094,
"text": " Which is a very very simple equation which always always says you just to do the following take a families of elliptic curves order them by this conductor range and and average over different elliptic curves but at a fixed prime."
},
{
"end_time": 5887.534,
"index": 241,
"start_time": 5858.729,
"text": " This is come to known as a vertical average. It's a very strange thing to do because traditionally what you would do is to average for fixed elliptic curves or average over different primes, right? You know, things like take the product formulas for a fixed elliptic curve and average over different primes. But the PCA told us no, you do the opposite. You take different elliptic curves over a range and average over a fixed curve."
},
{
"end_time": 5911.647,
"index": 242,
"start_time": 5887.91,
"text": " over a fixed prime and plot the thing against prime. Once you plot it, you see exactly what this is. So the red bit are all of these zero rank ones and the blue bit are exactly all the rank one ones. So the reason that all of the neural networks and all of the stuff were able to fundamentally tell the difference between"
},
{
"end_time": 5925.043,
"index": 243,
"start_time": 5912.056,
"text": " Hmm."
},
{
"end_time": 5954.394,
"index": 244,
"start_time": 5925.708,
"text": " It's guiding us what to do. It's guiding us. You can do this to all of the other ranks and just isolate different ranks and plot them. And it turns out that all of the even parity ranks, all the even ranks oscillated in this way and all of the odd ranks oscillated in the other way. Remember, there is 0, 1, 0, 1, 2, 3. They're all these different ranks. So fundamentally, you can tell the parity of the rank just by the way that these oscillation patterns happen."
},
{
"end_time": 5981.732,
"index": 245,
"start_time": 5955.299,
"text": " This is the point. This was a plot that was produced by Lee and Posnikoff. Posnikoff did this and showed us this. I remember this Zoom chat very well. This is 2020-2021. There were still COVID times. All we did all day was to Zoom people. My friends were saying, this looks like what these birds do."
},
{
"end_time": 6003.012,
"index": 246,
"start_time": 5982.108,
"text": " I say I looks like my rations of my my rations that's what it is my rations starlings and i said that is a way we should absolutely call this phenomenon. Memorations we should call instead of calling it a boring oscillatory pattern which is a this is an emigration like because it's not quite oscillatory right because it's kind of there's there's noise around it."
},
{
"end_time": 6025.691,
"index": 247,
"start_time": 6003.626,
"text": " And this noise is very interesting. I'll tell you in a bit. So this noise is part of the statistical error that you get from doing with finite data, because we know we only had, you know, 3.6 million, whatever LMFDB. But the point was, when then we immediately wrote to Sarnak, and to Sutherland, who are the leaders in this field, thinking that all this is trivial."
},
{
"end_time": 6054.258,
"index": 248,
"start_time": 6025.93,
"text": " This is a typical kind of thing. You write to the expert and you get a reply within a day. And he says, oh yeah, this is trivial. And it is a consequence of this theorem that I proved 20 years ago in this paper. This is the usual story, like 100 times is this, right? But not only do we not get back a thing, this message, we got a long message back from Peter Sarnak, who wrote,"
},
{
"end_time": 6084.138,
"index": 249,
"start_time": 6054.94,
"text": " What the hell guys, this is bizarre that this pattern exists. So why? Right. And, and then there was this many, many emails back and forth. I mean, to be honest, a lot of these emails were way over my head because you know, I, this is my contribution was, was an AI guided algebraic geometry. I'm not a number theorist. So there's lots of back and forth and, and, and then, and then this became the memorization, memorization phenomenon. And then there were all these conferences organized."
},
{
"end_time": 6101.203,
"index": 250,
"start_time": 6084.565,
"text": " So we're back to this birth test, right? So it failed the A because it wasn't automatic. We needed human. We were mucking around all the time with human experts. It passed the I test because it became interpretable. This was a precise formula."
},
{
"end_time": 6115.367,
"index": 251,
"start_time": 6101.51,
"text": " Most importantly, it passed the end test. This is the first time that an end test was passed because it actually galvanized a field of study number three. Now there's a whole field called murmuration phenomena."
},
{
"end_time": 6140.913,
"index": 252,
"start_time": 6115.691,
"text": " Wow. This is totally out of like, what the hell? I mean, this is above my pay grade because I don't know the number three community that well at all. So that's why Quanto was so exciting. There were conferences organized in ICERM. There are all these people, there are workshops apparently in Bristol and various universities. They're like, oh yeah, there's this kind of memorization workshop on this."
},
{
"end_time": 6156.834,
"index": 253,
"start_time": 6141.544,
"text": " so i'm not gonna i need to wrap up because i think it's getting too too much detail i want to tell you that there are other parts of this memorization thing and now but but now you can you can this is a precise conjecture this is a precise conjecture that was raised"
},
{
"end_time": 6172.244,
"index": 254,
"start_time": 6156.834,
"text": " Buy guided by a explorations with let humans and peter sun access is really well he says this is a conjecture that versions went and i could have raised themselves but they didn't because i never thought to take this average."
},
{
"end_time": 6202.961,
"index": 255,
"start_time": 6173.029,
"text": " because it's a bizarre thing to do. The AI doesn't know what it's doing. It always is doing this spotting patterns. So, and just to emphasize the parts of this conjecture now of the phenomena, which is expected to be true for all L functions. So this is why, so in other words, the, the memorization phenomena should be a general phenomenon in the entire Langlands program. So it's, it's now proven for Dirichlet character, the memorization Dirichlet character is now proved. So this memorization actually converges to a precise curve."
},
{
"end_time": 6230.418,
"index": 256,
"start_time": 6202.961,
"text": " And this was proven by, I wasn't involved in this because this was an actual number theory paper. And then Nina Zubrilina, who is a Sanax PhD student, who was a Sanax student, and then Alex Cohen, and now they've proved they've awaited two modular forms as well, and this was all in 2023. In 2024, there are more results being precisely proven. And what it really is, and this goes back to"
},
{
"end_time": 6256.391,
"index": 257,
"start_time": 6230.418,
"text": " Gauss and to Riemann zeta is that this memorization fundamentally generalizes a bias in the distribution of primes. That's also quite striking. So this is an interesting fact. So Chebyshev noticed this factor before. So Chebyshev noticed it's called the Chebyshev bias. If you take all primes,"
},
{
"end_time": 6278.985,
"index": 258,
"start_time": 6256.766,
"text": " If you take primes and find the remainder of these primes upon division by four, you're going to get remainder either one or three because you're modulo four and you would have thought that it's 50% one and 50% three in the large limit."
},
{
"end_time": 6288.046,
"index": 259,
"start_time": 6279.565,
"text": " But Chebyshev in the 19th century already noticed there's just a tiny bit of a bias towards three than one."
},
{
"end_time": 6316.152,
"index": 260,
"start_time": 6289.172,
"text": " and and that was just a conjecture of his and this is again mucking around with data this platonic data like why is it the primes are are more biased towards one one of the ones to upon division by four this is known as chebyshev's bias and this was proven actually by sarnak and and um rubinstein rubinstein in the 90s but only conditional on the riemann hypothesis being true"
},
{
"end_time": 6345.947,
"index": 261,
"start_time": 6316.493,
"text": " Interesting, right? There is this fundamental bias, so there's no unconditional proof of this, but it's conditional on the Riemann hypothesis that this is true. But what is interesting is that the memorization phenomenon is a generalization of this Chebyshev bias to all of the L-function world. So it's a generalization from prime biases to the biases in all L-functions because there is this underlined oscillatory behavior."
},
{
"end_time": 6365.469,
"index": 262,
"start_time": 6345.947,
"text": " Add that also has this deep relation with the bsd conjecture so that's where this whole this whole world um and that's why so so so so it it it passes i passes n people still work on it but it doesn't pass a because you know human expertise were constantly involved in interpreting you choosing pca"
},
{
"end_time": 6383.251,
"index": 263,
"start_time": 6365.981,
"text": " Anyhow, to wrap up this whole story, where are we in terms of mathematical conjectures in formulating problems in this top-down, guided mathematical discovery over the centuries? And I will say that in the 19th century, the eyes of Gauss were good enough to come up with this conjecture."
},
{
"end_time": 6410.725,
"index": 264,
"start_time": 6384.275,
"text": " But the 20th century BST already needed a computer to come up with a groundbreaking conjecture. And we are now just at the castle. But that's why so exciting that AI guided human intuition led to this deep mind paper and led to the memorization thing and this new in a new matrix multiplication. So obviously where we're going is this is this is this is a combination of these three directions."
},
{
"end_time": 6440.606,
"index": 265,
"start_time": 6411.578,
"text": " I mentioned earlier, in these very rooms that I'm talking, that Faraday would have had the conversation with Maxwell in this room about 150 years ago. Our institute has devoted one of our four themes to this AI for mathematical physics because this is really a paradigm shift in terms of how science is done."
},
{
"end_time": 6456.015,
"index": 266,
"start_time": 6440.981,
"text": " So just to promise, this is the last slide. So what is the current state of the art? Where are we with this AI guided thing? Let's drop this human guided intuition for the time being."
},
{
"end_time": 6483.712,
"index": 267,
"start_time": 6456.271,
"text": " Alpha G02 DeepMind has now reached silver medal. I think we had this long joke and discussion about how to beat the mind, the 16 year old Terry, sorry, the 12 year old Terry Tao. When you beat the 16 year old Terry Tao, it's game over. That's a new age of scientific experiments. But we're making like these guys or this field, we as a field, as a community, we're making progress every couple of months."
},
{
"end_time": 6492.585,
"index": 268,
"start_time": 6483.712,
"text": " the U.S. military thing."
},
{
"end_time": 6516.459,
"index": 269,
"start_time": 6493.166,
"text": " But you were not defined military. So the military wing, the DARPA, which is the Defense Advanced Research Institute, I was just at a meeting with them two weeks ago, just launched eXp math, where they literally are saying, exponentiating mathematics, you can Google this, the eXp math project that DARPA is funding now is how to benchmark"
},
{
"end_time": 6539.684,
"index": 270,
"start_time": 6516.459,
"text": " Proving accelerating mathematics by proving but unfortunately, they're not funding in this AI guided discovery area, which I think should be funded. And that's the way, you know, combination of all these three directions. Anyhow, so that's alpha due to alpha proof again, deep mind is on proving theorems at an almost"
},
{
"end_time": 6564.599,
"index": 271,
"start_time": 6540.981,
"text": " almost research level. And I think we even joked last time that at the silver medal level, AlphaGo2 is high school level, AlphaProof is maybe college level, right? But this is the interesting one. I think this is very just to wrap up the EPOC AI Frontier Math Project"
},
{
"end_time": 6592.892,
"index": 272,
"start_time": 6564.991,
"text": " And you can Google this. And this is really on professional mathematical problems. The kind of problems you would give it to a colleague or to a researcher, a very advanced postdoc or graduate student. And they're benchmarking this now. In fact, I'm flying to Berkeley on Thursday to help. There's a whole team of us we're flying in to benchmark the tier four problems. And this is happening this weekend, actually. So by"
},
{
"end_time": 6616.988,
"index": 273,
"start_time": 6593.933,
"text": " As of December 2024, EPOC AI is capable of solving only 2% of advanced research level mathematical problems. But by March, they finished their tier 1 to tier 3 problems, and they gave this division of tier 1 to 3 problems and gradual level problems. You can go to this website to look how hard"
},
{
"end_time": 6638.677,
"index": 274,
"start_time": 6616.988,
"text": " These problems are and terry tau gave some problems and i'm and i can all know these are all really research problems and um i gave a problem to the tier four which is which is about to appear um and there's there's still soliciting more and just just go to this link you can just look oh my god this is the kind of problem i would give to my to my research student"
},
{
"end_time": 6656.254,
"index": 275,
"start_time": 6638.677,
"text": " My order to the kind of problems i would work on with with a with a collaborator like that we would write papers about and their gate and their benchmark is about ten to twenty five percent on tier one to three so once you did their next benchmark is tier four so hard."
},
{
"end_time": 6680.538,
"index": 276,
"start_time": 6656.493,
"text": " that you know humans are not likely to solve it not because they're they're true not that not because they're tricky like you know um olympia problem but but because they're actual research level problems and and we are actually together as a community are attacking this kind of problems so this is where the the state of the art state of the artist so in terms of where the future of mathematics is i think"
},
{
"end_time": 6698.968,
"index": 277,
"start_time": 6680.896,
"text": " So I think I try to summarize in this picture. So I'm using an old picture of Terry Tao because he's the best human mathematician. And so how would it go? So you would have literature, the corpus of literature from scientific papers,"
},
{
"end_time": 6724.172,
"index": 278,
"start_time": 6699.377,
"text": " You go back and forth where human and I would process it together and then use top-down mathematics to formulate conjectures from platonic data that's gathered from the literature or process it directly from the literature and formulate the kind of problems. Once the problem is formulated, you would go to auto-formalization"
},
{
"end_time": 6743.677,
"index": 279,
"start_time": 6724.701,
"text": " where you it's only a matter of time before lean the lean community gets it get you know by also formalization i mean you take a math paper in latech and just hit return you know translated to a lean uh to a lean format we're very far from that right now um having conversations with buzzard"
},
{
"end_time": 6764.275,
"index": 280,
"start_time": 6743.677,
"text": " Not because of the technology is not there but because there's not enough lean data to train a large language model on interesting we only have millions of lines of. Lean so far available just millions we need billions in order to have and it's millions of you know this poor guys type in everything right yeah so."
},
{
"end_time": 6780.606,
"index": 281,
"start_time": 6765.179,
"text": " So conjecture formulation to this meta formalization and then you go through and find pathways through through mathly lib in this bottom up approach where you would have a combination of LLMs and that would generate your proof."
},
{
"end_time": 6796.766,
"index": 282,
"start_time": 6780.776,
"text": " I think it's it's where we already heading toward where where."
},
{
"end_time": 6812.193,
"index": 283,
"start_time": 6797.159,
"text": " It's not immediately foreseeable future that all of this is going to be automated, but what is remarkable is that this is within reach and I can't put a date on it, maybe 10 years, maybe 5 to 10 years."
},
{
"end_time": 6831.254,
"index": 284,
"start_time": 6812.193,
"text": " Because every single step of this, we are being helped by AI. Like for me, for example, the the memorization, which is taking available data and formulating in this way. And already last week, I'm meeting people who are actually coming up with proof pathways by even by by chat GPT and then humanly verifying that so."
},
{
"end_time": 6855.776,
"index": 285,
"start_time": 6831.254,
"text": " This is the brave new world of mathematics i mean by a theory of new discovery and where we were so lucky to be in this age where where has advanced enough we could actually help us with genuine new discoveries. Young thank you so much for bringing us to literally the front here the bleeding edge of math and also the future."
},
{
"end_time": 6873.387,
"index": 286,
"start_time": 6856.561,
"text": " It's a great pleasure talking to you. Thank you for listening. I get very excited about this. I'm parking everything else so I could devote to this new community and it's a growing community of mathematicians who believe in this and I think there was a recent"
},
{
"end_time": 6895.555,
"index": 287,
"start_time": 6874.002,
"text": " There's a recent Quanta report. I'm not involved in that one. It involves experts like Andrew Granville, who talked about what is a beautiful proof and how AI can help us with it. And they are also amazed at how fast this is going. Interesting. Thank you. Thank you very much."
},
{
"end_time": 6911.903,
"index": 288,
"start_time": 6895.93,
"text": " I've received several messages, emails and comments from professors saying that they recommend theories of everything to their students and that's fantastic. If you're a professor or lecturer and there's a particular standout episode that your students can benefit from, please do share and as always feel free to contact me."
},
{
"end_time": 6939.428,
"index": 289,
"start_time": 6912.363,
"text": " New update! Started a sub stack. Writings on there are currently about language and ill-defined concepts as well as some other mathematical details. Much more being written there. This is content that isn't anywhere else. It's not on theories of everything. It's not on Patreon. Also, full transcripts will be placed there at some point in the future. Several people ask me, hey Kurt, you've spoken to so many people in the fields of theoretical physics, philosophy, and consciousness. What are your thoughts?"
},
{
"end_time": 6951.63,
"index": 290,
"start_time": 6939.838,
"text": " While I remain impartial in interviews, this substack is a way to peer into my present deliberations on these topics. Also, thank you to our partner, The Economist."
},
{
"end_time": 6976.271,
"index": 291,
"start_time": 6953.882,
"text": " Firstly, thank you for watching, thank you for listening. If you haven't subscribed or clicked that like button, now is the time to do so. Why? Because each subscribe, each like helps YouTube push this content to more people like yourself, plus it helps out Kurt directly, aka me. I also found out last year that external links count plenty toward the algorithm,"
},
{
"end_time": 7002.295,
"index": 292,
"start_time": 6976.271,
"text": " Which means that whenever you share on Twitter, say on Facebook or even on Reddit, et cetera, it shows YouTube. Hey, people are talking about this content outside of YouTube, which in turn greatly aids the distribution on YouTube. Thirdly, you should know this podcast is on iTunes. It's on Spotify. It's on all of the audio platforms. All you have to do is type in theories of everything and you'll find it. Personally, I gained from rewatching lectures and podcasts."
},
{
"end_time": 7022.278,
"index": 293,
"start_time": 7002.295,
"text": " I also read in the comments"
},
{
"end_time": 7048.422,
"index": 294,
"start_time": 7022.278,
"text": " and donating with whatever you like. There's also PayPal. There's also crypto. There's also just joining on YouTube. Again, keep in mind it's support from the sponsors and you that allow me to work on toe full time. You also get early access to ad free episodes, whether it's audio or video. It's audio in the case of Patreon video in the case of YouTube. For instance, this episode that you're listening to right now was released a few days earlier. Every dollar helps far more than you think."
},
{
"end_time": 7076.954,
"index": 295,
"start_time": 7048.643,
"text": " Either way, your viewership is generosity enough. Thank you so much. Think Verizon, the best 5G network is expensive? Think again. Bring in your AT&T or T-Mobile bill to a Verizon store today and we'll give you a better deal. Now what to do with your unwanted bills? Ever seen an origami version of the Miami Bull?"
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{
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]
}No transcript available.