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Yang-Hui He: Math Will Never Be The Same Again...
January 3, 2025
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Yonghui He, welcome to the podcast. I'm so excited to speak with you. You have an energetic humility and your expertise and your passion comes across whenever I watch any of your lectures. So it's an honor. It's a great pleasure and great honor to be here. In fact, I'm a great admirer of yours. You've interviewed several of my very distinguished colleagues like, you know, Roger Penrose and Edik Franco. I actually watched some of them. It's actually really nice.
Wonderful, wonderful. Well, that's humbling to hear. So firstly, people should know that we're going to talk about or you're going to give a presentation on AI and machine learning mathematics and the relationship between them as well as the three different levels of what math is in terms of production and understanding bottom up top down and then the meta. But prior to that, what specific math and physics disciplines initially sparked your interest and how did the collaboration with Roger come about?
so my my you know my bread and butter was mathematical physics especially you know sort of the interface between algebra geometry and string theory so that's my background uh what i did my phd on and um so at some point uh i was editing the book uh with cn young
who is an absolute legend. You know, he's 102, he's still alive and he's the world's oldest, um, living Nobel laureate. You know, Penrose is a mere 93 or something. So CN Yang of the Yang-Mills theory. So it's an absolute legend. He got the Nobel prize in 1957. So at some point I got involved, uh, in editing a book with CN, CN called Topology in Physics. And, you know, with a name like that, you can just invite anybody you want and they'll say, probably say, yes,
And that was my initial friendship with Roger Pembroke started through working together on that editorial. I mean, I have Roger as a colleague in Oxford, then I've known him on and off for a number of years, but that's when we really started getting working together. So when Roger snickers at string theorists, what do you say? What do you do? How does that make you feel? That's totally fine. I mean, I'm not a diehard string theorist and, you know, I'm just generally interested
You just happened to study the mathematics that would be interesting to string theorists, though you're not one.
I talk a public lecture in Dublin about the interactions between physics and mathematics, and I still find that string theory is still very much a field that gives the best cross-disciplinary feedback. I've been doing that for decades. It's a fun thing. I talk to my friends in pure mathematics, especially in algebraic geometry. 100% of them are convinced that string theory is correct.
because for them it's inconceivable for a physics theory to give so much interest in mathematics. Interesting. And that's kind of a, I think that's a story that hasn't been told so much, you know, in the media. You know, if you talk to a physicist, they were like, you know, string theory doesn't predict anything, this and the other thing. But there's a big chapter of string theory, you know, to me, more than 50% of the story, backstory of string theory, is just constantly giving new ideas in mathematics.
And, you know, historically when a physical theory does that, it's very unlikely for it to be completely wrong. Yeah. You watched the podcast with Edward Frankel and he takes the opposite view. Although he initially took the former view that, okay, string theory must be on the correct track because of the positive externalities. It's like the opposite of fossil fuels. It doesn't give you what you want for your field like physics, but it gives you what you want for other fields as a serendipitous outgrowth. But then
He's no longer convinced after being at a string conference. So you still feel like the pure mathematicians that you interact with see string theory as on the correct track as a physical theory, not just as a mathematical theory. Yeah. So yeah, absolutely. He does make a good point. And so, like, I think, you know, Franco and, you know, algebra geometers like Richard Thomas and various people, they they appreciate what string theory is constantly doing in terms of mathematics.
and the challenges um you know where whether it is a you know whether it is a theory of physics based on the fact that it's giving so much mathematics i guess you know you got to be a mystic some of them some of some of them are mystics some of us are mystics and i actually i don't i don't personally have an opinion on that i just you know some days i'm like well you know it's this is such a cool mathematical structure and there's so much
internal consistency it's got to be there's got to be something there so it's just a gut feeling but of course you know it being a science you know you need the experimental evidence you know you need to go through the scientific process and that i have absolutely no idea it could take years and decades wouldn't you also have to weight the field like w e i g h t weight the
Whatever feel like the sub-discipline of string theory with how much iq power has been poured into how much rock talent has been poured into it versus others so you would imagine that if it was the big daddy field which happens to be that it should produce.
more and more insights and it's unclear to me at least if that this much time and effort went into asymptotic safety or loop quantum gravity or what have you or causal set theory if that would produce mathematical insights of the same level of quality we don't have a comparison I mean I don't know I want to know what your thoughts are on that I think the reason for that is just that you know we follow our nose as a community
The contending theories like low quantum gravity and stuff,
You know, there are people who do it. There are communities of people who do it. And, you know, there's a reason why the top mathematicians are going to do string related stuff is because, you know, you follow the right notes. You feel like it is actually giving the kind of the right mathematics. Things like, you know, mirror symmetry, you know, or vertex algebras that's kind of giving the right ideas constantly. And it's been doing this since the very beginning. So,
and people do the alternative theories of everything, but so far it hasn't produced new math. You can certainly prove us wrong, but I think there's a reason why Witten is the one who gets the Fields Medal.
Uh, because it's just somehow is at the right interface of the right ideas in geometry, number theory, representation theory, algebra, that this idea tends to produce the right, you know, the, the right, the right mathematics, whether, whether it is a theory of physics, that's still, you know, that's the next mystical level. Um, but, you know, it's, it's kind of, it's, it's, it's, it's an interesting, it's an exciting time actually. Witten didn't get the Fields Medal for string theory though.
It was his work on the Jones polynomial and turn Simon's theory and Morse theory with supersymmetry and topological quantum field theory, but not specifically string theory. That's right. That's right. But he certainly is a champion for string theory. And, um, for him, I mean, you know, that, that idea of funny, so he was able to do, um,
you know the Morse theory stuff he was able to get because of his work at supersymmetry he was able to realize this was a supersymmetric index theorem that generated this idea and that that's really a super symmetry really is a cornerstone for string theory even though there's no experimental evidence for it so i think that's one of the reasons that's guiding him towards this direction so what's cool is that
Just prior, the podcast that I filmed just prior to yours was Peter White, as you know, is a critic of string theory and Joseph Conlin, who is a defender of string theory and he has a book even called Why String Theory. That's right. I think it was the first time that publicly someone like Peter White, along with a defender of string theory, were just on a podcast of this length.
Speaking about in a technical manner. What are both of their likes and dislikes of string theory and then the string community? There's there's three issues string theory as a physical theory string theory as a tool for mathematical insight and then three string theory as a sociological phenomenon of overhype and does it see itself as the only game in town is the arrogance should there be arrogance?
was an interesting conversation.
slightly orthogonal to the main string theory community. I'm just happy because it's constantly giving me good problems to work on. Yes. Including what I'm about to talk about in AI. Wonderful. I'll mention a little bit about it because I got into this precisely because I had a huge database of Clavier manifolds and I wouldn't have done that without the string community. It's again one of those accidents that you know no other you know the other theoretical physicists didn't happen to have this
It didn't happen to be thinking about this problem. There's this proliferation of Klabial manifolds. And I'll mention that bit in my lecture later on why this is such an interesting problem, why Klabialness is interesting inherently, regardless whether you're a string theorist. And that kind of launched me in this direction of AI assisted mathematical discovery. So this is kind of really nice. And I think for me, the most exciting thing about this whole community is that, you know,
Science and especially theoretical science, including theoretical science has become so compartmentalized. Everyone is doing their tiny little bit of thing.
String theory has been breaking that mold for the last decades. It's constantly going, let's take a piece of algebraic geometry, let's take a bit of number theory here, elliptic curves, let's take a bit of quantum information, entanglement, whatever, entropy, black holes. And it's the only field that I know that different expertise are talking to each other. I mean, this doesn't happen in any other field that I know of.
Well let's hear more about what you like thinking about and what you're enthusiastic about these days. Let's get to the presentation.
Sure. Well, thank you very much for having me here. And I'm going to talk about work I've been thinking about stuff I've been thinking about for the last seven years, which is how AI can help us do mathematical discovery, you know, in theoretical physics and pure mathematics. I recently wrote this review for Nature.
which is trying to summarize a lot of these ideas that I've been thinking about. And there's an earlier review that I wrote in 2021 about how machine learning can help us with understanding mathematics.
So let me just take it away and think about, oh, by the way, please feel free to interrupt me. I know this is one of these lectures. I always like to make my lectures interactive. So please, if you have any questions, just interrupt me anytime. And I will just pretend there's a big audience out there and just make it. So firstly, you're likely going to get to this, but what's the definition of meta-mathematics? OK, great. So, Rafi, of course, you know,
How does one actually do mathematics? In this review I tried to divide it into three directions. These three directions are interlaced and it's very hard to pull them apart.
but roughly you can think about you know bottom-up mathematics which is you know mathematics is a formal logical system you know definition and uh you know lemma proof and you know theorem proof and that's certainly how mathematics is presented in in you know papers and there's another one which i like to call top-down mathematics is where you know where the practitioner looks from you know above that's why i say topped out from like a bird's eye view you see different
Ideas and subfields of mathematics and you try to do this as a sort of an intuitive creative art You know, you've got some experience and then you're trying to see oh well Maybe I can take a little bit of peace from here and a piece from there and I'm trying to create a new idea or or maybe a method of proof or attack or derivation Yes, so these are these two so that's that's you know, so complementary directions of research
And the third one, meta, that's just because it was short of any other creative words, because there's, you know, words like meta science and meta philosophy or metaphysics. I'm just thinking about mathematics as purely as a language, you know, whether the person understands what's going on underneath.
Okay. I don't know if you know of this experiment called the Chinese room experiment. Yeah. Okay. So in that,
The person in the center who doesn't actually understand Chinese but is just symbol pushing or pattern matching, I don't know if it's exactly pattern, rule following, that would be the better way of saying it. They would be an example of bottom up or meta in this. So I would say that's meta.
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Head over to their website www.economist.com slash totoe to get started. Thanks for tuning in and now back to our explorations of the mysteries of the universe. So I would, I would say that's meta in the sense that the person doesn't even have to be a mathematician. You're just simply taking symbols, a large language modeling for math, if you wish. Got it.
Of course, you know, there's a bit of component rather as you know that you can see there's a little bit of component on bottom up because you are taking mathematics as you know a sequence of symbols But I will mainly call that matter if that's okay. I mean this this definitions are just you know things that I've just I'm using yes. Yes, but in any case I would talk mostly about this this bit which is this What I've been thinking mostly about
One thing i just made just just to set the scene you know 20th century of course you know computers have been playing an increasingly important role in mathematical discovery.
and of course you know it speeds up computation all that stuff goes without saying but something that's perhaps not so emphasized and appreciated is the fact that there there are actually fundamental and major results in in mathematics that could no longer have been done without the help of the computer and so this you know there's famous examples even back in 1976 this is the famous apal haken cock proof of the four-color theorem
you know that every map you only takes four every map in a plane only takes four colors to completely color it with no neighbors it was a problem this is a problem that was posed I think probably by by Euler right and this was finally settled by reducing this whole topology problem to thousands of cases and then they ran it through a computer and checked it case by case so and then other major things like you know the Kepler conjecture which is you know that that stacking
and
Yes. Wasn't there a recent breakthrough in the generalized Kepler conjecture? Absolutely. So this is what Marina Wiozowska got the Fields Medal for. So the Kepler conjecture is in three dimensions, our world. Wiozowska showed in dimensions 8, 16, and 24 with the best possible patina. And she gave a beautiful proof of that fact. And to my knowledge, I don't think she actually used the computer. There's some optimization
Actually, what I'm referring to is that there are some researchers who generalized this for any n, not just 8, not just 24, who used methods in graph theory of selecting edges to maximize packing density to solve a sphere packing problem probabilistically for any n, though I don't believe they used machine learning.
The proof is, you know, it took 200 years
The final definitive volume was by Dittgenstein 2008. What's really interesting, the lore in the finite group theory community is that nobody's actually read the entire proof. It's just not possible. It takes longer for people to actually read the entire proof than a lifetime. This is kind of interesting that we have reached the cusp
In mathematical research, where mathematics, you know, the computers are not just becoming, you know, computational tools, but it's increasingly becoming an integral part of of who we are. So this is just just set the scene. So we're very much in this, you know, we're now in the early stages of the 21st century. And this is increasingly the case where we have this where computers can help us or I can help us in these three different directions. Great.
So let me just begin with this bottom up and you're sort of to to summarize this. This is probably the oldest attempt in in in where computers can can help us. So so this is where I'm going to define bottom up, which is I guess it goes back to
The modern version of this is this classic paper, the classic book of Russell Whitehead on the Principia Mathematica, which is 1910s, where they try to axiomize, axiomatize mathematics, you know, from the very beginning. You know, it took like 300 pages for them to prove that one plus one is good at two, famously. Nobody has read this. So this is this is one of these impenetrable books. But I mean, this but this tradition goes back to, you know, Leibniz or to Euclid even, you know, that the idea that mathematics should be axiomatized.
Of course, this program took only about 20 years before he was completely killed, in some sense, because of Gödel and Churchill's incompleteness theorems. This very idea of trying to axiomatize mathematics by constructing layer by layer is proven to be logically impossible within every order of logic.
I'd like to quote my very distinguished colleague, Professor Minyong Kim. He says the practice of mathematician hardly ever worries about Gödel.
Because if you have to worry about whether your axioms are valid to your day-to-day, if an algebraic geometry has to worry about this, then you're sunk. You get depressed about everything you do. So the two bots kind of cancel out. But the reason I mention this is that because of the fact that these two bots cancel each other out, these two negatives cancel each other out, this idea of using computers to check
I'm
Even back in 1956, Noah Simon and Shaw devised this logical theory machine. I have no idea how they did it because this is really very, very, very primitive computers. And they were actually able to prove some certain theorems of Principia by building this bottom up, you know, take these axioms and use the computer to prove.
This is becoming an entire field of itself with this very distinguished history. And just to mention that this 1956 is actually a very interesting year because it's the same year, 56-57, that the first neural networks emerged from the basement of Penn and MIT. And that's really interesting, right? So people in the 50s were really thinking about the beginnings of AI,
you know, because neural networks is what we now call, you know, you know, goes under the rubric of AI. And at the same time, they were really thinking about computers to prove theorems and mathematics. So it's 56 was a kind of a magical year. And, you know, this this neural network really was a neural network in the sense that, you know, they put
Cadmium sulfide cells in a in a in a basement. It's a it's a wall size of photoreceptors and they were Using, you know flashlights to try to stimulate neurons literally to try to simulate computation. Uh-huh. That's part of impressive Impressive thing and then this this thing really developed right and then and now, you know You know a half a century later We have very advanced and very very sophisticated
computer-aided proof, automated theorem provers, things like the Coq system, the Lean system, and they were able to create, so Coq was what was used in this four-color, the full verification of the four-color, the proof of the four-color theorem was through the Coq system. And, you know, then there's the Phy-Thomson theorem, which got Thompson the Fields Medal, again, was, they got the proof through this system.
A Lean is very good. I do a little bit of Lean, but also Lean, the true champion of Lean is Kevin Buzzard at Imperial, 30 minutes down the road from here, from this spot. And he's been very much a champion for what he calls the Xena project, and using Lean to formulate, to formalize all of mathematics. That's the dream. What Lean has done now,
is that it has, um, uh, Kevin tells me that all of the undergraduate level mathematics at Imperial, which is a user, it's a non-trivial set of mathematics, but still a very, very tiny bit of actual mathematics and they can check it and everything that we've been taught so far at undergraduate level is good and self-consistent. So nobody needs to cry about that one. Wonderful. And so that's all good. And then more recent breakthroughs is the beautiful work of, um, you know,
So three Fields Medalists here, also two Fields Medalists, Gowers Green, Manners and Tao, and when they prove this conjecture, which I don't know the details of, but they were actually using Lean to prove, to help prove in this. And I think Terry Tao in this public lecture, which he gave recently in 2024 in Oxford, he calls this whole idea of AI co-pilot, which I very much like this word.
I was with Tao in August in Barcelona. We were at this conference and he's very much into this very well. And of course, you know, Tao, Terry Tao for us is, you know, is a godlike figure. So the fact that he's championing this idea of AI co-pilots for mathematics is very, very encouraging for all of us.
Yes, and for people who are unfamiliar with Terry Tao, but are familiar with Ed Whitten, Terry Tao is considered the Ed Whitten of math and Ed Whitten is considered the Terry Tao of physics. Yeah, I've never heard that expression. That's kind of interesting. At Barcelona, when Terry was being introduced,
I'm
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the gauss of mathematics or the Mozart. But I think a more appropriate thing to describe him is to describe him as the Leonardo da Vinci of mathematics, because he has such a broad impact on all fields mathematics. And that's very rare thing.
Yeah, I remember he said something like topology is my weakest field and by weakest field to him, it means I can only write one or two graduate textbooks off of the top of my head on the subject of topology. Yeah, exactly. Exactly. I guess he's very he's his intuitions are more analytic. He's he's very much in that, you know, world of, you know, analytic number three functional analysis. He's not very pictorial, surprisingly.
I like Roger Penrose has to do everything has to be picked has to be in terms of pictures but Terry is a is a symbol symbolic matcher you know we can just look at equations extremely long complicated equations and just see which pieces should go together that's kind of very interesting.
Speaking of Eva Miranda, you and I, we have several lines of connection. Eva's coming on the podcast in a week or two to talk about geometric quantization. Eva is super fun. She's filled with energy. She's a good friend of mine. In this academic world of math and physics, I think we're at most one degree of separation from anyone else. It's a very small community, relatively small community.
This back to this thing about, of course, you know, one could one could get over optimistic. I was told by my friends in DeepMind that that Shagady, who I think he's one of the one of the on the on this AI math team, he says, you know, he was instructing that computers beat humans in chess in the 90s, beat humans go at 2018. So you should beat humans in beating proving theorems in 2030. I have no idea how he extrapolated how he extrapolated these points. I only three data points.
But you know but you know but deep mind has seen has a product to sell so it's very good for them to be over optimistic but i wouldn't be surprised that that this number you know you well i'm not sure to beat humans but it might give ideas that humans have not thought about before so that's that's possible.
Just moving on so that that's the bottom up and I said this is very much a a blossoming or not blossoming it's very much a long distinguished field of automated theorem computing of theorem provers and verifications of formalization mathematics which which tau calls the AI copilot. Just to mention a bit with your question a bit earlier about meta mathematics. So this is just
Kind of I like your analogy. This is like the Chinese the Chinese room, right? Can you do mathematics without actually understanding anything? You know personally I'm a little biased because having interacted with so many undergraduate students before I moved to the London Institute so I don't have to teach anymore Teach undergraduates. I've noticed, you know, I I
Maybe one can say the vast majority of undergraduates are just pattern matching, whether there's any understanding. I think this is one of the reasons why chat GPT does things so well. It's not just because LLMs are great, large language models are great. It's more that most things that humans do are so without comprehension anyway. So that's why it's kind of this pattern matching idea. And this is also true for mathematics.
What's funny is that my brother's a professor of math in the University of Toronto for machine learning, but for finance. And I recall 10 years ago, he would lament to me students that came to him who wanted to be PhD students. And he would say, okay, but Kurt, some of them, they don't have an understanding. They have a pattern matching understanding. He didn't want that at the time, but now he's into machine learning, which is effectively that times 10 to the power of 10. Right.
The vast majority of us doing most of the stuff is just pattern matching. So that's why and this is even this is true even for mathematics.
So here I just want to mention something which is a fun project that I did with my friends Vishnu Jigala and Brent Nelson back in 2018 before LLM. So before all this LLM for science thing. And this is a very fun thing because what we did, we took the archive and we took all the titles of the archive. This is the preprint server for contemporary research in theoretical sciences.
What's really interesting, this is my favorite bit, we took to benchmark the archive, we took Vixra. So Vixra is a very interesting repository
because it's of archive spelled backwards and he has all kinds of crazy stuff. I'm not saying everything on Vixra is crazy, but certainly he has everything that archive rejects because he thinks it's crazy. Things like, you know, three page proof of the Riemann hypothesis or Albert Einstein is wrong. It's got filled with that. It's interesting to study the linguistics, even at a title level, you could see that, you know, what they call the distinctions of quantum gravity versus the other things, they have the right words in Vixra.
But the word order is already quite random that you know the another with the classification matrix the confusion matrix for vixra is certainly not as distinct as archive which you know so kind of interesting you get all the right buzzwords is like you know kind of thing that's what i think is a good benchmark that linguistically is not as sophisticated as you know real research articles.
But this idea, so this is something much more serious is this very beautiful work of Chitoyan at all in nature, where they actually took all of material science and they did a large language model for that and they were able to actually generate new reactions in material science. So this, I think this paper in 2019, this paper by Chitoyan
is really the beginnings of l l m l l m for for scientific discovery this is quite early it is 2019 right yeah and it's remarkable how we can even say that that's quite early the field is exploding so quickly absolutely five years ago is considered quite some time ago yeah absolutely even five years ago i you know i was still very much in a lot of i've evolved in thinking a lot about this thing
I would also like to get to your personal use cases for LLMs, chat GPT, Claude, and what you see as the pros and cons between the different sorts like Gemini was just released at 2.0.
And then there's a one and there's a variety. So at some point I would like to get to you personally, how you use LLMs both as a researcher and then your personal use cases. Well, you know, I can mention, I can mention a little bit. So one of the very, very first, um, things would charge you BT three came out in what? 2018, something 2019, something like that. Oh, three. You mean GPT three?
GPT-3 like the really early baby versions. Yeah, that was during just before the pandemic just before pandemic. So that was just like so I got into this AI for for math through this club y'all manifold wasn't going to mention bit bit bit later. And then this GPT came out when I was just thinking about this large language model. So this is a great conversation. So I was typing and
problems in calculus, freshman calculus. And you will solve that fairly well. I mean, it's really quite impressive what you can do. So, you know, it's fairly sophisticated because, you know, things like, you know, I was typing questions like, you know, take vector fields, blah, blah, blah, on a sphere, you know, find me the grad or the curve. I mean, it's like
you know first second year stuff and you have to do a lot of computation and it was actually doing this kind of thing correctly you know partially because there's just so many example sheets of this type out there on the internet and so he's kind of learned all of that so i was getting very excited and i was um
Trying to sell this to everybody at lunch. I was having lunch with my usual team of colleagues in Oxford over this. And of course, lo and behold, who was at lunch was the great Andrew Wiles. So I felt like I was being a peddler for GPT, LLM for mathematics to perhaps the greatest living legend.
in mathematics. Andrew is super nice and he's a lovely guy and he just instantly asks me, he says, how about you try something much simpler? Two problems he tried. The first one was tell me the rank of a certain elliptic curve and he just typed it down a certain elliptic curve or rational points of a very simple elliptic curve, which is his baby.
I typed it when it was complete got completely wrong. Yes It was just even you started very quickly started things saying things like, you know, five over seven is an integer Partially because it's very this is a very hard thing to do You can't you can't really guess integer points by choosing I like in calculus where there's there's a routine of what you need to do, right? And then and then very quickly we we we converge on even simpler problem
how about find the 17th digit in the in the in the decimal expansion of 22 divided by 29 like whatever and that it's completely random because you can't train you actually have to do long division this is you know this is you know primary school level stuff and yet GP teacher simply cannot do and it's inconceivable that he could do it because no language model could possibly do this
But GPT now 0010201 is already clever enough. When you ask him a question like this, linguistically, he knows to go to war from alpha. And then it's okay. Then he's actually doing them. But so something so basic like this, you just can't train the language model to do you get, you know, one in 10, right? And it's just a randomly distributed thing. Yes.
where sophisticated things, there are seemingly sophisticated things like solving differential equations or doing very complicated integrals. It can do because there's somewhat of a routine and there are enough samples out there. So that's my user case, two user cases. That's also not terribly different than the way that you and I or the average person or people in general think.
So for instance, we're speaking right now in terms of conversation. And then if we ask each other a math question, we move to a math part of our brain. We recognize this is a math question. So there's some modularity in terms of how we think. It's not like we're going to solve long division using Shakespeare, even if we're in a Shakespeare class and someone just jumps in and then ask that question. We're like, OK, that's a difference. That's of a different sort of mechanism. Yeah, that's a good analogy. Yeah, yeah.
When you first encountered chat GPT or something more sophisticated that could answer even larger mathematical problems, did you get a sense of awe or terror initially? So I'll give you an example. There was this meeting with some other math friends of mine and I was showing them chat GPT when it first came out. And then one of the friends said, explain, can you get it to explain some inequality or prove some inequality?
And then it did and then explained it step by step that he then everyone just had their hand over their mouth like, are you serious? Can you do this? And then they're like, and one said, one friend said, this is like speaking to God. And then another friend said, had the thought like, what am I even doing? What's the point of even working if this can just do my job for me? So did you ever get that sense? Like, yes, we're excited about the future and it as an assistant, but did you ever feel any sense of dread?
I'm by nature a very optimistic person. So I think it was just all an excitement. I don't think I've ever felt that I was threatened or the community is being threatened. I could totally be wrong. But so far it just seems like this is such an awesome thing because it'll save me so much time looking up references and stuff like this. Yeah, I was happy.
I was just like, wow, this is kind of cool. I mean, I guess if I were an educator, I might get a bit of a dread because there's like, you know, you know, undergraduate degrees, you know, if you do an undergraduate degree, it's just basically one chat GPT being fed to another. You know, a lot of my colleagues started setting questions in exams with chat GPT with fully locked act out equation. I mean, this is becoming the standard thing to do. I guess even if you're an educator, you would probably worry.
Wonderful. All right, let's move forward. Yeah, sure.
I'm so twenty twenty two was a great yeah i'm surprised this was like every single newspaper i don't know why i'm at least i was told after some obscuring outlet i can't remember some some expert friends in the in the community told me that the chat gpt has passed the turing test.
This is a big deal, but I'm I don't know why it hasn't been. I was hoping to see some BBC and every year, every major newsletter, but it didn't didn't catch on. But anyhow, I believe that in 2022, Chachi BTS passed the Turing test.
And then you know where in the last and the last two two years you know this is obviously where we can you know this is a huge development now for LL large language models for mathematics and you know every every major company open AI matter AI epoch AI.
You know everything and they've been doing a tremendous, you know, working in trying to get LLM format. Basically, you know, take the archive, which is a great repository for mathematics and theoretical physics, pure mathematics and theoretical physics, and then just learn that and try to generate to see it to how much and I'm because this is very much work in progress.
And of course, you know, AlphaGeo, AlphaGeo 2, AlphaProof, this is all the DeepMind's success. It's kind of interesting within a year, you know, you've gone from 53% on Olympia level to 84%, which is part, you know, this is scary, right? Every, this is scary in the sense, like, impressively awesome that, you know, they could do so quickly. So basically in 2022, an AI is approximately equal to the 12-year-old Terence Tao.
In the sense that it does it could do a silver medal, but of course, this is a very specialized You know the the alpha alpha geo 2 was really just homing in on Euclidean geometry problems, which which to be fair extremely difficult, right? If you don't know how to add the right line or the right angle You have no idea how to attack this problem, but it's kind of learned how to do this. Yeah So it's kind of nice. So 20, you know, this is all within you know, a couple of years and There's this there's this
Very nice benchmark called Frontier Math that EPOC AI has put out. I think there was a white paper and they got Gowers and Tao, you know, the usual suspects, just a benchmark. Okay, fine. So we can do 84% on Math Olympiad, which is sort of high school level. What about truly advanced research problems? So to my knowledge, as of the beginning of this month, it was only doing 2%.
Alright so that's okay fine so it's not doing that great but the beginning of this week you learn that OpenAI 03 is doing 25% so we've gone 20% up we've got a fifth up within four weeks.
I love this right because it's exciting it's very rare to be I remember back in the day when I was a PhD student doing ADS CFT related algebra geometry
I remember that kind of excitement, the buzz in the string community. And you know, people are saying, you know, there was a paper every couple of days on the next, you know, that kind of excitement. And I haven't felt that kind of excitement for, you know, for a very long time just because. Wow.
and then this this is like that right you know every every week this is new benchmark and new new you know breakthrough so that's why i'm fine this field of ai system mathematics will be really really exciting can you explain perhaps it's just too small on my screen because i have to look over here but can you explain the graph to the left with terence tau oh gosh i'm not sure i can because i'm sure
I'm sure I can read this graph in detail. I think it's the year... What is it trying to convey? So it's the ranking of... No, this is just Terence Tal's individual performances over
Different years I over over different points. So he's retaking the test every year No, no his take you see his he's taken the three times ages 10 11 and 12 and when it was When it was 10, he got the bronze medal and then he got the silver medal then he got the gold medal within three years Okay, and age of 12 or something, but I can't I think what are those bars though? I think the bar is a good question
I think maybe it's to the the different question you're giving 60 questions and what do you what do you would take to get the gold medal I think or what it would take or to get the silver medal I think.
Okay, so it wasn't a foolish question of mine. It's actually. No, no, no, it's a good question. I have no recollection or maybe I never even looked at it. Somebody told me about this graph at some point. I forgot what it is. Okay, because it looks to me like Terrence Tao is retaking the same test and then this is just showing his score across time and he's only getting better. But that can't be it. Why would he retake the test? He's a professor. No, I think it goes to 66. It must be like
This is an open source graph. Oh, I thought you were going to say this is an open problem in the field. What does this graph mean? No, no, no. It's an open source. This graph is just you can take it from the Math Olympiad database. Got it. Which I shamelessly. See, again, perfect, right? I've just done something that I have absolutely no understanding of presented to you like a language model. And I just copy and paste it because it's got a nice, cute picture of Terry's style when he was a little boy.
So finally, I'll go back to the stuff that I that I be really thinking about, which is sort of top down mathematics, right? So and then this is kind of interesting. So the way we we do research, you know, practitioners is completely opposite to the way we write papers. I think that's important to point that out.
We mock about all the time. We look at my board, right? It's just filled with all kinds of stuff. And most of it is probably just wrong. And then once we got a perfectly good story, we write it backwards. And I think writing math papers backwards, and math generally defines math and theoretical physics papers backwards. Well, theoretical physics is a bit better. At least sometimes you write the process. But in pure math papers, everything is written in the style of Bobacki.
This very dry definition proof and which is completely not how it's actually done at all this is why you know i know that you know the great vladimir arnold says you know a boba key is criminal. He actually use this word that the boba the criminal boba key ization of mathematics.
Because it leaves out all human intuition experience. It just becomes this dry machine like presentation, which is exactly how things should not be done. But bovaki is extremely important because that's exactly the language that's most amenable to computers. So it's one way or another.
But we know human practitioners certainly don't do this kind of stuff, right? We mock about, you know, if we have to, and sometimes even rigorous sacrifice, right? If we have to wait for proper analysis in the 19th century to come about before Newton invented calculus, we won't even know how to compute the area of an ellipse.
Because you have to wait and formalize all of that, you don't just go all backwards. So kind of the historical progression of mathematics is exactly opposite to the to the way that is represented presented and it's fine. But the way it's presented is better. It's much more amenable to approve copilot system like lean than what we actually do.
even science in general is like that where we say it's the scientific method where you first come up with a hypothesis and then you just you test it against the world gather data and so on but the way that scientists not just in math and physics but biologists and chemists and so on work are based on hunches and
creative intuitions and conversations with colleagues and several dead ends. And then afterward you formalize it into a paper in terms of step by step, but it was highly nonlinear. You don't even have a recollection most of the time of how it came about. That's right. And I think one of the reasons I got so excited about all this, this AI format is, is this direction because this hazy idea of intuition or experience. And this is something that a neural network is actually a very, very good at. Wonderful.
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uh you could help you so i'm going to give concrete examples um later on about you know how we give guides humans but uh but just to just to um give some classical examples you know i always i've given this i've said this joke so many times i think
So, um, what's the best work neural network of the 18th century? Well, it's clearly the brain of gals. I mean, that's a perfectly functioning, perhaps the greatest neural network of all time. And this is, I mean, I want to use this as an example because you know, what, what did gals do? Now gals plotted the number of prime numbers less than a given positive real number. She used to give us sort of continuity.
And he plotted this and it's kind of a really, really, you know, jackety curve. And it's a step function. It's a step function because it jumps whenever you hit a prime. But Gauss was just able to look at this when he was 16. I said, well, this is clearly X over log X. How did he even do this experience? I mean, he had to compute this by hand. And he did. And he got some of the wrong even.
uh you know primes he had tables by his time the tables of primes were up in the tens and hundreds of thousands he has to go up in the hundred thousand range i think i just look at this is x or log of x and then but this is very beautiful this is very important because he was able to raise a conjecture before the method by which this conjecture is proved namely complex analysis was even conceived of by by kochi and rima and that's a very important fact
So he just kind of felt that this was X over log X and you had to wait for 50 years before Hadamard and Tel Aviv percent approved this fact because this technique known as which we now take for granted this technique called complex numbers complex analysis wasn't invented by Cauchy wasn't invented yet. You had to wait for that to happen. So that's kind of that's how it happens like this in mathematics all the time.
even major things on the of course you know this this is so it's now it's called the prime number theorem which is a cornerstone of all of mathematics right this is the first major result since euclid on the distribution of primes how did gauss say this was x over log x i don't hear that because he had a really great neural network and this happened it happens over and over again like you know the best swing die conjecture which i'm going to talk about later uh which is a is a it's one of the millennium problems
and it's still open and it's certainly one of the most important problems in mathematics of all time. And this is Birch and Swindon and Dyer in a basement, you know, in Cambridge in the 1960s. They just plotted ranks and conductors of lead curves. I'm going to define those in more detail later. And they will say, oh, well, that's kind of interesting. You know, the rank should be related to the conductor in some strange way. And that's now the BST conjecture, the Birch and Swindon and Dyer conjecture.
And what they were doing was computer aided conjectures. So this is so here was Gauss, the eyeballs of Gauss in the in the 19th century. But the 20th century really have seriously computer aided conjectures. And of course, the open of the proof of this is still open in general. There've been lots of nice progress in this. And, you know, where we're going to go is very much what technique do we need to wait
to prove something like this. Now is there a reason that you chose Gauss and not Euler? Like is it just because Gauss had this example of data points and guessing a form of a function? I'm sure Gauss, I'm sure Euler who is certainly is great had conjectures maybe
That's an interesting quote. I'll mention Euler later, but I think there is not an example as striking as this one. In fact, what's interesting as a byproduct of Gauss inventing this, because it was kind of mucking around with statistics, right? This is before statistics existed as a field as well, right? This is like early 1800s. And Gauss, I think, and you can check me on this, Gauss
got the idea of statistics and the Gaussian distribution because he was thinking about this problem. So it's kind of interesting. So he's laying foundations to both analytic number theory and modern statistics in one go. He was doing regression. So he, he, he, I think he, he essentially invented, you know, regression and the curve fitting, which is our, you know, this is like one on one of modern modern society. He was trying to fit a curve.
Yeah What was the curve that really fit this and you know in in the process he he got x over log x and in addition He got this idea of regression and a impressive guide. What can we say? He's a god to us all And then um, so the the the the the upshot of this is like I I love this again there's something I found on the internet and um Just to emphasize, you know that this idea of thinking of god
Yes, this idea of marking about with data in pure mathematics is a very ancient thing. Once you formulate something like this in conjecture,
You will write your paper imagine you know writing a paper you will say conjecture you know definition prime definition pie of x then conjecture pie of x evidence rather than all of the failed stuff about inventing regression and mocking about all that stuff just gets not written at all that intuitive creative process is not written down anywhere.
So so here is a it's great i'm glad i'm glad i'm chatting to you about right because it's nice to have a have an audience with this right so you know if you look at like so pattern recognition so what do we do right in terms of pure mathematical data if i gave you a sequence like this you can immediately tell me what the next number is to some confidence yeah zero zero one zeroes is just you know this is just multiple three or not this one i've tried this with many audiences and you know after a few minutes of struggle
You can you can get the answer and then this turns out to be the prime characteristic function so what I've done here is to mark all the odd integers and Evens obviously you're gonna get zero so it's kind of pointless you just all just a sequence of odd integers and Then it's a one if it's a prime is zero if it's not so three five seven eight eight
And so on and so forth. No, sorry. Three, five, seven, nine, 11. And you mark all the odd ones, which are one. Okay. And you can probably after a while, you can muck about and you can, you can see where, where, where this is going. Uh, the next sequence is much harder. So I'm not, I'm going to give away. So we won't have to spend like a couple of hours staring at it. Um, so this one is, um, the, what's called the shifted Moebius function. What this is just, you take an integer.
and you take the parity of the number of prime factors it has up to multiplicity starting from two. I think I didn't start from one here and then if it's one if it's a if it's and so maybe I did start one it's zero if it's an odd odd number of prime factors it's one if it's an even number of prime factors for all the sequence of integers.
And I hope now I've gotten this right. So it's if I think I start with two two has So that's all no, let's see two three Yeah, so so I did start with what I'm gonna mark one for one just to start this kick off the sequence And then two is a prime number. He has only one prime factor It's an odd number three is an odd number prime factors four is two because it's two squared So it has an even number of prime factors and so on so forth
So five is prime, it has one odd number. Six is two times three, so it has two, an even number of prime factors, so on and so forth. It looks kind of harmless. What's really interesting, so this is even number. So I've been staring at this for a while, it was very, very hard to recognize a pattern. And what's really interesting is that to know the parity of the next number, if you have an algorithm that can tell me the parity of this in an efficient way,
You will have an equivalent formulation of the Riemann hypothesis. So that's actually an extremely hard sequence to predict. So if you can tell me with some confidence more than 50% what the next number is without looking up some table, then you can probably end up cracking every bank in the world. Interesting. Because this is equivalent to the Riemann hypothesis. So I'm just giving three. So trivial kind of
Okay. Um, really, really, really hard. Yes. So now you can think about a question. How, if I were to feed sequences like this into some neural network, how would a neural network do? Um, so one way to do it. So this goes a bend. So we go way back to the very beginning to the question of what is mathematics and, you know, Hardy,
in this beautiful in this beautiful apology says you know what mathematicians do is essentially we are um pattern recognizers that's that's probably the best definition of what mathematics is or is is that it's a study of patterns finding regularity in patterns and in fact you know if there's one thing that AI can do better than us is pattern detection because you know we we evolved in being able to detect patterns in three dimensions and no more
So in this sense, if you have the right representation of data, you're sure that AI can do better than that. I mean, you know, generate a lot of stuff, but filtering out what is better is a very interesting problem of itself. So let's try to do one. I mean, there are various ways to do this representation. One way you can do it is to do a problem which is
Maybe best fit for for an AI system, which is binary classification of binary vectors. So what you do is what you know sequence prediction is kind of difficult. So what one thing you can do is just take this infinite sequence and just take say a window of a hundred a thousand with fixed window size and then label it with the one immediately outside the window and then shift label shift label. So then you can generate a lot of training training data this way. Uh huh.
So for this sequence, I think I've just taken here, you know, whatever the sequence is, and I might just, with a fixed window size, and with this label. So now you have a perfectly supervised, perfectly defined binary supervised machine learning problem. Then you pass it to your standard AI, you know, algorithm, that they're, you know, just, you know, out of the box ones, nothing. You don't even have to tune your particular architecture. Just take your favorite one.
And then,
that you know any any neural network or whatever base classifies would do it a hundred percent accuracy as you should because you'll be really dumb if you didn't because this is just a linear transformation so even if you have a single neuron that's just doing linear transform that's good enough to do it the prime q problem i did some experiment um some oh gosh i've been like seven years ago it got 80 accuracy
And I was like, wow, that's kind of, this was a wow moment. I was like, why, why is it doing it? I have, I don't have a good answer to this. Um, why is it doing 80% accuracy to this? How is it learning? Maybe it's doing some sieve method, uh, which is kind of interesting. I somehow, um, the second number is just to Chi square, just to double test that the, uh, what's called MCC, um, which is Matthew's correlation coefficient.
um these are just buzzwords in stats i i never learned stats but now i've learned i'm relearning i took cross sarah in 2017 so i can relearn all these buzzwords um it's great it's really useful and then this shifted over lambda function it's um uh sorry i think i made a i yeah i i mistakenly called this called this merbius mu function it's not i mean it's related but it's not it's the it's the shift in the over your lambda function
Sorry, one of my neurons died when I said Möbius mu, but it's Leuville lambda. You were subject to the one pixel attack. Yeah. But so this one I couldn't break 50%. Right. Point five just means it's coin toss. It's not doing any better guessing than whatever. And this chi-square is zero point zero zero. That means I'm up to statistical error. So which means I couldn't find an AI system which could break, which could do better than random guess.
I'm not saying there isn't one. It would be great if there were one. It's life. If I do break it, I might actually stand a good chance.
Breaking every bank in the world. All right, but I don't I haven't made it worse. Let's remain close friends. Yeah, that's right. That's right So I was very proud of this because this experiment I'm gonna mention a bit later this little lambda was suggesting I was just trying like way back when
But apparently Peter Sarnak, whom I really admire, he's one of the world's greatest number theorists currently, current number theorist. And I got to know him through this memorization thing that I'm going to talk about later. And I reminded him that I almost became his undergraduate research student. I ended up doing, I was an undergrad at Princeton where I had two paths I could follow.
um, for, you know, to, to, you know, it kind of defines your union, your undergraduate thesis, right? So one was in mathematical physics was one was that's with, uh, Alexander McDowell and the other one was with, uh, you know, two, two problems. And the other one was, was actually offered by Peter Sinek on arithmetic problems. And I somehow just, I, because I was, I wanted to understand the nature of space and time.
I went through the Alexander McDowell path to do mathematical physics, which led to do string theory. After 20, 30 years, I came full back to be in Peter Sondagwald again. I met him at this conference, I reminded him of this, and he was very happy. What's really interesting is that he was asking DeepMind the same question a few years ago about the deluvial lambda, whether DeepMind could do better than 50%.
So I was glad that I thought along the similar lines as a great expert in number theory and somebody who could have potentially have been my supervisor. And then I would have gone into another theory instead of swing theory, which is whatever. It's how life happens. So perhaps you're going to get to this later on in the talk. But I notice here you have the word classifier and the recent buzz since 2020 or so has been with architecture, the transformer architecture in specific.
So is there anything regarding mathematics, not just LLMs, that has to do with transformer architecture that's going to come up in your talk? Not specifically. I'm actually, it's interesting, I'm one of my colleagues here at the London Institute. He's Mikhail Berdsov. He's an AI, he's our Institute's AI fellow, and he's an expert on transformer architecture.
So I've been talking to him and we're trying to devise a nice transform architecture to address problems in finite group theory. It's in the works, but nothing so far, even with the memorization stuff, it's very basic neural networks that we didn't use anything more sophisticated than that.
So to be determined whether it will outperform the standard ones will be kind of interesting. Got it. Yeah. So actually now we go way back to the beginning of our conversation is how I got into this stuff. And that, I don't know, completely coincidentally was through string theory.
So at this point, maybe I'll just give a bit of a background of how all this stuff came about. At least personally. Why was I even thinking about this? Because I knew nothing about AI seven, eight years ago. Literally zero. I knew nothing more than to read it from the news.
I know this is actually a very interesting story which shows again the kind of ideas that the string theory community is capable of generating just because you got all these experts looking on a kind of interesting problems. So let's go way back and again you know I quoted Gauss right I gotta cook I have to say something about Euler. So this is a problem again this is a you can see I'm very influenced by three.
The number three, you know, I'm a total numerologist, right? Trinity, name the three, three is something, right? And then there is called the trichotomy classification theorem by Euler. This dates to 1736. So if you look at, so I'm going to say the buzzword, which is connected compact orientable surfaces. So these are
you know i mean the words explain themselves you know they have no boundaries yes and they're uh you know topologically you know whatever the the topological surfaces so euler was able to realize that a single integer uh characterizes
All such surfaces. So this is the standard thing that people see in topology, right? So the surface of a ball is the surface of a ball and you can deform it. You know, the surface of a football is the same as an American football. It can deform without cutting or tearing. And then the surface of a donut is the same as, you know, your cup.
right because you know is everything that everyone understand the thing you know this is it has one handle and then so the surface of a donut is exactly the topologically the what they call topologically homeomorphic to to the the cup and then you got the you know the the pretzel so i think that's a pretzel
Or maybe I think this is like the German pretzel and it gets more and more complicated but the oil oilers because you know either invented to the field of topology so you realize this this idea of topological equivalence in the sense that there's a single topological invariant called which we now call the oil number.
which characterizes these things. Another way to an equivalent way to say is the genus of these surfaces is, you know, no, no handles, one handle, two handle, three handles, and so on and so forth. It turns out that the Euler number, what we now call the Euler number is two minus twice the genus. So two, two, two minus two G. Okay, that's great. So this is, that's the classic Euler's theorem.
And then, you know, comes in Gauss, right? Once you got these three names next to each other, Euler, Gauss, and Riemann, you know, this is, it's got to be some serious theorem. So Euler did this in topology and then Gauss did this incredible work, which he calls him, he himself calls him the Theorema Grigium, the great theorem, which he considers this is his personal favorite. And this is Gauss, right?
and Gauss said, you can relate this number to, which is, this number is purely topological. You can relate this number to metric geometry. So he came up with this concept, which we now call Gaussian curvature. It's just some complicated stuff. You can characterize this curvature, which you can define on this. This is even before the word manifold existed on the surface.
And then you can integrate using calculus and the integral of this Gaussian curvature divided by four pi is exactly equal to this topological number. And that's incredible, right? The fact that you can do an integral, it comes out to be an integer. And that integer is exactly topology. So this idea, this Gauss related geometry to topology in this one suite
And then the next level comes Riemann. Riemann says, well, what you can do is to complexify. So these are no longer real connected compact orientable surfaces, but you can think about these as complex objects. So what do we mean by that is, well, if you think about the real Cartesian plane, that's a two-dimensional object.
But you can equally think of that as a one complex dimensional object, namely the complex plane. Or the complex line. Yeah, the complex line. Exactly. So with R2, Riemann would call C. And then Riemann realized that you can put similar structure on all of these things as well. So all of a sudden these things are no longer two dimensional real or interval surfaces, but one complex dimensional would
It's a terrible name, so a complex curve is actually a two-real dimensional surface. And it turns out that all complex curves
are orientable. So you already rule out things like applying bottles and stuff like that or Möbius strips. So the complex structure requires orientability and that's partly because of Cauchy-Riemann relations. It puts a direction. You can't get away. But the interesting thing is all of this now should be thought of as one complex dimensional curves. They're called curves because they're one complex dimension, but they're not curves. They're surfaces in the real sense.
So now here comes, so if you apply this to the Gauss thing, you get this amazing trichotomy theorem. And the theorem says, if you do this to the curvature, you can see this. I mean, the number here is two, right? You get the only number two, which is a positive curvature thing, right? And that's consistent with the fact that the sphere is a positively curved object. Locally, everywhere, it has positive curvature.
If you do it to a torus or the surface of a donut, which is just called the algebraic donut, you integrate that, you get zero curvature. And this is not a surprise because you have a sheet of paper, you fold it once, you get a cylinder and you fold it again, you glue it again, you get this torus, this donut. And this sheet of paper is inherently flat. Yes.
So if you just take a piece of paper, you take this piece of paper and you roll it up, you get a cylinder. And then you do it again, and you get the surface of a donut, like a rubber tire. And that is currently zero curvature. And then you can do this, and this is a consequence of what's known as Riemann uniformization theorem. If you do anything that has more than one handle, you get zero curvature.
So now you have the trichotomy, right? You have positive curvature, zero curvature and negative curvature. The one in the middle is really, obviously is interesting. It's the boundary case in complex algebra geometry. These things are called final varieties. Earlier you said if you have anything that's more than one handle, you have zero curvature. You meant negative curvature. Sorry, sorry. I meant negative curve. Okay. So these fidget spinners on the right,
They all have negative curvature. Everything here has negative curvature. Got it. Yeah. So now in the world of complex algebra geometry, these positive curvature things are called final varieties. After this Italian guy final, the, these negative curvature objects, which proliferate are called varieties of general type. And this boundary case are called zero, zero curvature objects.
And it just so happens we now call things in the middle clavia These zero curvature objects. Yes So so far this has nothing to do with physics. I mean it's just the fact of topology, right? But this is such a beautiful diagram that you know took from 1736 until Riemann Riemann what died in in the 1860s I think or something like that. So it took you know 100 120 years to really formulate just this table to relate
Metric Geometry to Topology to Algebraic Geometry is kind of a beautiful thing, right? So to generalize this table is the central piece of what's now called the Minimal Model Program in Algebraic Geometry for which there have been all these fields, you know, be a car a couple of years ago,
And then it started with Maury who got the Fields Medal and then this whole Mukai and this whole distinct, distinguished idea. So basically this minimal model program should just generalize this to higher dimension. This is dimension complex, dimension one, right? How do you do it? It's very hard. And once you have it, I won't bore you with the details. This is very nice, you know, there's topology, algebra, geometry, differential geometry, index theorem, they all get unified in this very beautiful way.
And you want to obviously want to generalize this to arbitrary dimension, arbitrary complex dimension. It'd be nice. It's still an open problem. How do you do it in general? It's a very nice problem. But at least for a class of complex manifold known as scalar manifolds, I won't bore you with the details, but scalar manifolds on which where the metric has very nice behavior, there's a potential for which you can have a double derivative that gets on the metric.
And then it was conjectured in by Calabi in the 50s. Again, you know, 54, 56, 57. It was a great year, right? That's all these different ideas. I mean, in three completely different worlds now come together because mathematical physicists have kind of tied it up, you know, the world of neural networks, the world of Calabi conjecture, the world of string theory to one.
I like, you know, when when things get bridged up in this way, you know, but, you know, again, this this the theorem itself is extremely technical. But the idea is for this killer manifold, there is an analog of this diagram. Basically, I love this slide. I saved this slide for my own. Okay. I keep a collection of dictionaries in physics and math. Yeah. I think this is beautiful. Yeah, me too. But you took me like it took me years.
to to do this table because you know it's not written down anywhere and it touches different things i think it's not written down anywhere precisely because mouth textbooks are written in the bobaki style uh-huh um but now it just becomes clear what would people be thinking about for the past 100 years you know after grodendieck
It's just trying to relay these ideas, you know, this is intersection theory of characteristic classes. So this is topology and you know, this is, I mean, this is over 200 years of work of, you know, sent, you know, the central part of analytics and mathematicians like churn, Richie, oil, everything, every, every, every, everyone, everybody was every involved in this diagram is an absolute legend. In fact, there is one more column to this diagram.
I think for short of, I think when I did this, this was a slide from some time ago, but when I was talking to a string audience, there is one more, one more, which is relations to L function. And that's when number theory comes in. So there is one more column. And to understand this world to this one more column of its behavior to L functions, that's the Langlands program.
Right so it's actually really magical that this this this table actually extends more as far as I mean that's just as far as we know now right of course L functions and its relations to modularity and I think this is of course obviously mathematics to me like mathematics is about extending this table as much as possible to let it go into different fields of mathematics
so but at least for sure we know there is one because of the langlands correspondence there is one more column and that column should be on on number theory and modularity
And soon there'll be another table on the yang invariant, the he invariant. No, I don't think I don't think I have enough talent to to create something that that but it could well be there should be something something new to to to. Right. That's that's to me. That's really the most fun part about mathematics. It's not not so I mean, they're like, you know, who is it? I think maybe it's Arnold as well because there's two types of mathematicians. There are the the the the hedgehogs and they're the the birds.
I mean, he's been saying his entire life
Just trying to think about can I bound, can I bound the, you know, the how many, you know, in the, in the, what is the, what's the limsop of the, of the distance between, between prime pairs. And the technique he uses is, is, is, is beautifully argued as analytic number theory technique, sieve methods, you know, kind of, you know, the, the, the Ben Green world of, of this, of, of, of sieves and James Maynard.
I'm 100% in the bird category.
I mean, once I see something, of course, sooner or later you need to dig like a hedgehog. But the most thrill that I get is when I say, oh, wow, this gets connected. So the results are proven when you dig, but the connections are seen when you get the overview. Yeah, yeah, absolutely. So I mean, of course, again, this is a division that's kind of artificial. In all of us, we do a bit of both. Yes.
the guy who really does a well is a great dimension of course it's like it's become like a grand well he passed away this John McKay who was a Canadian probably the greatest Canadian mathematician since Coxeter John McKay really saw unbelievable connections in fields that nobody will ever see and he passed away he became sort of in the last 10 years of life he became sort of like a
I like a grandfather to to me. He's so you know, he saw my kids grow up, you know over zoom I Know so the the the London Math Society asked me to write obituary I was very touched by this I know so I wrote his obituary for and I was just trying to say what this guy is the ultimate pattern, you know linker So so John McCain Absolute legend great
Um, moving on. Uh, I mean, this is a, this is very much, this is very much a huge digression for what I'm actually going to tell you about, which is, you know, the birch test for AI and that's great. Do you, you know, do you have a limit on how, what the, these videos are? No, just so you know, some of them are one hour. Some of them are four hours and people listen for all of it. Yeah, this is great fun. Great. Yeah, same. I'm loving this. Yeah, me too.
Because normally, you know, I have one up in the with 55 minute cutoff. Yes. Oh, right. And in like five minutes questions. And I'm like, oh, my God, I haven't said most of the stuff I wanted to say. Yeah. Yeah, exactly. Because the point of this channel is to give whoever I'm speaking to enough time to get through all of their points rather than their rushing and not covering something in depth. I want them to be technical and rigorous. So please.
Continue sure sounds good to me. So so club is so in that magical year of 1957 of neural networks the magical year of what it was as the the automated theorem prover world and the world of algebra geometry in three complete world different world They didn't even know of each other's names Let alone the results club the conjecture that it leads for scalar manifolds. This diagram is very much well defined this table
And Yao proved it 20 years later. So Shintong Yao, who is very much like a mentor to me. And he gets the Fields Medal immediately. So you can see why this is so important. He gets the Fields Medal because this idea of falling through clubby is trying to generalize this
this sequence of ideas of Euler, Riemann, and Euler, Gauss, and Riemann. So it's certainly very important. So there it is. We can park this idea. So Yao showed that there are these
Kailar manifolds that have this property that have the right metrical properties. So by metric, I mean distance, you know, something can integrate over that because here, you know, you never think you that this this integral is messy, right? Even if we do this on a sphere, right? This, this R has all these cosines and signs that have got the, you know, they've all got to cancel at the end of the day to get four pi.
yes like what the hell and then divided by two pi get two and that's the only number which is kind of amazing stuff and now you can do this in general the the just as a caveat um y'all show that this metric exists he never actually gave you a metric so the only currently no metric on these things
What's interesting is that these automated theorem provers, they seem computational, and it's my understanding that computationalists, so people who use intuitionist logic, they don't like constructive proofs.
Sorry, they like constructive proofs. They don't like non-constructive proofs. In other words, existence proofs without showing the specific construction. So it's interesting to me that all of undergraduate math, which has some non-constructive proofs, are included in Lean. So I don't know the relationship between Lean and non-constructive proofs, but that's an aside. Yeah, that's an aside. I probably won't have too much to say about it.
Cool. So back to, I don't know why I went on this diatribe on digression on string theory, but I just want to say this is a side comment. So this is a, this is something since seven, since 1736, which is kind of nice, which is, uh, you know, Oh, by the way, that's actually kind of interesting. I'm going to have to check this again. Um, just down the street from, from the, from the Institute is, it's the famous department store, Fortman, Fortman masons.
which I think is established in 17 something. It's a great department store. It's not usually, it's not where I usually do my shopping, but it's just a beautiful department store where, you know, Mozart would have and Haydn might have, you know, called and did their Christmas shopping. But anyhow, just random, random, random thought.
So back, so string theory was just one slide, right? I mean, I'm not, in some sense, I'm not, I'm not a string theorist in the sense, you know, I don't go quantize strings, you know, the kind of stuff that I'm more interested is like, I didn't grow up writing conformal field theories and do do all that stuff. It's just that it's for me, it's an input so I can play with a little more problems in geometry. Yeah. So string theory is this theory of
Space-time that unifies quantum gravity blah blah blah and then is it works in in ten dimensions and we've got to get down to Four dimensions, so we're missing six dimensions. So that's what I want to say and this this this this amazing paper in 1985 by Candela's Horowitz Strominger and Witten They were thinking about what are the properties of the six six extra dimensions?
What is interesting is that by imposing supersymmetry, and this is why supersymmetry is so interesting to me, by imposing supersymmetry and other anomaly cancellation, not too stringent conditions, they hit on the condition
that this six extra dimensions has to be richy flat richy flat is you can understand because it's vacuum style solutions is you know you want the vacuum string solution and then the condition which you've never seen before which just happens to be this caler condition they didn't know about this no physicists until 1985 would know what a caler manifold was
and it's a complex it's complex and it's complex dimension three remember again i said complex dimension three means real dimension six right that's ten minus three is six ten minus four is six and six needs to be complexified into three and again this is just an amazing fact that in 1985 strominger was a physicist um was visiting yow at the institute of advanced study in princeton
And so he went to Yao and said, can you tell me what this strange condition, this, this technical condition I got? And Yao says, wow, you know, I just got the Fields Medal for this. I think I may know a few things. I was just amazed. It was again, it was a complete confluence of ideas. That's totally random. Um, and, and the rest is history. So in fact, these four guys named this Richie Flatt, Kayla Manifold, Klabi Yao.
So it wasn't the mathematicians who did it. This word Calabi-Yau came from physicists, so from string theorists, which now, you know, of course, Calabi-Yau is now one of the central pieces. And so this is, so Philip Candelas was my mentor at Oxford.
and he when i when i was a junior fellow there and he tells me this story he's a very lively guy he tells me about how this whole story came about and it's very interesting and um but so so so he and the uh these these four guys came up with the word club yeah
So so all of a sudden we now have a have a name for this boundary case in complex geometry. This this this bounding case is now known as a club. So remember we had names before right. This was the final variety. This was varieties of general type and this bounding case is now called clubby. Uh huh.
So what we're seeing with the Taurus here is a Colabia 1. Exactly. So exactly, exactly. In fact, the Taurus is the only Colabia 1. So it's the only one that's richly flat. I mean, by this classification, it's the only one that's topologically possible.
So that's kind of interesting, right? And then this is just a comment. I like this title because I think your series is called TOE. This is a TOE on TOE. Love it. I just want to emphasize this is a nice confluence of ideas with mathematical physics. When we string theory really, what it really is, is this brainchild of interpreting
problems between interpreting and interpolating between problems in mathematics and physics. I see, for example, you know, we now, you know, GR should be phrased in differential geometry. The standard model gauge theory should be phrased in terms of
Algebraic geometry and representation theory of finite groups. And you know, condensed matter physics of topological insulators should be phrased in terms of algebraic topology. This idea, you know, I think, I think the greatest achievement of the 20th century physics is, to me, and I think something you would appreciate, since you like tables, is that here's a dictionary of a list of things
and then here's what they are in mathematics and then you know you can talk to mathematicians in this language and you can talk to physicists in language but they're actually the really same same same thing you know what's what's a fermion you know it's a spin representation of the lorenz group you know that i like that because it gives a precise definition of what we are seeing around then you have something you can purely play with in this platonic world and string theory is really just a brainchild of this translation this tradition of
What's on the left and what's on the right? And let's see what we can do. And sometimes you make progress on the left, you give insight and stuff on the right, and sometimes you make progress on the right and you give insight on the left. Why is it that you call the standard model algebraic geometry? Because bundles and connections are part of differential geometry, no? Oh, yeah, that's true. Well, I think that's, yeah, I mean, they're interlinked. And I think algebraic maybe I think maybe it's because of Atiyah and Hitchin.
Of course, they are fluid in both. They go either way. Algebraic in the sense that you can often work with bundles and connections without actually doing the integral in differential geometry. I think that's the part I want to emphasize.
You can understand bundles purely as algebraic objects without ever doing an integral. Like here, for example, this integral is obviously something you would do in differential geometry.
But this integral, the fact that it comes to, to, to be an integer was explained through the theory of churn classes to be, you know, to be, you know, this integral is a pairing between the churn class, between homology and cohomology, which is a purely algebraic thing. You know, we all try to avoid doing integrals because integrals are horrible because it's hard to do. And in, in this language, it really just becomes, um, polynomial manipulation and it becomes much simpler. Okay.
I like doing this diagram. If you look at the time lag between the mathematical idea and the physical realization of that idea, there really is a confluence.
It's getting closer. I mean, these things going up and down. I mean, I'm just saying in the past, if you take the last 200 years, last hundred years or so of the groundbreaking ideas in physics, there is this interesting, right? It gets shorter and shorter. So obviously Einstein took ideas of Riemann and, you know, there was a six year gap.
Dirac was able to come up with the equation of electron, essentially because of Clifford Algebras. Did, historically, was he motivated by Clifford Algebras or did it just, was it later realized, hey, Dirac, what you're doing is an example of a Clifford Algebra? So I believe, I believe the story goes, in order to write down the first time derivative version of the Klein-Gordon equation, which is a second order, you know, that's the bosonic one,
he had to
But what directed was, um, he, he, he, he said he was at St. John's in Cambridge at the time. He said, I have seen this in the textbook before someone, you know, this gamma mu gamma new thing. And then he said, I need to go to the library to check this. Uh, so he really knew about this. He, uh, and unfortunately the St. John's library was closed that, that evening. So he waited until the morning until the library was open to go to Clifford's book.
Or a book about Clifford. I can't remember whether it was Clifford's book or maybe it was one of one of these books and then he opened up and he really knew and that this gamma mu gamma nu anti-computation relation really was through the through so he knew about Clifford. Cool. It's kind of interesting. Yeah.
Just like Einstein knew about Riemann's work on curvature. But whether you say Dirac was really inspired by Clifford, well, he certainly did a funky factorization and then he knew how to justify it immediately by looking at the right source. And then similarly, you know, Yang-Mills theory depended on this Cybert's book on apology. And then, you know, by the time you get to Witten and Borchardt,
Really there's this this diagram for me is like what gets me excited about string theory Because string theory is a brainchild of this curve this this this orange curve And now it's getting mixed up I mean, of course, you know, you know people here hear about this this great quote that Witten says, you know string theory is a piece of 21st century mathematics that Happens to fall into 20th century and I think he means this Yes, you know that he was using supersymmetry
to prove, you know, there are theorems in Morse theory and vice versa. Richard Borchardt was using vertex algebras, which is sort of foundational thing, conformal field theory to prove some properties about the monster group. We're at this stage. And of course, you know, this was turn of last turn of the century. And now we're here and we have to where are we now? Are we are we crisscrossed or are we parallel? It's hard to it's hard to say.
And in a meta manner, you can even interpret this as the pair of pants in string theory with the world sheet. Yeah, cute. Very cute. Why not? Yeah, it is. But going back to what you were saying, how I got to. Oh, yeah. So just, yeah, this confluence idea, of course, you know, you know, everyone quotes these two book papers, you know, when Wigner was thinking of May 59.
Why mathematics is so effective in physics? And there's this maybe slightly less known paper, but certainly equally important paper by the great Leif Latia and then Dijkgraaf and Witten, Dijkgraaf and Hitchin, which is the other way around. Why is mathematics so, so why is physics so effective in giving ideas in mathematics? So this is a beautiful pair of essays. In this, this is like very much a, in the world of
A summary of the kind of physics. Ideas from string theory is making such beautiful advances in geometry. So this is a very beautiful pair of one given any other that needs to be, you know, sort of praised more.
And that's why you were you were mentioning earlier how I got to know, you know, Roger. So while he's through this editorials, we try to collect, you know, with my colleague, Molinka, who is a former director of the of the Charn Institute. You know, everybody's connected, right? So it just so happens that, you know, I, you know, I grew up in the West, but my after trip with with my parents after so many decades, my parents actually retired and went back to Tianjin.
uh... where uh... dunkeye university is where churn founded the what's now called the churn institute for mathematical sciences and that's an institute devoted to the dialogue between mathematics and physics in fact one-third of churn's ashes
is buried outside of the Math Institute. There's a great beautiful marble tomb. Once they're not because of any mathematical reason, it's just that he considered three parts of his home. His hometown in Zhejiang, China, and Berkeley, where he did most of his professional career,
And then Nankai University, where he retired to for the last 20 years of his life. So a third each. Yes. The number three comes up again. It's all about free. And in fact, I was going to joke. So in churn Simon's theory in three dimensions, there's this topological theory, churn Simon's theory. There's the there's a crucial factor of one third. I always joke, you know, that's that's why churn chose one third for his ashes. But that's not my complete coincidence.
But it's actually what is actually interesting is the That tomb that beautiful black marble tomb, you know for for somebody that's greatest turn It mentions nothing about you know, his he achieved on this done the other thing. It's just one page of his notebook I mean think about the poor guy where the chisel or that he have no idea what is chiseling right? The guy was chiseling this thing and it's the proof of this of this
the fact is such and of course it's a little you can you can you can look this on the internet just say the the grave of ss churn at nankai university well the whole conversation we've had is just about pattern matching without the intuitive understanding behind it so this chisel air may have had that yes that's what i do every day love it so the uh that chisel is essentially his proof why this is equal to this
you know, why this intersection product is the same as this integral. So he essentially, it's where the Gauss-Bonnet theorem is a corollary of this trick in algebraic geometry, which is his great achievement. But anyhow, so it just so, yeah, back to this coincidence. And it just so happens that my parents, after drifting all these years abroad, they retired back to Tianjin, where the China Institute is.
so that's why i became an honorary professor at nankai because i mean my my motivation was purely just so that i could see spend time to tie out with my parents but it just so happens that it happens there and i can just pay my homage to to churn just to see his grave i mean it's a great you know it's a it's it's it's a mind-blowing experience just to see the
Yeah, it's remarkable. And that he was, he was still doing, he wrote the preface to this when he was 99. These guys are unstoppable.
And you know Roger Roger Penrose sent he he sent his essay. Yeah to this one when he was what? 1992 Yeah, these guys are Anyhow, it's kind of I do you like tables, right? I love tables. So the tables are just here It's just like, you know, we're just a speculation of Western theory is going here's a list of you know, the annual conferences and
Like the series where string theory has been happening. So 1986 was the first string revolution where since then every year there's been a major string conference. I'm going to the one, the first one I'm going to for years in two weeks time. It happens to my Abu Dhabi. I guess I'm son.
And then, you know, there's a series of annual ones, the string phenol and the string math came in as late as in 2011. That's kind of interesting. So that's like, you know, 30 years after the first string conference and the various other ones. What's really interesting one is in 2017 is there's the first string data. This is what AI entered string theory.
And so it's kind of so so what i read the first paper in in in 2017 about ai sister stuff and there were there were three other groups independently mining different ai aspects and how to apply the string theory so that the reason i want to mention is was just how why was
You know with the string community even thinking about this problems problems in AI. Oh and also just to be clear briefly speaking I'm not a fan of tables per se. I'm a fan of dictionaries because they're like Rosetta stones So I'm a fan of Rosetta stones and translating between different languages. So you mentioned the siloing earlier and Mathematicians call if even physicists call them dictionaries, but technically they're the sources like a dictionary You just have a term and then you define it the translations like Rosetta stones
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Learn more at Capella.edu. Yes. Yeah. Yeah. No, absolutely. I guess that's why you like Langlands so much. Yeah. Yeah, for sure. Yeah, no. Yeah, absolutely. In some way, this whole channel is a project of a Rosetta stone between the different fields of math and physics and philosophy. Yeah. Yeah. That's fantastic. Love it. Big fan. Thank you.
okay so do you want to just I noticed it jump back to number 13 so it seems like I thought we were at 39 out of 40 no no no because I I've learned this non-linear structure it's like because you see like I've learned this this is really dangerous I've learned like the click button in PDF presentations like you click it it jumps to another one and you can have interludes so you know it's clearly an interlude and you say you jump back
To your main. So my my actual main presentation is only like, you know 30 pages, but it's good all these digressions Which is actually very typical of my personality So I gave you this big interlude about string about you know string theory clavier manifolds, right? So now we've already got to the point that clavier one-fold the one-dimensional complex clavier fund. There's only one example. That's just one of these
Right and then it turns out that in complex dimension two There are two of these There is the four dimensional torus Which is and then it's this crazy thing called the k3 Which is richie flat and keller. So you got one in complex measure one two in complex animation three you would think
In three dimensions, there's three of these things that are topologically distinct. And unfortunately, this is one of the sequences in mathematics that goes as one, two, we have absolutely no idea. And we know at least one billion. At least. So it's kind of, it goes one, two, a billion. And so starting from complex dimension three just goes crazy. It's still a conjecture of Yao that in every dimension, this number is finite.
So remember this positive curvature thing this final thing to the very third it's now it's it is a theorem that in every dimension final varieties is finite impossibility In topology that only a finite number of these that are distinct topologically It's also known that the negative curvatures is infinite in every dimension
And when it goes higher, it's like even uncountably infinite. Oh, interesting. But is this boundary case Yao conjectures in an ideal world? They're also finite, but we don't know. This is the open conjecture. Now the billion, are any of them constructed or is it just the existence? Yeah, that's it. Now that's exactly where we're getting. So it's gotten one, two and three. Three is like, you know, I'm going to list these things. Right. And then,
Algebra journal just never really bother listing on one mouth. This is not something they do. So it took on the physicists to go on the challenge. So Philip Candela's and France and then, um, Harold Scharke and Maximilian Kreutzer started just listing these. And that's why we have these billions. There is actually databases of these.
And they're presented in, you know, just like matrices like this. I won't bore you with the details of these matrices. You know, these algebraic varieties, you can define this as, you know, like intersections of polynomials. That's one way to present them. And in Kreutzer and Schakka's database, they put vertices of toric varieties. But the upshot is that, you know, there's a database of many, many gigabytes
That that really got done by the by by the certainly by the turn of the century but by year 2000 These guys were running on Pentium machines. I mean, this is an absolute feat Especially our courts and shkalka they were able to get 500 million of this is stored on a hard drive using a Pentium machine Of this car be our manifolds. Hmm And they were able to compute topological invariance of these
So that's, so I happen to have this database. I could access them and that was kind of fun. And I've been playing on and off with them for a number of years. So, and you know, a typical calculation is like, you know, you have something like, like a configuration of tensors of here is even in integers and you have some standard method in algebra geometry to compute topological invariance. And this topological invariance again,
in this dictionary means something. So, for example, H21 in some context is the number of generations of fermions in the low energy world. So that's a complete problem in this computing a topological invariant in algebraic geometry. And there are methods to do it. And in these databases, people took 10, 20 years to compile this database and you got these things in. And they're not easy. It's very complicated to compute these things.
So in twenty seventeen i was playing around with this and the reason is very why i was playing around with this was very simple is because my son was born. And i had infinite sleepless nights and i couldn't do anything right i had like you know there's the kid and then you know there's the kid there's the kid and you know and you wake you up at two.
Put him to bed and I was bottle feeding him and I had a daughter at the time so my wife's taking care of the daughter. They're passed out. And then I got this kid, I passed him out, put them into bed and I'm wide awake at this point. It's like 2 a.m. It's like I can't fall asleep anymore and I can't do real, you know, serious computation anymore because I'm just too tired.
So let's just play around with data. At least I can let the computer help me to do something. And that's what I learned. What's this thing that everybody's talking about? Well, you know, it's machine learning. Right.
So that's why I got through this. It's a very simple biological reason why I was trying to learn machine learning. So then I think I was hallucinating at some point, right? I was like, well, if you look at pictures like matrices, we're talking about 500 million of these things, right? Yes. Certainly I wasn't going through all of them. And they're being labeled by topological invariants.
How different is it if I just saw a pixelated one of these and label them by this? And all of a sudden this began to look like a problem in hand digit recognition, right? This is like, how different is this or image recognition? So, and I just literally started feeding in, I took 500, I mean, 500 million is too much, right? So I took like 5,000 of these, 10,000 of these, and then trained them to look and recognize this.
to
And it was recognizing it to great accuracy. And now, I mean, people have improved this, like loads of people, like in Affinatello, there's a group there that did some serious work on just trying to this problem. But this idea suddenly didn't seem so crazy anymore. The idea seemed completely crazy to me because I was hallucinating at 2 a.m. And it was so. But what's the upshot of this? The upshot is somehow the neural network was doing algebraic geometry like this kind of algebraic geometry.
Really sequence chasing very complicated bobacki style stuff without knowing anything about our drug geometry It somehow was just doing pattern recognition and somehow it's beating us because you know if you do this computation Seriously, it's it's it's double exponential complexity But it's just now but pattern recognition is bypassing all of that So then I became a fanatic right then I said well all of algebraic geometry is image processing
So far I have not been shot by the Algebraic Geometers because it's actually true if you really think about it.
You know, any algebraic, the point of algebraic geometry, the reason I like algebraic more than differential is because there's a very nice way to represent manifolds in this way. Manifolds in algebraic geometry. So in differential geometry, manifolds are defined in terms of Euclidean patches. Then you do, you know, transition functions, you know, which are differentiable, C infinity, you know, blah, blah, blah. But in algebraic geometry, they're just vanishing low-side polynomials.
and then once you have systems of polynomials you have a very good representation so for example here this is i'm just recording the list of polynomials the degrees of polynomials that are that are you know embedded in some space and and that really is algebraic geometry so basically any algebraic variety so that's fancy way of saying the this polynomial representation of a manifold
which is called an algebraic variety. This thing is representable as in terms of a matrix or a tensor, sometimes even an integer tensor. And then could be computation of invariance or topological invariance is the recognition problem of such tensor. But once you have a tensor, you can always pixelate it and, you know, picturize it. You know, at the end of the day is doing this because it's just image process and algebraic geometry.
Do you mean to say every problem in algebraic geometry is an image process or is an image processing problem or just problems involving invariance or image processing or even broader than that? I think it is really more broad. I think, you know, at some level, you know, I think in my view, I try to say, you know, bottom up mathematics is language processing.
And top-down mathematics is image processing. Interesting. Of course, this is, I mean, take with a caveat, but of course, at some level, there is truth in what I say. Of course, it's an extreme thing to say. But, you know, in terms of what mathematical discovery is, is that you're trying to take a pattern in mathematics. So in algebra, if you use perfect example, you can pixelate everything and you can just try to see certain images have certain properties. And so your image processing mathematics,
Where's bottom up you're building up like mathematics as a language. So it's like processing. Of course. All of this will be useless if you can't actually get human readable mathematics out of it, right? So this is the first surprise. The fact that it's even doing it at all to a certain degree of accuracy. Now we're talking about courses like now it's been improved to like 99.99% accuracy in these databases.
But that's the first level. That's the first surprise. The second surprise is that you can actually extract human understandable mathematics from it. And I think that's the next level surprise. So the memorization conjectures, this beautiful work in DeepMind that Jody Williamson's involved in, in this human guided intuition, you can actually get human mathematics out of it. And that's really quite something.
So that's maybe that's a good point to break for part two, which is an advertisement of, you know, here is like we've gone through many, many things about what mathematics is and to, you know, how it got this through doing, you know, this interaction between algebra, geometry and string theory. And then a second part would be how you can actually extrapolate and extract
mathematics, actual conjectures, things to prove from doing this kind of experimentation, which are summarized in these books. I keep on advertising my books because I get £50 for a year of, what do they call it, royalties, you know, so I don't have to sell my liver for my kids. But it's actually, it's kind of fun. It's a complete, I mean, academic publish is a joke, right? You get like, I don't know, like £100 a year.
because you don't actually make money out of it. But maybe that's a good place to break. And then for part two, how we try to formulate what the Birch test is for AI, which is sort of the Turing test plus. Because the Birch test is how to get actual meaningful human mathematics out of this kind of playing around with mathematical data.
I see two of your sentence that will be these maxims for the future will be that machine learning is the 22nd centuries math that fell into the 21st. So this machine learning assisted mathematics or that the bottom up is language processing and then the bottom the top down is image processing. Yeah, I like those two. Yeah.
Anyone who's watching, if you have questions for Yang Hui for part two, please leave them in the comments. Do you want to give just a brief overview? Oh, yeah, sure. So just so I'm going to talk about what the birch test is and what which which papers so far have have gone have come close. They've gone to the birch test. And then I'm going to talk about something more experiments. Number three. And the one that I really enjoyed doing with my collaborators, Lee Oliver, Postnikov,
which is to actually make something meaningful that's related to the Bletch-Thunwind-Dyer Conjecture just by just letting machine go crazy and finding a new pattern in elliptic curves which is fundamentally a new pattern in the prime numbers which is completely amazing. You mentioned quanta earlier so this quanta feature that featured this one considered this as one of the
The breakthroughs of of twenty twenty four great and that word murmuration which was used repeatedly throughout it was never defined but it will be in the part two. I'm looking forward to it me too we took okay thank you so much thank you this has been wonderful i could continue speaking to you for four hours both of us have to get going but. That's so much fun pleasure.
Don't go anywhere just yet. Now I have a recap of today's episode brought to you by The Economist. Just as The Economist brings clarity to complex concepts, we're doing the same with our new AI-powered episode recap. Here's a concise summary of the key insights from today's podcast. All right, let's dive in. We're talking about Kurt J. Mungle and his deep dives into all things mind-bending.
You know this guy puts in the hours like weeks prepping to grill guests like Roger Penrose on some wild topics. Yeah, it's amazing using his own background to dig in really challenging guests with his knowledge of mathematical physics pushes them beyond the usual.
Definitely and today we're focusing on his chat with mathematician yang we he they're getting into AI math Where those two worlds collide and it's fascinating because it really makes you think differently about how math works How we do math and where AI might fit into the picture?
You might think a mathematician's life is all formulas and proofs, but Yanqui, he actually started exploring AI-assisted math while dealing with sleepless nights with his newborn son. It's such a cool example of finding inspiration when you least expect it. Tired but inspired, he started messing around with machine learning in those quiet early morning hours. So let's break down this whole AI and math thing. Yanqui, he talks about three levels of math. Bottom-up, top-down, and meta. Bottom-up is like building with Legos. Very structured, rigorous proofs.
That's the foundation. But here's where things get really interesting. It has limitations. Right. And those limitations are highlighted by Gödel's incompleteness theorems. Basically, Gödel showed us that even in perfectly logical systems, there will always be true statements that can't be proven within that system. It's mind blowing. So if even our most rigorous math has these inherent limitations, it makes you think. Could AI discover truths that we as humans bound by our formal systems might miss? Could it explore uncharted territory?
That's a really deep thought, and it's really at the core of what makes this conversation revolutionary. It's not about AI just helping us with math faster. It's about AI possibly changing how we think about math altogether. So how is this all playing out? We've had computers in math for ages, from early theorem provers to AI assistants like Lean. But where are we now with AI actually doing math?
Well, AI is already making some big strides. It's tackling Olympiad-level problems and doing it well, which makes you ask, can AI really unlock the secrets of math? And that leads us to the big philosophical questions. Is AI really understanding these mathematical ideas, or is it just incredibly good at spotting patterns? It's like that famous Chinese room thought experiment.
You could follow rules to manipulate Chinese symbols without truly understanding the language. Yang Hui, he shared a story about Andrew Wiles, the guy who proved Fermat's last theorem, trying to challenge GPT-3 with some basic math problems. It highlights how early AI models, while excelling in tasks with clear rules and plenty of examples, struggled with things that needed real deep understanding. It seems like AI's strength right now is in pattern recognition.
And that ties into what Yan Kuo he calls top-down mathematics. It's where intuition and seeing connections between different parts of math are king. Like Gauss. He figured out the prime number theorem way before we had the tools to prove it. It shows how a knack for patterns can lead to big breakthroughs even before we have the rigorous structure. It's like AI is taking that intuitive leap, seeing connections that might have taken us humans years, even decades to figure out.
And it's all because AI can deal with such massive amounts of data. Which brings us back to Yang Hui. He's sleepless nights. He started thinking about Calabiao manifolds, super complex mathematical things key to string theory, as image processing problems. Wait, Calabiao manifolds? Those sound like something straight out of science fiction. They're pretty wild. Think six dimensions all curled up, nearly impossible to picture. They're vital to string theory, which tries to bring all the forces of nature together.
Now mathematicians typically use these really abstract algebraic geometry techniques for this. But Yang Wei? He had a different thought. So instead of equations and formulas, he starts thinking about pixels. Yeah. Like taking a Klabi-Yau manifold, breaking it down into a pixel grid, like you do with an image. He's taking abstract geometry and turning it into something a neural network built for image recognition can handle. That is a radical change in how we think about this.
It's like he's making something incredibly abstract, tangible, translating it for AI. Did it even work? The results blew people away. He fed these pixelated manifolds into a neural network and it predicted their topological properties really accurately. He basically showed AI could do algebraic geometry in a whole new way. So it's not just speeding up calculations. It's uncovering hidden patterns and connections that might've stayed hidden, like opening a new way of seeing math.
And that leads us to the big question. If AI can crack open complex math like this, what other secrets could it unlock? We're back.
Last time we were talking about AI, not just helping us with math, but actually coming up with new mathematical insights, which is where the Birch test comes in. It's like, can AI go from being a super calculator to actually being a math partner? Exactly. And now we'll look at how researchers like Yang Hui He are trying to answer that. Remember, the Turing test was about a machine being able to hold a conversation like a human. The Birch test is a whole other level. It's not about imitation. It's about creating completely new mathematical ideas. Think about Brian Birch back in the 60s.
He came up with this bowl conjecture about elliptic curves just from looking at patterns and numbers. So this test wants AI to do similar leaps to go through tons of data, find patterns and come up with conjectures that push math forward. Exactly. Can AI like Birch show us new mathematical landscapes? That's asking a lot. So how are we doing? Are there any signs AI might be on the right track? There have been some promising developments like in 2021 Davies and his team used AI to explore not theory,
knots like tying your shoelaces. What's that got to do with advanced math? It's more complex than you think. Knot theory is about how you can embed a loop in three-dimensional space and it actually connects to things like topology and even quantum physics. Okay, that's interesting. So how does AI come in? Well, every knot has certain mathematical properties called invariants. It's kind of like its fingerprint. Davey's team used machine learning to analyze a massive amount of these invariants. So was the AI just crunching numbers or was it doing something more?
What's amazing is the AI didn't just process the data, it actually found hidden relationships between these invariants, which led to new conjectures that mathematicians hadn't even considered before, like the AI was pointing the way to new mathematical truths. That's wild. Sounds like AI is becoming a powerful tool to spot patterns our human minds might miss. Absolutely. Another cool example is Lample and Charton's work in 2019. They trained AI on a massive data set of math formulas. And what did they find?
Well, this AI could accurately predict the next formula in a sequence, even for really complex ones. It was like the AI was learning the grammar of math and could guess what might come next. So we might not have AI writing full-blown proofs yet, but it's getting really good at understanding the structure of math and suggesting new directions.
And that brings us back to Yang Kuhi. His work with those Calabiao manifolds, analyzing them as pixelated forms, that was a huge breakthrough. Showed that AI could take on algebraic geometry problems in a totally new way. Like bridging abstract math in the world of data and algorithms. Exactly. And that bridge leads to some really mind-bending possibilities.
Yang Hui and his colleagues started exploring something they call murmuration. It's a great analogy. Think of a flock of birds moving together like one. Each bird reacts to the ones around it, and you get these complex, beautiful patterns. Well, Yang Hui
He sees a parallel between how birds navigate together in a murmuration and how AI can guide mathematicians towards new insights by sifting through tons of math data. So the AI is like the flock, exploring math and showing us where things get interesting. Yeah, and they've actually used this murmuration idea to look into a famous problem in number theory, the Birch and Swinerton-Dyer conjecture. That name sounds a bit intimidating. What's it all about? Imagine a donut shape, but in the world of numbers, these are called elliptic curves.
Maths
and a specific math function, like linking the geometry of these curves to number theory. I think these are definitely getting complex now. And it's a big deal in math. It's actually one of the Clay Mathematics Institute's millennium price problems. Solve it, you win a million bucks. Now that's some serious math street cred. So how did Yang Hu, his team, use AI for this? They trained an AI on this massive data set of elliptic curves and their functions.
The AI didn't actually solve the whole conjecture, but it found this new pattern, this correlation that mathematicians hadn't noticed before. So the AI was like a digital explorer mapping out this math territory and showing mathematicians what to look at more closely. Exactly. This discovery, while not a complete proof, gives more support to the conjecture and opens up some exciting new areas for research. It shows how AI can help with even the hardest problems in mathematics. It feels like we're on the edge of something new in math.
AI is not just a tool, it's a partner in figuring out the truth. What does all this mean for math in the future? That's a great question and it's something we'll dig into in the final part of this deep dive. We'll look at the philosophical and ethical stuff around AI in math. We'll ask if AI is really understanding the math it's working with or if it's just manipulating symbols in a really fancy way. See you there. Welcome back to our deep dive. We've been exploring how AI is changing the game in math.
from solving tough problems to finding hidden patterns in complex structures. But what does it all mean? What are the implications of all of this? We've touched on this question of understanding. Does AI really understand the math it's dealing with, or is it just a master of pattern matching? Yeah, we can get caught up in the cool stuff AI is doing, but we can't forget about those implications. If AI is going to be a real collaborator in mathematics, this whole understanding question is huge.
It goes way back to the Chinese room thought experiment. Imagine someone who doesn't speak Chinese has this rule book for moving Chinese symbols around. They can follow the rules to make grammatically correct sentences, but do they actually get the meaning?
So is AI like that, just manipulating symbols and math without grasping the deeper concepts? That's the big question, and there's no easy answer. Some people say that because AI gets meaningful results, like we've talked about, it shows some kind of understanding, even if it's different from how we understand things.
Others say AI doesn't have that intuitive grasp of math concepts that we humans have. It's a debate that's probably going to keep going as AI gets better and better at math. Makes you wonder how it's going to affect the foundations of mathematics itself. That's a key point. Traditionally, mathematical proof has been all about logic, building arguments step by step using established axioms and theorems. But AI brings something new, inductive reasoning, finding patterns and extrapolating from those patterns.
So could we see a change in how mathematicians approach proof? Could we move toward a way of doing math that's driven by data? It's possible. Some mathematicians are already using AI as a partner in the proving process. AI can help generate potential theorems or find good strategies for tackling conjectures. But others are more cautious, worried that relying too much on AI could make math less rigorous, more prone to errors. It's like with any new tool. There's good and bad.
Finding that balance is important. We need to be aware of the limitations and not rely on AI too much. Right. And as AI becomes more important in math, it's crucial to have open and honest conversations. We need to talk about what AI means, not just for math, but for everything we do. It's not just about the tech. It's about how we choose to use it.
We need to make sure AI helps humanity and the benefits are shared. That's everyone's responsibility. A responsibility that goes way beyond just mathematicians and computer scientists. We need philosophers, ethicists, social scientists, and most importantly, the public. We need all sorts of voices and perspectives to guide us as we go into this uncharted territory. This has been an amazing journey into the world of AI and math. From sleepless nights to those mind-bending manifolds, we've seen how AI is pushing the boundaries of what's possible.
And as we wrap up, we encourage you to keep thinking about these things. What does it really mean for a machine to understand math? How will AI change the way we prove things and make discoveries in math? How can we make sure we're using AI responsibly and ethically in our search for knowledge? These are tough questions, but they're worth asking. The future of mathematics is being shaped right now, and AI is a major player. Thanks for joining us on this deep dive.
We'll catch you next time, ready to explore some other fascinating corner of the universe of knowledge.
New update! Started a substack. 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?
Also, thank you to our partner, The Economist.
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"text": " Yonghui He, welcome to the podcast. I'm so excited to speak with you. You have an energetic humility and your expertise and your passion comes across whenever I watch any of your lectures. So it's an honor. It's a great pleasure and great honor to be here. In fact, I'm a great admirer of yours. You've interviewed several of my very distinguished colleagues like, you know, Roger Penrose and Edik Franco. I actually watched some of them. It's actually really nice."
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"text": " Wonderful, wonderful. Well, that's humbling to hear. So firstly, people should know that we're going to talk about or you're going to give a presentation on AI and machine learning mathematics and the relationship between them as well as the three different levels of what math is in terms of production and understanding bottom up top down and then the meta. But prior to that, what specific math and physics disciplines initially sparked your interest and how did the collaboration with Roger come about?"
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"text": " so my my you know my bread and butter was mathematical physics especially you know sort of the interface between algebra geometry and string theory so that's my background uh what i did my phd on and um so at some point uh i was editing the book uh with cn young"
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"text": " who is an absolute legend. You know, he's 102, he's still alive and he's the world's oldest, um, living Nobel laureate. You know, Penrose is a mere 93 or something. So CN Yang of the Yang-Mills theory. So it's an absolute legend. He got the Nobel prize in 1957. So at some point I got involved, uh, in editing a book with CN, CN called Topology in Physics. And, you know, with a name like that, you can just invite anybody you want and they'll say, probably say, yes,"
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"text": " And that was my initial friendship with Roger Pembroke started through working together on that editorial. I mean, I have Roger as a colleague in Oxford, then I've known him on and off for a number of years, but that's when we really started getting working together. So when Roger snickers at string theorists, what do you say? What do you do? How does that make you feel? That's totally fine. I mean, I'm not a diehard string theorist and, you know, I'm just generally interested"
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"text": " You just happened to study the mathematics that would be interesting to string theorists, though you're not one."
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"text": " I talk a public lecture in Dublin about the interactions between physics and mathematics, and I still find that string theory is still very much a field that gives the best cross-disciplinary feedback. I've been doing that for decades. It's a fun thing. I talk to my friends in pure mathematics, especially in algebraic geometry. 100% of them are convinced that string theory is correct."
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"text": " because for them it's inconceivable for a physics theory to give so much interest in mathematics. Interesting. And that's kind of a, I think that's a story that hasn't been told so much, you know, in the media. You know, if you talk to a physicist, they were like, you know, string theory doesn't predict anything, this and the other thing. But there's a big chapter of string theory, you know, to me, more than 50% of the story, backstory of string theory, is just constantly giving new ideas in mathematics."
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"text": " And, you know, historically when a physical theory does that, it's very unlikely for it to be completely wrong. Yeah. You watched the podcast with Edward Frankel and he takes the opposite view. Although he initially took the former view that, okay, string theory must be on the correct track because of the positive externalities. It's like the opposite of fossil fuels. It doesn't give you what you want for your field like physics, but it gives you what you want for other fields as a serendipitous outgrowth. But then"
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"text": " He's no longer convinced after being at a string conference. So you still feel like the pure mathematicians that you interact with see string theory as on the correct track as a physical theory, not just as a mathematical theory. Yeah. So yeah, absolutely. He does make a good point. And so, like, I think, you know, Franco and, you know, algebra geometers like Richard Thomas and various people, they they appreciate what string theory is constantly doing in terms of mathematics."
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"text": " and the challenges um you know where whether it is a you know whether it is a theory of physics based on the fact that it's giving so much mathematics i guess you know you got to be a mystic some of them some of some of them are mystics some of us are mystics and i actually i don't i don't personally have an opinion on that i just you know some days i'm like well you know it's this is such a cool mathematical structure and there's so much"
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"text": " internal consistency it's got to be there's got to be something there so it's just a gut feeling but of course you know it being a science you know you need the experimental evidence you know you need to go through the scientific process and that i have absolutely no idea it could take years and decades wouldn't you also have to weight the field like w e i g h t weight the"
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"text": " Whatever feel like the sub-discipline of string theory with how much iq power has been poured into how much rock talent has been poured into it versus others so you would imagine that if it was the big daddy field which happens to be that it should produce."
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"text": " more and more insights and it's unclear to me at least if that this much time and effort went into asymptotic safety or loop quantum gravity or what have you or causal set theory if that would produce mathematical insights of the same level of quality we don't have a comparison I mean I don't know I want to know what your thoughts are on that I think the reason for that is just that you know we follow our nose as a community"
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"text": " The contending theories like low quantum gravity and stuff,"
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"text": " You know, there are people who do it. There are communities of people who do it. And, you know, there's a reason why the top mathematicians are going to do string related stuff is because, you know, you follow the right notes. You feel like it is actually giving the kind of the right mathematics. Things like, you know, mirror symmetry, you know, or vertex algebras that's kind of giving the right ideas constantly. And it's been doing this since the very beginning. So,"
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"text": " and people do the alternative theories of everything, but so far it hasn't produced new math. You can certainly prove us wrong, but I think there's a reason why Witten is the one who gets the Fields Medal."
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"text": " Uh, because it's just somehow is at the right interface of the right ideas in geometry, number theory, representation theory, algebra, that this idea tends to produce the right, you know, the, the right, the right mathematics, whether, whether it is a theory of physics, that's still, you know, that's the next mystical level. Um, but, you know, it's, it's kind of, it's, it's, it's, it's an interesting, it's an exciting time actually. Witten didn't get the Fields Medal for string theory though."
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"text": " It was his work on the Jones polynomial and turn Simon's theory and Morse theory with supersymmetry and topological quantum field theory, but not specifically string theory. That's right. That's right. But he certainly is a champion for string theory. And, um, for him, I mean, you know, that, that idea of funny, so he was able to do, um,"
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"text": " you know the Morse theory stuff he was able to get because of his work at supersymmetry he was able to realize this was a supersymmetric index theorem that generated this idea and that that's really a super symmetry really is a cornerstone for string theory even though there's no experimental evidence for it so i think that's one of the reasons that's guiding him towards this direction so what's cool is that"
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"text": " Just prior, the podcast that I filmed just prior to yours was Peter White, as you know, is a critic of string theory and Joseph Conlin, who is a defender of string theory and he has a book even called Why String Theory. That's right. I think it was the first time that publicly someone like Peter White, along with a defender of string theory, were just on a podcast of this length."
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"text": " Speaking about in a technical manner. What are both of their likes and dislikes of string theory and then the string community? There's there's three issues string theory as a physical theory string theory as a tool for mathematical insight and then three string theory as a sociological phenomenon of overhype and does it see itself as the only game in town is the arrogance should there be arrogance?"
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"text": " was an interesting conversation."
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"text": " slightly orthogonal to the main string theory community. I'm just happy because it's constantly giving me good problems to work on. Yes. Including what I'm about to talk about in AI. Wonderful. I'll mention a little bit about it because I got into this precisely because I had a huge database of Clavier manifolds and I wouldn't have done that without the string community. It's again one of those accidents that you know no other you know the other theoretical physicists didn't happen to have this"
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"text": " It didn't happen to be thinking about this problem. There's this proliferation of Klabial manifolds. And I'll mention that bit in my lecture later on why this is such an interesting problem, why Klabialness is interesting inherently, regardless whether you're a string theorist. And that kind of launched me in this direction of AI assisted mathematical discovery. So this is kind of really nice. And I think for me, the most exciting thing about this whole community is that, you know,"
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"text": " Science and especially theoretical science, including theoretical science has become so compartmentalized. Everyone is doing their tiny little bit of thing."
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"text": " String theory has been breaking that mold for the last decades. It's constantly going, let's take a piece of algebraic geometry, let's take a bit of number theory here, elliptic curves, let's take a bit of quantum information, entanglement, whatever, entropy, black holes. And it's the only field that I know that different expertise are talking to each other. I mean, this doesn't happen in any other field that I know of."
},
{
"end_time": 756.067,
"index": 31,
"start_time": 742.483,
"text": " Well let's hear more about what you like thinking about and what you're enthusiastic about these days. Let's get to the presentation."
},
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"end_time": 779.189,
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"start_time": 756.527,
"text": " Sure. Well, thank you very much for having me here. And I'm going to talk about work I've been thinking about stuff I've been thinking about for the last seven years, which is how AI can help us do mathematical discovery, you know, in theoretical physics and pure mathematics. I recently wrote this review for Nature."
},
{
"end_time": 793.951,
"index": 33,
"start_time": 779.616,
"text": " which is trying to summarize a lot of these ideas that I've been thinking about. And there's an earlier review that I wrote in 2021 about how machine learning can help us with understanding mathematics."
},
{
"end_time": 818.763,
"index": 34,
"start_time": 794.394,
"text": " So let me just take it away and think about, oh, by the way, please feel free to interrupt me. I know this is one of these lectures. I always like to make my lectures interactive. So please, if you have any questions, just interrupt me anytime. And I will just pretend there's a big audience out there and just make it. So firstly, you're likely going to get to this, but what's the definition of meta-mathematics? OK, great. So, Rafi, of course, you know,"
},
{
"end_time": 835.435,
"index": 35,
"start_time": 818.831,
"text": " How does one actually do mathematics? In this review I tried to divide it into three directions. These three directions are interlaced and it's very hard to pull them apart."
},
{
"end_time": 864.787,
"index": 36,
"start_time": 835.759,
"text": " but roughly you can think about you know bottom-up mathematics which is you know mathematics is a formal logical system you know definition and uh you know lemma proof and you know theorem proof and that's certainly how mathematics is presented in in you know papers and there's another one which i like to call top-down mathematics is where you know where the practitioner looks from you know above that's why i say topped out from like a bird's eye view you see different"
},
{
"end_time": 889.121,
"index": 37,
"start_time": 865.162,
"text": " Ideas and subfields of mathematics and you try to do this as a sort of an intuitive creative art You know, you've got some experience and then you're trying to see oh well Maybe I can take a little bit of peace from here and a piece from there and I'm trying to create a new idea or or maybe a method of proof or attack or derivation Yes, so these are these two so that's that's you know, so complementary directions of research"
},
{
"end_time": 912.466,
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"start_time": 889.787,
"text": " And the third one, meta, that's just because it was short of any other creative words, because there's, you know, words like meta science and meta philosophy or metaphysics. I'm just thinking about mathematics as purely as a language, you know, whether the person understands what's going on underneath."
},
{
"end_time": 941.783,
"index": 39,
"start_time": 912.705,
"text": " Okay. I don't know if you know of this experiment called the Chinese room experiment. Yeah. Okay. So in that,"
},
{
"end_time": 958.422,
"index": 40,
"start_time": 942.415,
"text": " The person in the center who doesn't actually understand Chinese but is just symbol pushing or pattern matching, I don't know if it's exactly pattern, rule following, that would be the better way of saying it. They would be an example of bottom up or meta in this. So I would say that's meta."
},
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"end_time": 982.432,
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"start_time": 959.104,
"text": " As you know, on Theories of Everything, we delve into some of the most reality-spiraling concepts from theoretical physics and consciousness to AI and emerging technologies. To stay informed, in an ever-evolving landscape, I see The Economist as a wellspring of insightful analysis and in-depth reporting on the various topics we explore here and beyond."
},
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"text": " The Economist's commitment to rigorous journalism means you get a clear picture of the world's most significant developments, whether it's in scientific innovation or the shifting tectonic plates of global politics. The Economist provides comprehensive coverage that goes beyond the headlines. What sets the Economist apart is their ability to make complex issues accessible and engaging, much like we strive to do in this podcast."
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"text": " If you're passionate about expanding your knowledge and gaining a deeper understanding of the forces that shape our world, then I highly recommend subscribing to The Economist. It's an investment into intellectual growth, one that you won't regret. As a listener of Toe, you get a special 20% off discount. Now you can enjoy The Economist and all it has to offer for less."
},
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"text": " Head over to their website www.economist.com slash totoe to get started. Thanks for tuning in and now back to our explorations of the mysteries of the universe. So I would, I would say that's meta in the sense that the person doesn't even have to be a mathematician. You're just simply taking symbols, a large language modeling for math, if you wish. Got it."
},
{
"end_time": 1080.486,
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"start_time": 1054.326,
"text": " Of course, you know, there's a bit of component rather as you know that you can see there's a little bit of component on bottom up because you are taking mathematics as you know a sequence of symbols But I will mainly call that matter if that's okay. I mean this this definitions are just you know things that I've just I'm using yes. Yes, but in any case I would talk mostly about this this bit which is this What I've been thinking mostly about"
},
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"end_time": 1092.688,
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"start_time": 1080.828,
"text": " One thing i just made just just to set the scene you know 20th century of course you know computers have been playing an increasingly important role in mathematical discovery."
},
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"end_time": 1122.039,
"index": 47,
"start_time": 1093.046,
"text": " and of course you know it speeds up computation all that stuff goes without saying but something that's perhaps not so emphasized and appreciated is the fact that there there are actually fundamental and major results in in mathematics that could no longer have been done without the help of the computer and so this you know there's famous examples even back in 1976 this is the famous apal haken cock proof of the four-color theorem"
},
{
"end_time": 1150.998,
"index": 48,
"start_time": 1122.329,
"text": " you know that every map you only takes four every map in a plane only takes four colors to completely color it with no neighbors it was a problem this is a problem that was posed I think probably by by Euler right and this was finally settled by reducing this whole topology problem to thousands of cases and then they ran it through a computer and checked it case by case so and then other major things like you know the Kepler conjecture which is you know that that stacking"
},
{
"end_time": 1180.964,
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"start_time": 1151.357,
"text": " and"
},
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"end_time": 1210.503,
"index": 50,
"start_time": 1181.271,
"text": " Yes. Wasn't there a recent breakthrough in the generalized Kepler conjecture? Absolutely. So this is what Marina Wiozowska got the Fields Medal for. So the Kepler conjecture is in three dimensions, our world. Wiozowska showed in dimensions 8, 16, and 24 with the best possible patina. And she gave a beautiful proof of that fact. And to my knowledge, I don't think she actually used the computer. There's some optimization"
},
{
"end_time": 1230.503,
"index": 51,
"start_time": 1210.947,
"text": " Actually, what I'm referring to is that there are some researchers who generalized this for any n, not just 8, not just 24, who used methods in graph theory of selecting edges to maximize packing density to solve a sphere packing problem probabilistically for any n, though I don't believe they used machine learning."
},
{
"end_time": 1249.053,
"index": 52,
"start_time": 1232.125,
"text": " The proof is, you know, it took 200 years"
},
{
"end_time": 1272.858,
"index": 53,
"start_time": 1249.394,
"text": " The final definitive volume was by Dittgenstein 2008. What's really interesting, the lore in the finite group theory community is that nobody's actually read the entire proof. It's just not possible. It takes longer for people to actually read the entire proof than a lifetime. This is kind of interesting that we have reached the cusp"
},
{
"end_time": 1300.299,
"index": 54,
"start_time": 1273.183,
"text": " In mathematical research, where mathematics, you know, the computers are not just becoming, you know, computational tools, but it's increasingly becoming an integral part of of who we are. So this is just just set the scene. So we're very much in this, you know, we're now in the early stages of the 21st century. And this is increasingly the case where we have this where computers can help us or I can help us in these three different directions. Great."
},
{
"end_time": 1321.561,
"index": 55,
"start_time": 1300.828,
"text": " So let me just begin with this bottom up and you're sort of to to summarize this. This is probably the oldest attempt in in in where computers can can help us. So so this is where I'm going to define bottom up, which is I guess it goes back to"
},
{
"end_time": 1351.817,
"index": 56,
"start_time": 1321.852,
"text": " The modern version of this is this classic paper, the classic book of Russell Whitehead on the Principia Mathematica, which is 1910s, where they try to axiomize, axiomatize mathematics, you know, from the very beginning. You know, it took like 300 pages for them to prove that one plus one is good at two, famously. Nobody has read this. So this is this is one of these impenetrable books. But I mean, this but this tradition goes back to, you know, Leibniz or to Euclid even, you know, that the idea that mathematics should be axiomatized."
},
{
"end_time": 1378.439,
"index": 57,
"start_time": 1352.892,
"text": " Of course, this program took only about 20 years before he was completely killed, in some sense, because of Gödel and Churchill's incompleteness theorems. This very idea of trying to axiomatize mathematics by constructing layer by layer is proven to be logically impossible within every order of logic."
},
{
"end_time": 1387.637,
"index": 58,
"start_time": 1378.933,
"text": " I'd like to quote my very distinguished colleague, Professor Minyong Kim. He says the practice of mathematician hardly ever worries about Gödel."
},
{
"end_time": 1415.299,
"index": 59,
"start_time": 1388.336,
"text": " Because if you have to worry about whether your axioms are valid to your day-to-day, if an algebraic geometry has to worry about this, then you're sunk. You get depressed about everything you do. So the two bots kind of cancel out. But the reason I mention this is that because of the fact that these two bots cancel each other out, these two negatives cancel each other out, this idea of using computers to check"
},
{
"end_time": 1426.92,
"index": 60,
"start_time": 1415.708,
"text": " I'm"
},
{
"end_time": 1446.647,
"index": 61,
"start_time": 1427.227,
"text": " Even back in 1956, Noah Simon and Shaw devised this logical theory machine. I have no idea how they did it because this is really very, very, very primitive computers. And they were actually able to prove some certain theorems of Principia by building this bottom up, you know, take these axioms and use the computer to prove."
},
{
"end_time": 1472.79,
"index": 62,
"start_time": 1447.432,
"text": " This is becoming an entire field of itself with this very distinguished history. And just to mention that this 1956 is actually a very interesting year because it's the same year, 56-57, that the first neural networks emerged from the basement of Penn and MIT. And that's really interesting, right? So people in the 50s were really thinking about the beginnings of AI,"
},
{
"end_time": 1492.381,
"index": 63,
"start_time": 1473.012,
"text": " you know, because neural networks is what we now call, you know, you know, goes under the rubric of AI. And at the same time, they were really thinking about computers to prove theorems and mathematics. So it's 56 was a kind of a magical year. And, you know, this this neural network really was a neural network in the sense that, you know, they put"
},
{
"end_time": 1520.828,
"index": 64,
"start_time": 1492.892,
"text": " Cadmium sulfide cells in a in a in a basement. It's a it's a wall size of photoreceptors and they were Using, you know flashlights to try to stimulate neurons literally to try to simulate computation. Uh-huh. That's part of impressive Impressive thing and then this this thing really developed right and then and now, you know You know a half a century later We have very advanced and very very sophisticated"
},
{
"end_time": 1547.858,
"index": 65,
"start_time": 1521.203,
"text": " computer-aided proof, automated theorem provers, things like the Coq system, the Lean system, and they were able to create, so Coq was what was used in this four-color, the full verification of the four-color, the proof of the four-color theorem was through the Coq system. And, you know, then there's the Phy-Thomson theorem, which got Thompson the Fields Medal, again, was, they got the proof through this system."
},
{
"end_time": 1576.459,
"index": 66,
"start_time": 1548.643,
"text": " A Lean is very good. I do a little bit of Lean, but also Lean, the true champion of Lean is Kevin Buzzard at Imperial, 30 minutes down the road from here, from this spot. And he's been very much a champion for what he calls the Xena project, and using Lean to formulate, to formalize all of mathematics. That's the dream. What Lean has done now,"
},
{
"end_time": 1603.933,
"index": 67,
"start_time": 1576.869,
"text": " is that it has, um, uh, Kevin tells me that all of the undergraduate level mathematics at Imperial, which is a user, it's a non-trivial set of mathematics, but still a very, very tiny bit of actual mathematics and they can check it and everything that we've been taught so far at undergraduate level is good and self-consistent. So nobody needs to cry about that one. Wonderful. And so that's all good. And then more recent breakthroughs is the beautiful work of, um, you know,"
},
{
"end_time": 1632.637,
"index": 68,
"start_time": 1604.258,
"text": " So three Fields Medalists here, also two Fields Medalists, Gowers Green, Manners and Tao, and when they prove this conjecture, which I don't know the details of, but they were actually using Lean to prove, to help prove in this. And I think Terry Tao in this public lecture, which he gave recently in 2024 in Oxford, he calls this whole idea of AI co-pilot, which I very much like this word."
},
{
"end_time": 1655.725,
"index": 69,
"start_time": 1633.234,
"text": " I was with Tao in August in Barcelona. We were at this conference and he's very much into this very well. And of course, you know, Tao, Terry Tao for us is, you know, is a godlike figure. So the fact that he's championing this idea of AI co-pilots for mathematics is very, very encouraging for all of us."
},
{
"end_time": 1675.367,
"index": 70,
"start_time": 1656.254,
"text": " Yes, and for people who are unfamiliar with Terry Tao, but are familiar with Ed Whitten, Terry Tao is considered the Ed Whitten of math and Ed Whitten is considered the Terry Tao of physics. Yeah, I've never heard that expression. That's kind of interesting. At Barcelona, when Terry was being introduced,"
},
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"end_time": 1687.329,
"index": 71,
"start_time": 1676.015,
"text": " I'm"
},
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"start_time": 1688.592,
"text": " Hola, Miami! When's the last time you've been in Burlington? We've updated, organized, and added fresh fashion. See for yourself Friday, November 14th to Sunday, November 16th at our Big Deal event. You can enter for a chance to win free wawa gas for a year, plus more surprises in your Burlington. Miami, that means so many ways and days to save. Burlington. Deals. Brands. Wow! No purchase necessary. Visit BigDealEvent.com for more details."
},
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"end_time": 1744.019,
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"start_time": 1718.268,
"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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"end_time": 1764.633,
"index": 74,
"start_time": 1747.039,
"text": " the gauss of mathematics or the Mozart. But I think a more appropriate thing to describe him is to describe him as the Leonardo da Vinci of mathematics, because he has such a broad impact on all fields mathematics. And that's very rare thing."
},
{
"end_time": 1788.404,
"index": 75,
"start_time": 1765.674,
"text": " Yeah, I remember he said something like topology is my weakest field and by weakest field to him, it means I can only write one or two graduate textbooks off of the top of my head on the subject of topology. Yeah, exactly. Exactly. I guess he's very he's his intuitions are more analytic. He's he's very much in that, you know, world of, you know, analytic number three functional analysis. He's not very pictorial, surprisingly."
},
{
"end_time": 1804.36,
"index": 76,
"start_time": 1788.882,
"text": " I like Roger Penrose has to do everything has to be picked has to be in terms of pictures but Terry is a is a symbol symbolic matcher you know we can just look at equations extremely long complicated equations and just see which pieces should go together that's kind of very interesting."
},
{
"end_time": 1833.626,
"index": 77,
"start_time": 1805.043,
"text": " Speaking of Eva Miranda, you and I, we have several lines of connection. Eva's coming on the podcast in a week or two to talk about geometric quantization. Eva is super fun. She's filled with energy. She's a good friend of mine. In this academic world of math and physics, I think we're at most one degree of separation from anyone else. It's a very small community, relatively small community."
},
{
"end_time": 1861.34,
"index": 78,
"start_time": 1834.019,
"text": " This back to this thing about, of course, you know, one could one could get over optimistic. I was told by my friends in DeepMind that that Shagady, who I think he's one of the one of the on the on this AI math team, he says, you know, he was instructing that computers beat humans in chess in the 90s, beat humans go at 2018. So you should beat humans in beating proving theorems in 2030. I have no idea how he extrapolated how he extrapolated these points. I only three data points."
},
{
"end_time": 1877.688,
"index": 79,
"start_time": 1861.34,
"text": " But you know but you know but deep mind has seen has a product to sell so it's very good for them to be over optimistic but i wouldn't be surprised that that this number you know you well i'm not sure to beat humans but it might give ideas that humans have not thought about before so that's that's possible."
},
{
"end_time": 1904.889,
"index": 80,
"start_time": 1878.507,
"text": " Just moving on so that that's the bottom up and I said this is very much a a blossoming or not blossoming it's very much a long distinguished field of automated theorem computing of theorem provers and verifications of formalization mathematics which which tau calls the AI copilot. Just to mention a bit with your question a bit earlier about meta mathematics. So this is just"
},
{
"end_time": 1927.005,
"index": 81,
"start_time": 1905.384,
"text": " Kind of I like your analogy. This is like the Chinese the Chinese room, right? Can you do mathematics without actually understanding anything? You know personally I'm a little biased because having interacted with so many undergraduate students before I moved to the London Institute so I don't have to teach anymore Teach undergraduates. I've noticed, you know, I I"
},
{
"end_time": 1957.056,
"index": 82,
"start_time": 1927.619,
"text": " Maybe one can say the vast majority of undergraduates are just pattern matching, whether there's any understanding. I think this is one of the reasons why chat GPT does things so well. It's not just because LLMs are great, large language models are great. It's more that most things that humans do are so without comprehension anyway. So that's why it's kind of this pattern matching idea. And this is also true for mathematics."
},
{
"end_time": 1983.148,
"index": 83,
"start_time": 1957.91,
"text": " What's funny is that my brother's a professor of math in the University of Toronto for machine learning, but for finance. And I recall 10 years ago, he would lament to me students that came to him who wanted to be PhD students. And he would say, okay, but Kurt, some of them, they don't have an understanding. They have a pattern matching understanding. He didn't want that at the time, but now he's into machine learning, which is effectively that times 10 to the power of 10. Right."
},
{
"end_time": 2012.807,
"index": 84,
"start_time": 1983.797,
"text": " The vast majority of us doing most of the stuff is just pattern matching. So that's why and this is even this is true even for mathematics."
},
{
"end_time": 2041.203,
"index": 85,
"start_time": 2013.473,
"text": " So here I just want to mention something which is a fun project that I did with my friends Vishnu Jigala and Brent Nelson back in 2018 before LLM. So before all this LLM for science thing. And this is a very fun thing because what we did, we took the archive and we took all the titles of the archive. This is the preprint server for contemporary research in theoretical sciences."
},
{
"end_time": 2061.015,
"index": 86,
"start_time": 2041.647,
"text": " What's really interesting, this is my favorite bit, we took to benchmark the archive, we took Vixra. So Vixra is a very interesting repository"
},
{
"end_time": 2090.828,
"index": 87,
"start_time": 2061.237,
"text": " because it's of archive spelled backwards and he has all kinds of crazy stuff. I'm not saying everything on Vixra is crazy, but certainly he has everything that archive rejects because he thinks it's crazy. Things like, you know, three page proof of the Riemann hypothesis or Albert Einstein is wrong. It's got filled with that. It's interesting to study the linguistics, even at a title level, you could see that, you know, what they call the distinctions of quantum gravity versus the other things, they have the right words in Vixra."
},
{
"end_time": 2115.725,
"index": 88,
"start_time": 2090.828,
"text": " But the word order is already quite random that you know the another with the classification matrix the confusion matrix for vixra is certainly not as distinct as archive which you know so kind of interesting you get all the right buzzwords is like you know kind of thing that's what i think is a good benchmark that linguistically is not as sophisticated as you know real research articles."
},
{
"end_time": 2137.261,
"index": 89,
"start_time": 2116.442,
"text": " But this idea, so this is something much more serious is this very beautiful work of Chitoyan at all in nature, where they actually took all of material science and they did a large language model for that and they were able to actually generate new reactions in material science. So this, I think this paper in 2019, this paper by Chitoyan"
},
{
"end_time": 2163.541,
"index": 90,
"start_time": 2137.517,
"text": " is really the beginnings of l l m l l m for for scientific discovery this is quite early it is 2019 right yeah and it's remarkable how we can even say that that's quite early the field is exploding so quickly absolutely five years ago is considered quite some time ago yeah absolutely even five years ago i you know i was still very much in a lot of i've evolved in thinking a lot about this thing"
},
{
"end_time": 2175.196,
"index": 91,
"start_time": 2163.951,
"text": " I would also like to get to your personal use cases for LLMs, chat GPT, Claude, and what you see as the pros and cons between the different sorts like Gemini was just released at 2.0."
},
{
"end_time": 2198.609,
"index": 92,
"start_time": 2175.64,
"text": " And then there's a one and there's a variety. So at some point I would like to get to you personally, how you use LLMs both as a researcher and then your personal use cases. Well, you know, I can mention, I can mention a little bit. So one of the very, very first, um, things would charge you BT three came out in what? 2018, something 2019, something like that. Oh, three. You mean GPT three?"
},
{
"end_time": 2227.176,
"index": 93,
"start_time": 2198.609,
"text": " GPT-3 like the really early baby versions. Yeah, that was during just before the pandemic just before pandemic. So that was just like so I got into this AI for for math through this club y'all manifold wasn't going to mention bit bit bit later. And then this GPT came out when I was just thinking about this large language model. So this is a great conversation. So I was typing and"
},
{
"end_time": 2252.961,
"index": 94,
"start_time": 2227.176,
"text": " problems in calculus, freshman calculus. And you will solve that fairly well. I mean, it's really quite impressive what you can do. So, you know, it's fairly sophisticated because, you know, things like, you know, I was typing questions like, you know, take vector fields, blah, blah, blah, on a sphere, you know, find me the grad or the curve. I mean, it's like"
},
{
"end_time": 2269.718,
"index": 95,
"start_time": 2253.575,
"text": " you know first second year stuff and you have to do a lot of computation and it was actually doing this kind of thing correctly you know partially because there's just so many example sheets of this type out there on the internet and so he's kind of learned all of that so i was getting very excited and i was um"
},
{
"end_time": 2295.606,
"index": 96,
"start_time": 2270.845,
"text": " Trying to sell this to everybody at lunch. I was having lunch with my usual team of colleagues in Oxford over this. And of course, lo and behold, who was at lunch was the great Andrew Wiles. So I felt like I was being a peddler for GPT, LLM for mathematics to perhaps the greatest living legend."
},
{
"end_time": 2318.933,
"index": 97,
"start_time": 2295.964,
"text": " in mathematics. Andrew is super nice and he's a lovely guy and he just instantly asks me, he says, how about you try something much simpler? Two problems he tried. The first one was tell me the rank of a certain elliptic curve and he just typed it down a certain elliptic curve or rational points of a very simple elliptic curve, which is his baby."
},
{
"end_time": 2343.78,
"index": 98,
"start_time": 2319.462,
"text": " I typed it when it was complete got completely wrong. Yes It was just even you started very quickly started things saying things like, you know, five over seven is an integer Partially because it's very this is a very hard thing to do You can't you can't really guess integer points by choosing I like in calculus where there's there's a routine of what you need to do, right? And then and then very quickly we we we converge on even simpler problem"
},
{
"end_time": 2368.848,
"index": 99,
"start_time": 2344.36,
"text": " how about find the 17th digit in the in the in the decimal expansion of 22 divided by 29 like whatever and that it's completely random because you can't train you actually have to do long division this is you know this is you know primary school level stuff and yet GP teacher simply cannot do and it's inconceivable that he could do it because no language model could possibly do this"
},
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"index": 100,
"start_time": 2369.462,
"text": " But GPT now 0010201 is already clever enough. When you ask him a question like this, linguistically, he knows to go to war from alpha. And then it's okay. Then he's actually doing them. But so something so basic like this, you just can't train the language model to do you get, you know, one in 10, right? And it's just a randomly distributed thing. Yes."
},
{
"end_time": 2417.329,
"index": 101,
"start_time": 2392.875,
"text": " where sophisticated things, there are seemingly sophisticated things like solving differential equations or doing very complicated integrals. It can do because there's somewhat of a routine and there are enough samples out there. So that's my user case, two user cases. That's also not terribly different than the way that you and I or the average person or people in general think."
},
{
"end_time": 2441.732,
"index": 102,
"start_time": 2417.585,
"text": " So for instance, we're speaking right now in terms of conversation. And then if we ask each other a math question, we move to a math part of our brain. We recognize this is a math question. So there's some modularity in terms of how we think. It's not like we're going to solve long division using Shakespeare, even if we're in a Shakespeare class and someone just jumps in and then ask that question. We're like, OK, that's a difference. That's of a different sort of mechanism. Yeah, that's a good analogy. Yeah, yeah."
},
{
"end_time": 2467.756,
"index": 103,
"start_time": 2442.176,
"text": " When you first encountered chat GPT or something more sophisticated that could answer even larger mathematical problems, did you get a sense of awe or terror initially? So I'll give you an example. There was this meeting with some other math friends of mine and I was showing them chat GPT when it first came out. And then one of the friends said, explain, can you get it to explain some inequality or prove some inequality?"
},
{
"end_time": 2497.312,
"index": 104,
"start_time": 2468.507,
"text": " And then it did and then explained it step by step that he then everyone just had their hand over their mouth like, are you serious? Can you do this? And then they're like, and one said, one friend said, this is like speaking to God. And then another friend said, had the thought like, what am I even doing? What's the point of even working if this can just do my job for me? So did you ever get that sense? Like, yes, we're excited about the future and it as an assistant, but did you ever feel any sense of dread?"
},
{
"end_time": 2524.224,
"index": 105,
"start_time": 2498.439,
"text": " I'm by nature a very optimistic person. So I think it was just all an excitement. I don't think I've ever felt that I was threatened or the community is being threatened. I could totally be wrong. But so far it just seems like this is such an awesome thing because it'll save me so much time looking up references and stuff like this. Yeah, I was happy."
},
{
"end_time": 2552.21,
"index": 106,
"start_time": 2524.582,
"text": " I was just like, wow, this is kind of cool. I mean, I guess if I were an educator, I might get a bit of a dread because there's like, you know, you know, undergraduate degrees, you know, if you do an undergraduate degree, it's just basically one chat GPT being fed to another. You know, a lot of my colleagues started setting questions in exams with chat GPT with fully locked act out equation. I mean, this is becoming the standard thing to do. I guess even if you're an educator, you would probably worry."
},
{
"end_time": 2568.933,
"index": 107,
"start_time": 2552.944,
"text": " Wonderful. All right, let's move forward. Yeah, sure."
},
{
"end_time": 2587.295,
"index": 108,
"start_time": 2569.377,
"text": " I'm so twenty twenty two was a great yeah i'm surprised this was like every single newspaper i don't know why i'm at least i was told after some obscuring outlet i can't remember some some expert friends in the in the community told me that the chat gpt has passed the turing test."
},
{
"end_time": 2601.92,
"index": 109,
"start_time": 2587.585,
"text": " This is a big deal, but I'm I don't know why it hasn't been. I was hoping to see some BBC and every year, every major newsletter, but it didn't didn't catch on. But anyhow, I believe that in 2022, Chachi BTS passed the Turing test."
},
{
"end_time": 2620.316,
"index": 110,
"start_time": 2602.978,
"text": " And then you know where in the last and the last two two years you know this is obviously where we can you know this is a huge development now for LL large language models for mathematics and you know every every major company open AI matter AI epoch AI."
},
{
"end_time": 2643.148,
"index": 111,
"start_time": 2620.776,
"text": " You know everything and they've been doing a tremendous, you know, working in trying to get LLM format. Basically, you know, take the archive, which is a great repository for mathematics and theoretical physics, pure mathematics and theoretical physics, and then just learn that and try to generate to see it to how much and I'm because this is very much work in progress."
},
{
"end_time": 2673.131,
"index": 112,
"start_time": 2643.763,
"text": " And of course, you know, AlphaGeo, AlphaGeo 2, AlphaProof, this is all the DeepMind's success. It's kind of interesting within a year, you know, you've gone from 53% on Olympia level to 84%, which is part, you know, this is scary, right? Every, this is scary in the sense, like, impressively awesome that, you know, they could do so quickly. So basically in 2022, an AI is approximately equal to the 12-year-old Terence Tao."
},
{
"end_time": 2703.541,
"index": 113,
"start_time": 2673.882,
"text": " In the sense that it does it could do a silver medal, but of course, this is a very specialized You know the the alpha alpha geo 2 was really just homing in on Euclidean geometry problems, which which to be fair extremely difficult, right? If you don't know how to add the right line or the right angle You have no idea how to attack this problem, but it's kind of learned how to do this. Yeah So it's kind of nice. So 20, you know, this is all within you know, a couple of years and There's this there's this"
},
{
"end_time": 2730.759,
"index": 114,
"start_time": 2703.848,
"text": " Very nice benchmark called Frontier Math that EPOC AI has put out. I think there was a white paper and they got Gowers and Tao, you know, the usual suspects, just a benchmark. Okay, fine. So we can do 84% on Math Olympiad, which is sort of high school level. What about truly advanced research problems? So to my knowledge, as of the beginning of this month, it was only doing 2%."
},
{
"end_time": 2749.77,
"index": 115,
"start_time": 2731.732,
"text": " Alright so that's okay fine so it's not doing that great but the beginning of this week you learn that OpenAI 03 is doing 25% so we've gone 20% up we've got a fifth up within four weeks."
},
{
"end_time": 2772.056,
"index": 116,
"start_time": 2750.094,
"text": " I love this right because it's exciting it's very rare to be I remember back in the day when I was a PhD student doing ADS CFT related algebra geometry"
},
{
"end_time": 2792.363,
"index": 117,
"start_time": 2772.688,
"text": " I remember that kind of excitement, the buzz in the string community. And you know, people are saying, you know, there was a paper every couple of days on the next, you know, that kind of excitement. And I haven't felt that kind of excitement for, you know, for a very long time just because. Wow."
},
{
"end_time": 2816.852,
"index": 118,
"start_time": 2792.739,
"text": " and then this this is like that right you know every every week this is new benchmark and new new you know breakthrough so that's why i'm fine this field of ai system mathematics will be really really exciting can you explain perhaps it's just too small on my screen because i have to look over here but can you explain the graph to the left with terence tau oh gosh i'm not sure i can because i'm sure"
},
{
"end_time": 2830.247,
"index": 119,
"start_time": 2817.21,
"text": " I'm sure I can read this graph in detail. I think it's the year... What is it trying to convey? So it's the ranking of... No, this is just Terence Tal's individual performances over"
},
{
"end_time": 2857.073,
"index": 120,
"start_time": 2830.93,
"text": " Different years I over over different points. So he's retaking the test every year No, no his take you see his he's taken the three times ages 10 11 and 12 and when it was When it was 10, he got the bronze medal and then he got the silver medal then he got the gold medal within three years Okay, and age of 12 or something, but I can't I think what are those bars though? I think the bar is a good question"
},
{
"end_time": 2871.015,
"index": 121,
"start_time": 2857.841,
"text": " I think maybe it's to the the different question you're giving 60 questions and what do you what do you would take to get the gold medal I think or what it would take or to get the silver medal I think."
},
{
"end_time": 2900.538,
"index": 122,
"start_time": 2871.425,
"text": " Okay, so it wasn't a foolish question of mine. It's actually. No, no, no, it's a good question. I have no recollection or maybe I never even looked at it. Somebody told me about this graph at some point. I forgot what it is. Okay, because it looks to me like Terrence Tao is retaking the same test and then this is just showing his score across time and he's only getting better. But that can't be it. Why would he retake the test? He's a professor. No, I think it goes to 66. It must be like"
},
{
"end_time": 2930.913,
"index": 123,
"start_time": 2901.22,
"text": " This is an open source graph. Oh, I thought you were going to say this is an open problem in the field. What does this graph mean? No, no, no. It's an open source. This graph is just you can take it from the Math Olympiad database. Got it. Which I shamelessly. See, again, perfect, right? I've just done something that I have absolutely no understanding of presented to you like a language model. And I just copy and paste it because it's got a nice, cute picture of Terry's style when he was a little boy."
},
{
"end_time": 2950.452,
"index": 124,
"start_time": 2931.374,
"text": " So finally, I'll go back to the stuff that I that I be really thinking about, which is sort of top down mathematics, right? So and then this is kind of interesting. So the way we we do research, you know, practitioners is completely opposite to the way we write papers. I think that's important to point that out."
},
{
"end_time": 2979.445,
"index": 125,
"start_time": 2950.862,
"text": " We mock about all the time. We look at my board, right? It's just filled with all kinds of stuff. And most of it is probably just wrong. And then once we got a perfectly good story, we write it backwards. And I think writing math papers backwards, and math generally defines math and theoretical physics papers backwards. Well, theoretical physics is a bit better. At least sometimes you write the process. But in pure math papers, everything is written in the style of Bobacki."
},
{
"end_time": 2998.626,
"index": 126,
"start_time": 2980.128,
"text": " This very dry definition proof and which is completely not how it's actually done at all this is why you know i know that you know the great vladimir arnold says you know a boba key is criminal. He actually use this word that the boba the criminal boba key ization of mathematics."
},
{
"end_time": 3019.753,
"index": 127,
"start_time": 2999.019,
"text": " Because it leaves out all human intuition experience. It just becomes this dry machine like presentation, which is exactly how things should not be done. But bovaki is extremely important because that's exactly the language that's most amenable to computers. So it's one way or another."
},
{
"end_time": 3044.411,
"index": 128,
"start_time": 3020.077,
"text": " But we know human practitioners certainly don't do this kind of stuff, right? We mock about, you know, if we have to, and sometimes even rigorous sacrifice, right? If we have to wait for proper analysis in the 19th century to come about before Newton invented calculus, we won't even know how to compute the area of an ellipse."
},
{
"end_time": 3065.913,
"index": 129,
"start_time": 3044.855,
"text": " Because you have to wait and formalize all of that, you don't just go all backwards. So kind of the historical progression of mathematics is exactly opposite to the to the way that is represented presented and it's fine. But the way it's presented is better. It's much more amenable to approve copilot system like lean than what we actually do."
},
{
"end_time": 3083.558,
"index": 130,
"start_time": 3066.357,
"text": " even science in general is like that where we say it's the scientific method where you first come up with a hypothesis and then you just you test it against the world gather data and so on but the way that scientists not just in math and physics but biologists and chemists and so on work are based on hunches and"
},
{
"end_time": 3113.473,
"index": 131,
"start_time": 3083.985,
"text": " creative intuitions and conversations with colleagues and several dead ends. And then afterward you formalize it into a paper in terms of step by step, but it was highly nonlinear. You don't even have a recollection most of the time of how it came about. That's right. And I think one of the reasons I got so excited about all this, this AI format is, is this direction because this hazy idea of intuition or experience. And this is something that a neural network is actually a very, very good at. Wonderful."
},
{
"end_time": 3122.21,
"index": 132,
"start_time": 3114.138,
"text": " 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 joy? I don't."
},
{
"end_time": 3144.155,
"index": 133,
"start_time": 3122.705,
"text": " Then Hulu has National Lampoon's Christmas Vacation. We're all in for a very big Christmas treat. All of these and more streaming this holiday season. And right now, save big with our special Black Friday offer. Bundle Disney Plus and Hulu for just $4.99 a month for one year. Savings compared to current regular monthly price. Ends 12-1. Offer for ad-supported Disney Plus Hulu bundle only. Then $12.99 a month or then current regular monthly price. 18 Plus terms apply."
},
{
"end_time": 3173.524,
"index": 134,
"start_time": 3144.565,
"text": " Close your eyes, exhale, feel your body relax, and let go of whatever you're carrying today. Well, I'm letting go of the worry that I wouldn't get my new contacts in time for this class. I got them delivered free from 1-800-CONTACTS. Oh my gosh, they're so fast! And breathe. Oh, sorry. I almost couldn't breathe when I saw the discount they gave me on my first order. Oh, sorry. Namaste. Visit 1-800-CONTACTS.COM today to save on your first order."
},
{
"end_time": 3193.507,
"index": 135,
"start_time": 3174.411,
"text": " uh you could help you so i'm going to give concrete examples um later on about you know how we give guides humans but uh but just to just to um give some classical examples you know i always i've given this i've said this joke so many times i think"
},
{
"end_time": 3221.681,
"index": 136,
"start_time": 3194.087,
"text": " So, um, what's the best work neural network of the 18th century? Well, it's clearly the brain of gals. I mean, that's a perfectly functioning, perhaps the greatest neural network of all time. And this is, I mean, I want to use this as an example because you know, what, what did gals do? Now gals plotted the number of prime numbers less than a given positive real number. She used to give us sort of continuity."
},
{
"end_time": 3248.746,
"index": 137,
"start_time": 3222.193,
"text": " And he plotted this and it's kind of a really, really, you know, jackety curve. And it's a step function. It's a step function because it jumps whenever you hit a prime. But Gauss was just able to look at this when he was 16. I said, well, this is clearly X over log X. How did he even do this experience? I mean, he had to compute this by hand. And he did. And he got some of the wrong even."
},
{
"end_time": 3276.357,
"index": 138,
"start_time": 3249.258,
"text": " uh you know primes he had tables by his time the tables of primes were up in the tens and hundreds of thousands he has to go up in the hundred thousand range i think i just look at this is x or log of x and then but this is very beautiful this is very important because he was able to raise a conjecture before the method by which this conjecture is proved namely complex analysis was even conceived of by by kochi and rima and that's a very important fact"
},
{
"end_time": 3298.882,
"index": 139,
"start_time": 3276.903,
"text": " So he just kind of felt that this was X over log X and you had to wait for 50 years before Hadamard and Tel Aviv percent approved this fact because this technique known as which we now take for granted this technique called complex numbers complex analysis wasn't invented by Cauchy wasn't invented yet. You had to wait for that to happen. So that's kind of that's how it happens like this in mathematics all the time."
},
{
"end_time": 3328.302,
"index": 140,
"start_time": 3299.343,
"text": " even major things on the of course you know this this is so it's now it's called the prime number theorem which is a cornerstone of all of mathematics right this is the first major result since euclid on the distribution of primes how did gauss say this was x over log x i don't hear that because he had a really great neural network and this happened it happens over and over again like you know the best swing die conjecture which i'm going to talk about later uh which is a is a it's one of the millennium problems"
},
{
"end_time": 3357.995,
"index": 141,
"start_time": 3328.797,
"text": " and it's still open and it's certainly one of the most important problems in mathematics of all time. And this is Birch and Swindon and Dyer in a basement, you know, in Cambridge in the 1960s. They just plotted ranks and conductors of lead curves. I'm going to define those in more detail later. And they will say, oh, well, that's kind of interesting. You know, the rank should be related to the conductor in some strange way. And that's now the BST conjecture, the Birch and Swindon and Dyer conjecture."
},
{
"end_time": 3388.131,
"index": 142,
"start_time": 3358.524,
"text": " And what they were doing was computer aided conjectures. So this is so here was Gauss, the eyeballs of Gauss in the in the 19th century. But the 20th century really have seriously computer aided conjectures. And of course, the open of the proof of this is still open in general. There've been lots of nice progress in this. And, you know, where we're going to go is very much what technique do we need to wait"
},
{
"end_time": 3409.343,
"index": 143,
"start_time": 3388.49,
"text": " to prove something like this. Now is there a reason that you chose Gauss and not Euler? Like is it just because Gauss had this example of data points and guessing a form of a function? I'm sure Gauss, I'm sure Euler who is certainly is great had conjectures maybe"
},
{
"end_time": 3437.381,
"index": 144,
"start_time": 3409.599,
"text": " That's an interesting quote. I'll mention Euler later, but I think there is not an example as striking as this one. In fact, what's interesting as a byproduct of Gauss inventing this, because it was kind of mucking around with statistics, right? This is before statistics existed as a field as well, right? This is like early 1800s. And Gauss, I think, and you can check me on this, Gauss"
},
{
"end_time": 3467.602,
"index": 145,
"start_time": 3438.012,
"text": " got the idea of statistics and the Gaussian distribution because he was thinking about this problem. So it's kind of interesting. So he's laying foundations to both analytic number theory and modern statistics in one go. He was doing regression. So he, he, he, I think he, he essentially invented, you know, regression and the curve fitting, which is our, you know, this is like one on one of modern modern society. He was trying to fit a curve."
},
{
"end_time": 3495.503,
"index": 146,
"start_time": 3467.961,
"text": " Yeah What was the curve that really fit this and you know in in the process he he got x over log x and in addition He got this idea of regression and a impressive guide. What can we say? He's a god to us all And then um, so the the the the the upshot of this is like I I love this again there's something I found on the internet and um Just to emphasize, you know that this idea of thinking of god"
},
{
"end_time": 3511.937,
"index": 147,
"start_time": 3496.032,
"text": " Yes, this idea of marking about with data in pure mathematics is a very ancient thing. Once you formulate something like this in conjecture,"
},
{
"end_time": 3533.319,
"index": 148,
"start_time": 3512.432,
"text": " You will write your paper imagine you know writing a paper you will say conjecture you know definition prime definition pie of x then conjecture pie of x evidence rather than all of the failed stuff about inventing regression and mocking about all that stuff just gets not written at all that intuitive creative process is not written down anywhere."
},
{
"end_time": 3563.131,
"index": 149,
"start_time": 3534.565,
"text": " So so here is a it's great i'm glad i'm glad i'm chatting to you about right because it's nice to have a have an audience with this right so you know if you look at like so pattern recognition so what do we do right in terms of pure mathematical data if i gave you a sequence like this you can immediately tell me what the next number is to some confidence yeah zero zero one zeroes is just you know this is just multiple three or not this one i've tried this with many audiences and you know after a few minutes of struggle"
},
{
"end_time": 3584.923,
"index": 150,
"start_time": 3563.712,
"text": " You can you can get the answer and then this turns out to be the prime characteristic function so what I've done here is to mark all the odd integers and Evens obviously you're gonna get zero so it's kind of pointless you just all just a sequence of odd integers and Then it's a one if it's a prime is zero if it's not so three five seven eight eight"
},
{
"end_time": 3612.91,
"index": 151,
"start_time": 3585.247,
"text": " And so on and so forth. No, sorry. Three, five, seven, nine, 11. And you mark all the odd ones, which are one. Okay. And you can probably after a while, you can muck about and you can, you can see where, where, where this is going. Uh, the next sequence is much harder. So I'm not, I'm going to give away. So we won't have to spend like a couple of hours staring at it. Um, so this one is, um, the, what's called the shifted Moebius function. What this is just, you take an integer."
},
{
"end_time": 3639.36,
"index": 152,
"start_time": 3613.37,
"text": " and you take the parity of the number of prime factors it has up to multiplicity starting from two. I think I didn't start from one here and then if it's one if it's a if it's and so maybe I did start one it's zero if it's an odd odd number of prime factors it's one if it's an even number of prime factors for all the sequence of integers."
},
{
"end_time": 3668.217,
"index": 153,
"start_time": 3639.838,
"text": " And I hope now I've gotten this right. So it's if I think I start with two two has So that's all no, let's see two three Yeah, so so I did start with what I'm gonna mark one for one just to start this kick off the sequence And then two is a prime number. He has only one prime factor It's an odd number three is an odd number prime factors four is two because it's two squared So it has an even number of prime factors and so on so forth"
},
{
"end_time": 3698.763,
"index": 154,
"start_time": 3669.104,
"text": " So five is prime, it has one odd number. Six is two times three, so it has two, an even number of prime factors, so on and so forth. It looks kind of harmless. What's really interesting, so this is even number. So I've been staring at this for a while, it was very, very hard to recognize a pattern. And what's really interesting is that to know the parity of the next number, if you have an algorithm that can tell me the parity of this in an efficient way,"
},
{
"end_time": 3728.268,
"index": 155,
"start_time": 3699.309,
"text": " You will have an equivalent formulation of the Riemann hypothesis. So that's actually an extremely hard sequence to predict. So if you can tell me with some confidence more than 50% what the next number is without looking up some table, then you can probably end up cracking every bank in the world. Interesting. Because this is equivalent to the Riemann hypothesis. So I'm just giving three. So trivial kind of"
},
{
"end_time": 3755.213,
"index": 156,
"start_time": 3728.592,
"text": " Okay. Um, really, really, really hard. Yes. So now you can think about a question. How, if I were to feed sequences like this into some neural network, how would a neural network do? Um, so one way to do it. So this goes a bend. So we go way back to the very beginning to the question of what is mathematics and, you know, Hardy,"
},
{
"end_time": 3785.367,
"index": 157,
"start_time": 3756.032,
"text": " in this beautiful in this beautiful apology says you know what mathematicians do is essentially we are um pattern recognizers that's that's probably the best definition of what mathematics is or is is that it's a study of patterns finding regularity in patterns and in fact you know if there's one thing that AI can do better than us is pattern detection because you know we we evolved in being able to detect patterns in three dimensions and no more"
},
{
"end_time": 3811.271,
"index": 158,
"start_time": 3787.739,
"text": " So in this sense, if you have the right representation of data, you're sure that AI can do better than that. I mean, you know, generate a lot of stuff, but filtering out what is better is a very interesting problem of itself. So let's try to do one. I mean, there are various ways to do this representation. One way you can do it is to do a problem which is"
},
{
"end_time": 3840.282,
"index": 159,
"start_time": 3811.834,
"text": " Maybe best fit for for an AI system, which is binary classification of binary vectors. So what you do is what you know sequence prediction is kind of difficult. So what one thing you can do is just take this infinite sequence and just take say a window of a hundred a thousand with fixed window size and then label it with the one immediately outside the window and then shift label shift label. So then you can generate a lot of training training data this way. Uh huh."
},
{
"end_time": 3869.804,
"index": 160,
"start_time": 3840.742,
"text": " So for this sequence, I think I've just taken here, you know, whatever the sequence is, and I might just, with a fixed window size, and with this label. So now you have a perfectly supervised, perfectly defined binary supervised machine learning problem. Then you pass it to your standard AI, you know, algorithm, that they're, you know, just, you know, out of the box ones, nothing. You don't even have to tune your particular architecture. Just take your favorite one."
},
{
"end_time": 3887.125,
"index": 161,
"start_time": 3870.418,
"text": " And then,"
},
{
"end_time": 3912.193,
"index": 162,
"start_time": 3887.705,
"text": " that you know any any neural network or whatever base classifies would do it a hundred percent accuracy as you should because you'll be really dumb if you didn't because this is just a linear transformation so even if you have a single neuron that's just doing linear transform that's good enough to do it the prime q problem i did some experiment um some oh gosh i've been like seven years ago it got 80 accuracy"
},
{
"end_time": 3935.606,
"index": 163,
"start_time": 3912.517,
"text": " And I was like, wow, that's kind of, this was a wow moment. I was like, why, why is it doing it? I have, I don't have a good answer to this. Um, why is it doing 80% accuracy to this? How is it learning? Maybe it's doing some sieve method, uh, which is kind of interesting. I somehow, um, the second number is just to Chi square, just to double test that the, uh, what's called MCC, um, which is Matthew's correlation coefficient."
},
{
"end_time": 3964.019,
"index": 164,
"start_time": 3935.981,
"text": " um these are just buzzwords in stats i i never learned stats but now i've learned i'm relearning i took cross sarah in 2017 so i can relearn all these buzzwords um it's great it's really useful and then this shifted over lambda function it's um uh sorry i think i made a i yeah i i mistakenly called this called this merbius mu function it's not i mean it's related but it's not it's the it's the shift in the over your lambda function"
},
{
"end_time": 3992.739,
"index": 165,
"start_time": 3964.326,
"text": " Sorry, one of my neurons died when I said Möbius mu, but it's Leuville lambda. You were subject to the one pixel attack. Yeah. But so this one I couldn't break 50%. Right. Point five just means it's coin toss. It's not doing any better guessing than whatever. And this chi-square is zero point zero zero. That means I'm up to statistical error. So which means I couldn't find an AI system which could break, which could do better than random guess."
},
{
"end_time": 4012.807,
"index": 166,
"start_time": 3993.217,
"text": " I'm not saying there isn't one. It would be great if there were one. It's life. If I do break it, I might actually stand a good chance."
},
{
"end_time": 4028.046,
"index": 167,
"start_time": 4013.353,
"text": " Breaking every bank in the world. All right, but I don't I haven't made it worse. Let's remain close friends. Yeah, that's right. That's right So I was very proud of this because this experiment I'm gonna mention a bit later this little lambda was suggesting I was just trying like way back when"
},
{
"end_time": 4056.305,
"index": 168,
"start_time": 4028.387,
"text": " But apparently Peter Sarnak, whom I really admire, he's one of the world's greatest number theorists currently, current number theorist. And I got to know him through this memorization thing that I'm going to talk about later. And I reminded him that I almost became his undergraduate research student. I ended up doing, I was an undergrad at Princeton where I had two paths I could follow."
},
{
"end_time": 4083.37,
"index": 169,
"start_time": 4056.63,
"text": " um, for, you know, to, to, you know, it kind of defines your union, your undergraduate thesis, right? So one was in mathematical physics was one was that's with, uh, Alexander McDowell and the other one was with, uh, you know, two, two problems. And the other one was, was actually offered by Peter Sinek on arithmetic problems. And I somehow just, I, because I was, I wanted to understand the nature of space and time."
},
{
"end_time": 4113.012,
"index": 170,
"start_time": 4083.831,
"text": " I went through the Alexander McDowell path to do mathematical physics, which led to do string theory. After 20, 30 years, I came full back to be in Peter Sondagwald again. I met him at this conference, I reminded him of this, and he was very happy. What's really interesting is that he was asking DeepMind the same question a few years ago about the deluvial lambda, whether DeepMind could do better than 50%."
},
{
"end_time": 4141.954,
"index": 171,
"start_time": 4113.422,
"text": " So I was glad that I thought along the similar lines as a great expert in number theory and somebody who could have potentially have been my supervisor. And then I would have gone into another theory instead of swing theory, which is whatever. It's how life happens. So perhaps you're going to get to this later on in the talk. But I notice here you have the word classifier and the recent buzz since 2020 or so has been with architecture, the transformer architecture in specific."
},
{
"end_time": 4165.674,
"index": 172,
"start_time": 4142.415,
"text": " So is there anything regarding mathematics, not just LLMs, that has to do with transformer architecture that's going to come up in your talk? Not specifically. I'm actually, it's interesting, I'm one of my colleagues here at the London Institute. He's Mikhail Berdsov. He's an AI, he's our Institute's AI fellow, and he's an expert on transformer architecture."
},
{
"end_time": 4187.858,
"index": 173,
"start_time": 4165.828,
"text": " So I've been talking to him and we're trying to devise a nice transform architecture to address problems in finite group theory. It's in the works, but nothing so far, even with the memorization stuff, it's very basic neural networks that we didn't use anything more sophisticated than that."
},
{
"end_time": 4208.319,
"index": 174,
"start_time": 4188.626,
"text": " So to be determined whether it will outperform the standard ones will be kind of interesting. Got it. Yeah. So actually now we go way back to the beginning of our conversation is how I got into this stuff. And that, I don't know, completely coincidentally was through string theory."
},
{
"end_time": 4231.203,
"index": 175,
"start_time": 4208.933,
"text": " So at this point, maybe I'll just give a bit of a background of how all this stuff came about. At least personally. Why was I even thinking about this? Because I knew nothing about AI seven, eight years ago. Literally zero. I knew nothing more than to read it from the news."
},
{
"end_time": 4256.425,
"index": 176,
"start_time": 4231.647,
"text": " I know this is actually a very interesting story which shows again the kind of ideas that the string theory community is capable of generating just because you got all these experts looking on a kind of interesting problems. So let's go way back and again you know I quoted Gauss right I gotta cook I have to say something about Euler. So this is a problem again this is a you can see I'm very influenced by three."
},
{
"end_time": 4281.886,
"index": 177,
"start_time": 4256.834,
"text": " The number three, you know, I'm a total numerologist, right? Trinity, name the three, three is something, right? And then there is called the trichotomy classification theorem by Euler. This dates to 1736. So if you look at, so I'm going to say the buzzword, which is connected compact orientable surfaces. So these are"
},
{
"end_time": 4298.473,
"index": 178,
"start_time": 4282.329,
"text": " you know i mean the words explain themselves you know they have no boundaries yes and they're uh you know topologically you know whatever the the topological surfaces so euler was able to realize that a single integer uh characterizes"
},
{
"end_time": 4322.056,
"index": 179,
"start_time": 4298.848,
"text": " All such surfaces. So this is the standard thing that people see in topology, right? So the surface of a ball is the surface of a ball and you can deform it. You know, the surface of a football is the same as an American football. It can deform without cutting or tearing. And then the surface of a donut is the same as, you know, your cup."
},
{
"end_time": 4341.664,
"index": 180,
"start_time": 4322.739,
"text": " right because you know is everything that everyone understand the thing you know this is it has one handle and then so the surface of a donut is exactly the topologically the what they call topologically homeomorphic to to the the cup and then you got the you know the the pretzel so i think that's a pretzel"
},
{
"end_time": 4362.892,
"index": 181,
"start_time": 4342.176,
"text": " Or maybe I think this is like the German pretzel and it gets more and more complicated but the oil oilers because you know either invented to the field of topology so you realize this this idea of topological equivalence in the sense that there's a single topological invariant called which we now call the oil number."
},
{
"end_time": 4390.401,
"index": 182,
"start_time": 4363.439,
"text": " which characterizes these things. Another way to an equivalent way to say is the genus of these surfaces is, you know, no, no handles, one handle, two handle, three handles, and so on and so forth. It turns out that the Euler number, what we now call the Euler number is two minus twice the genus. So two, two, two minus two G. Okay, that's great. So this is, that's the classic Euler's theorem."
},
{
"end_time": 4416.237,
"index": 183,
"start_time": 4391.101,
"text": " And then, you know, comes in Gauss, right? Once you got these three names next to each other, Euler, Gauss, and Riemann, you know, this is, it's got to be some serious theorem. So Euler did this in topology and then Gauss did this incredible work, which he calls him, he himself calls him the Theorema Grigium, the great theorem, which he considers this is his personal favorite. And this is Gauss, right?"
},
{
"end_time": 4446.305,
"index": 184,
"start_time": 4416.698,
"text": " and Gauss said, you can relate this number to, which is, this number is purely topological. You can relate this number to metric geometry. So he came up with this concept, which we now call Gaussian curvature. It's just some complicated stuff. You can characterize this curvature, which you can define on this. This is even before the word manifold existed on the surface."
},
{
"end_time": 4475.572,
"index": 185,
"start_time": 4447.193,
"text": " And then you can integrate using calculus and the integral of this Gaussian curvature divided by four pi is exactly equal to this topological number. And that's incredible, right? The fact that you can do an integral, it comes out to be an integer. And that integer is exactly topology. So this idea, this Gauss related geometry to topology in this one suite"
},
{
"end_time": 4506.51,
"index": 186,
"start_time": 4476.715,
"text": " And then the next level comes Riemann. Riemann says, well, what you can do is to complexify. So these are no longer real connected compact orientable surfaces, but you can think about these as complex objects. So what do we mean by that is, well, if you think about the real Cartesian plane, that's a two-dimensional object."
},
{
"end_time": 4534.923,
"index": 187,
"start_time": 4507.142,
"text": " But you can equally think of that as a one complex dimensional object, namely the complex plane. Or the complex line. Yeah, the complex line. Exactly. So with R2, Riemann would call C. And then Riemann realized that you can put similar structure on all of these things as well. So all of a sudden these things are no longer two dimensional real or interval surfaces, but one complex dimensional would"
},
{
"end_time": 4546.596,
"index": 188,
"start_time": 4535.23,
"text": " It's a terrible name, so a complex curve is actually a two-real dimensional surface. And it turns out that all complex curves"
},
{
"end_time": 4575.794,
"index": 189,
"start_time": 4547.022,
"text": " are orientable. So you already rule out things like applying bottles and stuff like that or Möbius strips. So the complex structure requires orientability and that's partly because of Cauchy-Riemann relations. It puts a direction. You can't get away. But the interesting thing is all of this now should be thought of as one complex dimensional curves. They're called curves because they're one complex dimension, but they're not curves. They're surfaces in the real sense."
},
{
"end_time": 4604.957,
"index": 190,
"start_time": 4576.22,
"text": " So now here comes, so if you apply this to the Gauss thing, you get this amazing trichotomy theorem. And the theorem says, if you do this to the curvature, you can see this. I mean, the number here is two, right? You get the only number two, which is a positive curvature thing, right? And that's consistent with the fact that the sphere is a positively curved object. Locally, everywhere, it has positive curvature."
},
{
"end_time": 4633.865,
"index": 191,
"start_time": 4605.265,
"text": " If you do it to a torus or the surface of a donut, which is just called the algebraic donut, you integrate that, you get zero curvature. And this is not a surprise because you have a sheet of paper, you fold it once, you get a cylinder and you fold it again, you glue it again, you get this torus, this donut. And this sheet of paper is inherently flat. Yes."
},
{
"end_time": 4661.647,
"index": 192,
"start_time": 4634.411,
"text": " So if you just take a piece of paper, you take this piece of paper and you roll it up, you get a cylinder. And then you do it again, and you get the surface of a donut, like a rubber tire. And that is currently zero curvature. And then you can do this, and this is a consequence of what's known as Riemann uniformization theorem. If you do anything that has more than one handle, you get zero curvature."
},
{
"end_time": 4688.439,
"index": 193,
"start_time": 4662.381,
"text": " So now you have the trichotomy, right? You have positive curvature, zero curvature and negative curvature. The one in the middle is really, obviously is interesting. It's the boundary case in complex algebra geometry. These things are called final varieties. Earlier you said if you have anything that's more than one handle, you have zero curvature. You meant negative curvature. Sorry, sorry. I meant negative curve. Okay. So these fidget spinners on the right,"
},
{
"end_time": 4714.189,
"index": 194,
"start_time": 4688.712,
"text": " They all have negative curvature. Everything here has negative curvature. Got it. Yeah. So now in the world of complex algebra geometry, these positive curvature things are called final varieties. After this Italian guy final, the, these negative curvature objects, which proliferate are called varieties of general type. And this boundary case are called zero, zero curvature objects."
},
{
"end_time": 4742.483,
"index": 195,
"start_time": 4714.718,
"text": " And it just so happens we now call things in the middle clavia These zero curvature objects. Yes So so far this has nothing to do with physics. I mean it's just the fact of topology, right? But this is such a beautiful diagram that you know took from 1736 until Riemann Riemann what died in in the 1860s I think or something like that. So it took you know 100 120 years to really formulate just this table to relate"
},
{
"end_time": 4764.309,
"index": 196,
"start_time": 4742.858,
"text": " Metric Geometry to Topology to Algebraic Geometry is kind of a beautiful thing, right? So to generalize this table is the central piece of what's now called the Minimal Model Program in Algebraic Geometry for which there have been all these fields, you know, be a car a couple of years ago,"
},
{
"end_time": 4789.94,
"index": 197,
"start_time": 4764.65,
"text": " And then it started with Maury who got the Fields Medal and then this whole Mukai and this whole distinct, distinguished idea. So basically this minimal model program should just generalize this to higher dimension. This is dimension complex, dimension one, right? How do you do it? It's very hard. And once you have it, I won't bore you with the details. This is very nice, you know, there's topology, algebra, geometry, differential geometry, index theorem, they all get unified in this very beautiful way."
},
{
"end_time": 4817.193,
"index": 198,
"start_time": 4790.333,
"text": " And you want to obviously want to generalize this to arbitrary dimension, arbitrary complex dimension. It'd be nice. It's still an open problem. How do you do it in general? It's a very nice problem. But at least for a class of complex manifold known as scalar manifolds, I won't bore you with the details, but scalar manifolds on which where the metric has very nice behavior, there's a potential for which you can have a double derivative that gets on the metric."
},
{
"end_time": 4841.408,
"index": 199,
"start_time": 4817.722,
"text": " And then it was conjectured in by Calabi in the 50s. Again, you know, 54, 56, 57. It was a great year, right? That's all these different ideas. I mean, in three completely different worlds now come together because mathematical physicists have kind of tied it up, you know, the world of neural networks, the world of Calabi conjecture, the world of string theory to one."
},
{
"end_time": 4871.459,
"index": 200,
"start_time": 4841.971,
"text": " I like, you know, when when things get bridged up in this way, you know, but, you know, again, this this the theorem itself is extremely technical. But the idea is for this killer manifold, there is an analog of this diagram. Basically, I love this slide. I saved this slide for my own. Okay. I keep a collection of dictionaries in physics and math. Yeah. I think this is beautiful. Yeah, me too. But you took me like it took me years."
},
{
"end_time": 4893.848,
"index": 201,
"start_time": 4872.108,
"text": " to to do this table because you know it's not written down anywhere and it touches different things i think it's not written down anywhere precisely because mouth textbooks are written in the bobaki style uh-huh um but now it just becomes clear what would people be thinking about for the past 100 years you know after grodendieck"
},
{
"end_time": 4918.882,
"index": 202,
"start_time": 4894.241,
"text": " It's just trying to relay these ideas, you know, this is intersection theory of characteristic classes. So this is topology and you know, this is, I mean, this is over 200 years of work of, you know, sent, you know, the central part of analytics and mathematicians like churn, Richie, oil, everything, every, every, every, everyone, everybody was every involved in this diagram is an absolute legend. In fact, there is one more column to this diagram."
},
{
"end_time": 4946.698,
"index": 203,
"start_time": 4919.889,
"text": " I think for short of, I think when I did this, this was a slide from some time ago, but when I was talking to a string audience, there is one more, one more, which is relations to L function. And that's when number theory comes in. So there is one more column. And to understand this world to this one more column of its behavior to L functions, that's the Langlands program."
},
{
"end_time": 4966.954,
"index": 204,
"start_time": 4947.432,
"text": " Right so it's actually really magical that this this this table actually extends more as far as I mean that's just as far as we know now right of course L functions and its relations to modularity and I think this is of course obviously mathematics to me like mathematics is about extending this table as much as possible to let it go into different fields of mathematics"
},
{
"end_time": 4977.568,
"index": 205,
"start_time": 4967.363,
"text": " so but at least for sure we know there is one because of the langlands correspondence there is one more column and that column should be on on number theory and modularity"
},
{
"end_time": 5006.596,
"index": 206,
"start_time": 4978.131,
"text": " And soon there'll be another table on the yang invariant, the he invariant. No, I don't think I don't think I have enough talent to to create something that that but it could well be there should be something something new to to to. Right. That's that's to me. That's really the most fun part about mathematics. It's not not so I mean, they're like, you know, who is it? I think maybe it's Arnold as well because there's two types of mathematicians. There are the the the the hedgehogs and they're the the birds."
},
{
"end_time": 5027.875,
"index": 207,
"start_time": 5007.295,
"text": " I mean, he's been saying his entire life"
},
{
"end_time": 5057.244,
"index": 208,
"start_time": 5028.643,
"text": " Just trying to think about can I bound, can I bound the, you know, the how many, you know, in the, in the, what is the, what's the limsop of the, of the distance between, between prime pairs. And the technique he uses is, is, is, is beautifully argued as analytic number theory technique, sieve methods, you know, kind of, you know, the, the, the Ben Green world of, of this, of, of, of sieves and James Maynard."
},
{
"end_time": 5079.377,
"index": 209,
"start_time": 5057.927,
"text": " I'm 100% in the bird category."
},
{
"end_time": 5106.971,
"index": 210,
"start_time": 5079.633,
"text": " I mean, once I see something, of course, sooner or later you need to dig like a hedgehog. But the most thrill that I get is when I say, oh, wow, this gets connected. So the results are proven when you dig, but the connections are seen when you get the overview. Yeah, yeah, absolutely. So I mean, of course, again, this is a division that's kind of artificial. In all of us, we do a bit of both. Yes."
},
{
"end_time": 5135.162,
"index": 211,
"start_time": 5107.346,
"text": " the guy who really does a well is a great dimension of course it's like it's become like a grand well he passed away this John McKay who was a Canadian probably the greatest Canadian mathematician since Coxeter John McKay really saw unbelievable connections in fields that nobody will ever see and he passed away he became sort of in the last 10 years of life he became sort of like a"
},
{
"end_time": 5159.189,
"index": 212,
"start_time": 5135.469,
"text": " I like a grandfather to to me. He's so you know, he saw my kids grow up, you know over zoom I Know so the the the London Math Society asked me to write obituary I was very touched by this I know so I wrote his obituary for and I was just trying to say what this guy is the ultimate pattern, you know linker So so John McCain Absolute legend great"
},
{
"end_time": 5187.79,
"index": 213,
"start_time": 5160.469,
"text": " Um, moving on. Uh, I mean, this is a, this is very much, this is very much a huge digression for what I'm actually going to tell you about, which is, you know, the birch test for AI and that's great. Do you, you know, do you have a limit on how, what the, these videos are? No, just so you know, some of them are one hour. Some of them are four hours and people listen for all of it. Yeah, this is great fun. Great. Yeah, same. I'm loving this. Yeah, me too."
},
{
"end_time": 5213.712,
"index": 214,
"start_time": 5188.387,
"text": " Because normally, you know, I have one up in the with 55 minute cutoff. Yes. Oh, right. And in like five minutes questions. And I'm like, oh, my God, I haven't said most of the stuff I wanted to say. Yeah. Yeah, exactly. Because the point of this channel is to give whoever I'm speaking to enough time to get through all of their points rather than their rushing and not covering something in depth. I want them to be technical and rigorous. So please."
},
{
"end_time": 5242.944,
"index": 215,
"start_time": 5214.002,
"text": " Continue sure sounds good to me. So so club is so in that magical year of 1957 of neural networks the magical year of what it was as the the automated theorem prover world and the world of algebra geometry in three complete world different world They didn't even know of each other's names Let alone the results club the conjecture that it leads for scalar manifolds. This diagram is very much well defined this table"
},
{
"end_time": 5264.889,
"index": 216,
"start_time": 5243.626,
"text": " And Yao proved it 20 years later. So Shintong Yao, who is very much like a mentor to me. And he gets the Fields Medal immediately. So you can see why this is so important. He gets the Fields Medal because this idea of falling through clubby is trying to generalize this"
},
{
"end_time": 5278.643,
"index": 217,
"start_time": 5265.401,
"text": " this sequence of ideas of Euler, Riemann, and Euler, Gauss, and Riemann. So it's certainly very important. So there it is. We can park this idea. So Yao showed that there are these"
},
{
"end_time": 5301.442,
"index": 218,
"start_time": 5279.36,
"text": " Kailar manifolds that have this property that have the right metrical properties. So by metric, I mean distance, you know, something can integrate over that because here, you know, you never think you that this this integral is messy, right? Even if we do this on a sphere, right? This, this R has all these cosines and signs that have got the, you know, they've all got to cancel at the end of the day to get four pi."
},
{
"end_time": 5320.367,
"index": 219,
"start_time": 5302.022,
"text": " yes like what the hell and then divided by two pi get two and that's the only number which is kind of amazing stuff and now you can do this in general the the just as a caveat um y'all show that this metric exists he never actually gave you a metric so the only currently no metric on these things"
},
{
"end_time": 5349.292,
"index": 220,
"start_time": 5320.367,
"text": " What's interesting is that these automated theorem provers, they seem computational, and it's my understanding that computationalists, so people who use intuitionist logic, they don't like constructive proofs."
},
{
"end_time": 5376.271,
"index": 221,
"start_time": 5349.514,
"text": " Sorry, they like constructive proofs. They don't like non-constructive proofs. In other words, existence proofs without showing the specific construction. So it's interesting to me that all of undergraduate math, which has some non-constructive proofs, are included in Lean. So I don't know the relationship between Lean and non-constructive proofs, but that's an aside. Yeah, that's an aside. I probably won't have too much to say about it."
},
{
"end_time": 5404.906,
"index": 222,
"start_time": 5377.21,
"text": " Cool. So back to, I don't know why I went on this diatribe on digression on string theory, but I just want to say this is a side comment. So this is a, this is something since seven, since 1736, which is kind of nice, which is, uh, you know, Oh, by the way, that's actually kind of interesting. I'm going to have to check this again. Um, just down the street from, from the, from the Institute is, it's the famous department store, Fortman, Fortman masons."
},
{
"end_time": 5423.78,
"index": 223,
"start_time": 5405.486,
"text": " which I think is established in 17 something. It's a great department store. It's not usually, it's not where I usually do my shopping, but it's just a beautiful department store where, you know, Mozart would have and Haydn might have, you know, called and did their Christmas shopping. But anyhow, just random, random, random thought."
},
{
"end_time": 5449.872,
"index": 224,
"start_time": 5424.36,
"text": " So back, so string theory was just one slide, right? I mean, I'm not, in some sense, I'm not, I'm not a string theorist in the sense, you know, I don't go quantize strings, you know, the kind of stuff that I'm more interested is like, I didn't grow up writing conformal field theories and do do all that stuff. It's just that it's for me, it's an input so I can play with a little more problems in geometry. Yeah. So string theory is this theory of"
},
{
"end_time": 5475.623,
"index": 225,
"start_time": 5450.435,
"text": " Space-time that unifies quantum gravity blah blah blah and then is it works in in ten dimensions and we've got to get down to Four dimensions, so we're missing six dimensions. So that's what I want to say and this this this this amazing paper in 1985 by Candela's Horowitz Strominger and Witten They were thinking about what are the properties of the six six extra dimensions?"
},
{
"end_time": 5493.336,
"index": 226,
"start_time": 5476.323,
"text": " What is interesting is that by imposing supersymmetry, and this is why supersymmetry is so interesting to me, by imposing supersymmetry and other anomaly cancellation, not too stringent conditions, they hit on the condition"
},
{
"end_time": 5513.217,
"index": 227,
"start_time": 5493.677,
"text": " that this six extra dimensions has to be richy flat richy flat is you can understand because it's vacuum style solutions is you know you want the vacuum string solution and then the condition which you've never seen before which just happens to be this caler condition they didn't know about this no physicists until 1985 would know what a caler manifold was"
},
{
"end_time": 5542.534,
"index": 228,
"start_time": 5514.138,
"text": " and it's a complex it's complex and it's complex dimension three remember again i said complex dimension three means real dimension six right that's ten minus three is six ten minus four is six and six needs to be complexified into three and again this is just an amazing fact that in 1985 strominger was a physicist um was visiting yow at the institute of advanced study in princeton"
},
{
"end_time": 5571.715,
"index": 229,
"start_time": 5543.951,
"text": " And so he went to Yao and said, can you tell me what this strange condition, this, this technical condition I got? And Yao says, wow, you know, I just got the Fields Medal for this. I think I may know a few things. I was just amazed. It was again, it was a complete confluence of ideas. That's totally random. Um, and, and the rest is history. So in fact, these four guys named this Richie Flatt, Kayla Manifold, Klabi Yao."
},
{
"end_time": 5589.991,
"index": 230,
"start_time": 5572.108,
"text": " So it wasn't the mathematicians who did it. This word Calabi-Yau came from physicists, so from string theorists, which now, you know, of course, Calabi-Yau is now one of the central pieces. And so this is, so Philip Candelas was my mentor at Oxford."
},
{
"end_time": 5608.524,
"index": 231,
"start_time": 5590.623,
"text": " and he when i when i was a junior fellow there and he tells me this story he's a very lively guy he tells me about how this whole story came about and it's very interesting and um but so so so he and the uh these these four guys came up with the word club yeah"
},
{
"end_time": 5629.206,
"index": 232,
"start_time": 5608.933,
"text": " So so all of a sudden we now have a have a name for this boundary case in complex geometry. This this this bounding case is now known as a club. So remember we had names before right. This was the final variety. This was varieties of general type and this bounding case is now called clubby. Uh huh."
},
{
"end_time": 5646.049,
"index": 233,
"start_time": 5629.411,
"text": " So what we're seeing with the Taurus here is a Colabia 1. Exactly. So exactly, exactly. In fact, the Taurus is the only Colabia 1. So it's the only one that's richly flat. I mean, by this classification, it's the only one that's topologically possible."
},
{
"end_time": 5669.292,
"index": 234,
"start_time": 5646.834,
"text": " So that's kind of interesting, right? And then this is just a comment. I like this title because I think your series is called TOE. This is a TOE on TOE. Love it. I just want to emphasize this is a nice confluence of ideas with mathematical physics. When we string theory really, what it really is, is this brainchild of interpreting"
},
{
"end_time": 5686.374,
"index": 235,
"start_time": 5669.718,
"text": " problems between interpreting and interpolating between problems in mathematics and physics. I see, for example, you know, we now, you know, GR should be phrased in differential geometry. The standard model gauge theory should be phrased in terms of"
},
{
"end_time": 5707.534,
"index": 236,
"start_time": 5686.852,
"text": " Algebraic geometry and representation theory of finite groups. And you know, condensed matter physics of topological insulators should be phrased in terms of algebraic topology. This idea, you know, I think, I think the greatest achievement of the 20th century physics is, to me, and I think something you would appreciate, since you like tables, is that here's a dictionary of a list of things"
},
{
"end_time": 5735.196,
"index": 237,
"start_time": 5707.841,
"text": " and then here's what they are in mathematics and then you know you can talk to mathematicians in this language and you can talk to physicists in language but they're actually the really same same same thing you know what's what's a fermion you know it's a spin representation of the lorenz group you know that i like that because it gives a precise definition of what we are seeing around then you have something you can purely play with in this platonic world and string theory is really just a brainchild of this translation this tradition of"
},
{
"end_time": 5764.087,
"index": 238,
"start_time": 5735.418,
"text": " What's on the left and what's on the right? And let's see what we can do. And sometimes you make progress on the left, you give insight and stuff on the right, and sometimes you make progress on the right and you give insight on the left. Why is it that you call the standard model algebraic geometry? Because bundles and connections are part of differential geometry, no? Oh, yeah, that's true. Well, I think that's, yeah, I mean, they're interlinked. And I think algebraic maybe I think maybe it's because of Atiyah and Hitchin."
},
{
"end_time": 5792.022,
"index": 239,
"start_time": 5764.906,
"text": " Of course, they are fluid in both. They go either way. Algebraic in the sense that you can often work with bundles and connections without actually doing the integral in differential geometry. I think that's the part I want to emphasize."
},
{
"end_time": 5807.125,
"index": 240,
"start_time": 5792.415,
"text": " You can understand bundles purely as algebraic objects without ever doing an integral. Like here, for example, this integral is obviously something you would do in differential geometry."
},
{
"end_time": 5837.056,
"index": 241,
"start_time": 5808.063,
"text": " But this integral, the fact that it comes to, to, to be an integer was explained through the theory of churn classes to be, you know, to be, you know, this integral is a pairing between the churn class, between homology and cohomology, which is a purely algebraic thing. You know, we all try to avoid doing integrals because integrals are horrible because it's hard to do. And in, in this language, it really just becomes, um, polynomial manipulation and it becomes much simpler. Okay."
},
{
"end_time": 5857.415,
"index": 242,
"start_time": 5838.097,
"text": " I like doing this diagram. If you look at the time lag between the mathematical idea and the physical realization of that idea, there really is a confluence."
},
{
"end_time": 5879.565,
"index": 243,
"start_time": 5857.91,
"text": " It's getting closer. I mean, these things going up and down. I mean, I'm just saying in the past, if you take the last 200 years, last hundred years or so of the groundbreaking ideas in physics, there is this interesting, right? It gets shorter and shorter. So obviously Einstein took ideas of Riemann and, you know, there was a six year gap."
},
{
"end_time": 5908.916,
"index": 244,
"start_time": 5879.906,
"text": " Dirac was able to come up with the equation of electron, essentially because of Clifford Algebras. Did, historically, was he motivated by Clifford Algebras or did it just, was it later realized, hey, Dirac, what you're doing is an example of a Clifford Algebra? So I believe, I believe the story goes, in order to write down the first time derivative version of the Klein-Gordon equation, which is a second order, you know, that's the bosonic one,"
},
{
"end_time": 5927.944,
"index": 245,
"start_time": 5909.309,
"text": " he had to"
},
{
"end_time": 5958.422,
"index": 246,
"start_time": 5928.592,
"text": " But what directed was, um, he, he, he, he said he was at St. John's in Cambridge at the time. He said, I have seen this in the textbook before someone, you know, this gamma mu gamma new thing. And then he said, I need to go to the library to check this. Uh, so he really knew about this. He, uh, and unfortunately the St. John's library was closed that, that evening. So he waited until the morning until the library was open to go to Clifford's book."
},
{
"end_time": 5980.742,
"index": 247,
"start_time": 5959.548,
"text": " Or a book about Clifford. I can't remember whether it was Clifford's book or maybe it was one of one of these books and then he opened up and he really knew and that this gamma mu gamma nu anti-computation relation really was through the through so he knew about Clifford. Cool. It's kind of interesting. Yeah."
},
{
"end_time": 6006.357,
"index": 248,
"start_time": 5981.135,
"text": " Just like Einstein knew about Riemann's work on curvature. But whether you say Dirac was really inspired by Clifford, well, he certainly did a funky factorization and then he knew how to justify it immediately by looking at the right source. And then similarly, you know, Yang-Mills theory depended on this Cybert's book on apology. And then, you know, by the time you get to Witten and Borchardt,"
},
{
"end_time": 6036.032,
"index": 249,
"start_time": 6006.869,
"text": " Really there's this this diagram for me is like what gets me excited about string theory Because string theory is a brainchild of this curve this this this orange curve And now it's getting mixed up I mean, of course, you know, you know people here hear about this this great quote that Witten says, you know string theory is a piece of 21st century mathematics that Happens to fall into 20th century and I think he means this Yes, you know that he was using supersymmetry"
},
{
"end_time": 6066.766,
"index": 250,
"start_time": 6036.886,
"text": " to prove, you know, there are theorems in Morse theory and vice versa. Richard Borchardt was using vertex algebras, which is sort of foundational thing, conformal field theory to prove some properties about the monster group. We're at this stage. And of course, you know, this was turn of last turn of the century. And now we're here and we have to where are we now? Are we are we crisscrossed or are we parallel? It's hard to it's hard to say."
},
{
"end_time": 6097.073,
"index": 251,
"start_time": 6067.466,
"text": " And in a meta manner, you can even interpret this as the pair of pants in string theory with the world sheet. Yeah, cute. Very cute. Why not? Yeah, it is. But going back to what you were saying, how I got to. Oh, yeah. So just, yeah, this confluence idea, of course, you know, you know, everyone quotes these two book papers, you know, when Wigner was thinking of May 59."
},
{
"end_time": 6126.92,
"index": 252,
"start_time": 6097.551,
"text": " Why mathematics is so effective in physics? And there's this maybe slightly less known paper, but certainly equally important paper by the great Leif Latia and then Dijkgraaf and Witten, Dijkgraaf and Hitchin, which is the other way around. Why is mathematics so, so why is physics so effective in giving ideas in mathematics? So this is a beautiful pair of essays. In this, this is like very much a, in the world of"
},
{
"end_time": 6146.527,
"index": 253,
"start_time": 6127.381,
"text": " A summary of the kind of physics. Ideas from string theory is making such beautiful advances in geometry. So this is a very beautiful pair of one given any other that needs to be, you know, sort of praised more."
},
{
"end_time": 6176.578,
"index": 254,
"start_time": 6147.295,
"text": " And that's why you were you were mentioning earlier how I got to know, you know, Roger. So while he's through this editorials, we try to collect, you know, with my colleague, Molinka, who is a former director of the of the Charn Institute. You know, everybody's connected, right? So it just so happens that, you know, I, you know, I grew up in the West, but my after trip with with my parents after so many decades, my parents actually retired and went back to Tianjin."
},
{
"end_time": 6194.497,
"index": 255,
"start_time": 6177.09,
"text": " uh... where uh... dunkeye university is where churn founded the what's now called the churn institute for mathematical sciences and that's an institute devoted to the dialogue between mathematics and physics in fact one-third of churn's ashes"
},
{
"end_time": 6219.019,
"index": 256,
"start_time": 6194.735,
"text": " is buried outside of the Math Institute. There's a great beautiful marble tomb. Once they're not because of any mathematical reason, it's just that he considered three parts of his home. His hometown in Zhejiang, China, and Berkeley, where he did most of his professional career,"
},
{
"end_time": 6247.551,
"index": 257,
"start_time": 6219.394,
"text": " And then Nankai University, where he retired to for the last 20 years of his life. So a third each. Yes. The number three comes up again. It's all about free. And in fact, I was going to joke. So in churn Simon's theory in three dimensions, there's this topological theory, churn Simon's theory. There's the there's a crucial factor of one third. I always joke, you know, that's that's why churn chose one third for his ashes. But that's not my complete coincidence."
},
{
"end_time": 6277.159,
"index": 258,
"start_time": 6247.551,
"text": " But it's actually what is actually interesting is the That tomb that beautiful black marble tomb, you know for for somebody that's greatest turn It mentions nothing about you know, his he achieved on this done the other thing. It's just one page of his notebook I mean think about the poor guy where the chisel or that he have no idea what is chiseling right? The guy was chiseling this thing and it's the proof of this of this"
},
{
"end_time": 6302.551,
"index": 259,
"start_time": 6277.91,
"text": " the fact is such and of course it's a little you can you can you can look this on the internet just say the the grave of ss churn at nankai university well the whole conversation we've had is just about pattern matching without the intuitive understanding behind it so this chisel air may have had that yes that's what i do every day love it so the uh that chisel is essentially his proof why this is equal to this"
},
{
"end_time": 6328.865,
"index": 260,
"start_time": 6303.712,
"text": " you know, why this intersection product is the same as this integral. So he essentially, it's where the Gauss-Bonnet theorem is a corollary of this trick in algebraic geometry, which is his great achievement. But anyhow, so it just so, yeah, back to this coincidence. And it just so happens that my parents, after drifting all these years abroad, they retired back to Tianjin, where the China Institute is."
},
{
"end_time": 6353.541,
"index": 261,
"start_time": 6328.865,
"text": " so that's why i became an honorary professor at nankai because i mean my my motivation was purely just so that i could see spend time to tie out with my parents but it just so happens that it happens there and i can just pay my homage to to churn just to see his grave i mean it's a great you know it's a it's it's it's a mind-blowing experience just to see the"
},
{
"end_time": 6382.602,
"index": 262,
"start_time": 6354.087,
"text": " Yeah, it's remarkable. And that he was, he was still doing, he wrote the preface to this when he was 99. These guys are unstoppable."
},
{
"end_time": 6407.363,
"index": 263,
"start_time": 6383.166,
"text": " And you know Roger Roger Penrose sent he he sent his essay. Yeah to this one when he was what? 1992 Yeah, these guys are Anyhow, it's kind of I do you like tables, right? I love tables. So the tables are just here It's just like, you know, we're just a speculation of Western theory is going here's a list of you know, the annual conferences and"
},
{
"end_time": 6430.111,
"index": 264,
"start_time": 6408.097,
"text": " Like the series where string theory has been happening. So 1986 was the first string revolution where since then every year there's been a major string conference. I'm going to the one, the first one I'm going to for years in two weeks time. It happens to my Abu Dhabi. I guess I'm son."
},
{
"end_time": 6452.159,
"index": 265,
"start_time": 6430.913,
"text": " And then, you know, there's a series of annual ones, the string phenol and the string math came in as late as in 2011. That's kind of interesting. So that's like, you know, 30 years after the first string conference and the various other ones. What's really interesting one is in 2017 is there's the first string data. This is what AI entered string theory."
},
{
"end_time": 6470.606,
"index": 266,
"start_time": 6453.012,
"text": " And so it's kind of so so what i read the first paper in in in 2017 about ai sister stuff and there were there were three other groups independently mining different ai aspects and how to apply the string theory so that the reason i want to mention is was just how why was"
},
{
"end_time": 6500.452,
"index": 267,
"start_time": 6470.606,
"text": " You know with the string community even thinking about this problems problems in AI. Oh and also just to be clear briefly speaking I'm not a fan of tables per se. I'm a fan of dictionaries because they're like Rosetta stones So I'm a fan of Rosetta stones and translating between different languages. So you mentioned the siloing earlier and Mathematicians call if even physicists call them dictionaries, but technically they're the sources like a dictionary You just have a term and then you define it the translations like Rosetta stones"
},
{
"end_time": 6530.111,
"index": 268,
"start_time": 6501.357,
"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."
},
{
"end_time": 6558.166,
"index": 269,
"start_time": 6530.759,
"text": " 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."
},
{
"end_time": 6578.609,
"index": 270,
"start_time": 6558.677,
"text": " Learn more at Capella.edu. Yes. Yeah. Yeah. No, absolutely. I guess that's why you like Langlands so much. Yeah. Yeah, for sure. Yeah, no. Yeah, absolutely. In some way, this whole channel is a project of a Rosetta stone between the different fields of math and physics and philosophy. Yeah. Yeah. That's fantastic. Love it. Big fan. Thank you."
},
{
"end_time": 6605.896,
"index": 271,
"start_time": 6579.053,
"text": " okay so do you want to just I noticed it jump back to number 13 so it seems like I thought we were at 39 out of 40 no no no because I I've learned this non-linear structure it's like because you see like I've learned this this is really dangerous I've learned like the click button in PDF presentations like you click it it jumps to another one and you can have interludes so you know it's clearly an interlude and you say you jump back"
},
{
"end_time": 6633.729,
"index": 272,
"start_time": 6606.288,
"text": " To your main. So my my actual main presentation is only like, you know 30 pages, but it's good all these digressions Which is actually very typical of my personality So I gave you this big interlude about string about you know string theory clavier manifolds, right? So now we've already got to the point that clavier one-fold the one-dimensional complex clavier fund. There's only one example. That's just one of these"
},
{
"end_time": 6657.09,
"index": 273,
"start_time": 6634.206,
"text": " Right and then it turns out that in complex dimension two There are two of these There is the four dimensional torus Which is and then it's this crazy thing called the k3 Which is richie flat and keller. So you got one in complex measure one two in complex animation three you would think"
},
{
"end_time": 6686.903,
"index": 274,
"start_time": 6657.944,
"text": " In three dimensions, there's three of these things that are topologically distinct. And unfortunately, this is one of the sequences in mathematics that goes as one, two, we have absolutely no idea. And we know at least one billion. At least. So it's kind of, it goes one, two, a billion. And so starting from complex dimension three just goes crazy. It's still a conjecture of Yao that in every dimension, this number is finite."
},
{
"end_time": 6710.708,
"index": 275,
"start_time": 6687.449,
"text": " So remember this positive curvature thing this final thing to the very third it's now it's it is a theorem that in every dimension final varieties is finite impossibility In topology that only a finite number of these that are distinct topologically It's also known that the negative curvatures is infinite in every dimension"
},
{
"end_time": 6740.401,
"index": 276,
"start_time": 6712.005,
"text": " And when it goes higher, it's like even uncountably infinite. Oh, interesting. But is this boundary case Yao conjectures in an ideal world? They're also finite, but we don't know. This is the open conjecture. Now the billion, are any of them constructed or is it just the existence? Yeah, that's it. Now that's exactly where we're getting. So it's gotten one, two and three. Three is like, you know, I'm going to list these things. Right. And then,"
},
{
"end_time": 6766.954,
"index": 277,
"start_time": 6743.029,
"text": " Algebra journal just never really bother listing on one mouth. This is not something they do. So it took on the physicists to go on the challenge. So Philip Candela's and France and then, um, Harold Scharke and Maximilian Kreutzer started just listing these. And that's why we have these billions. There is actually databases of these."
},
{
"end_time": 6789.599,
"index": 278,
"start_time": 6767.329,
"text": " And they're presented in, you know, just like matrices like this. I won't bore you with the details of these matrices. You know, these algebraic varieties, you can define this as, you know, like intersections of polynomials. That's one way to present them. And in Kreutzer and Schakka's database, they put vertices of toric varieties. But the upshot is that, you know, there's a database of many, many gigabytes"
},
{
"end_time": 6817.654,
"index": 279,
"start_time": 6789.991,
"text": " That that really got done by the by by the certainly by the turn of the century but by year 2000 These guys were running on Pentium machines. I mean, this is an absolute feat Especially our courts and shkalka they were able to get 500 million of this is stored on a hard drive using a Pentium machine Of this car be our manifolds. Hmm And they were able to compute topological invariance of these"
},
{
"end_time": 6846.988,
"index": 280,
"start_time": 6818.097,
"text": " So that's, so I happen to have this database. I could access them and that was kind of fun. And I've been playing on and off with them for a number of years. So, and you know, a typical calculation is like, you know, you have something like, like a configuration of tensors of here is even in integers and you have some standard method in algebra geometry to compute topological invariance. And this topological invariance again,"
},
{
"end_time": 6874.087,
"index": 281,
"start_time": 6847.363,
"text": " in this dictionary means something. So, for example, H21 in some context is the number of generations of fermions in the low energy world. So that's a complete problem in this computing a topological invariant in algebraic geometry. And there are methods to do it. And in these databases, people took 10, 20 years to compile this database and you got these things in. And they're not easy. It's very complicated to compute these things."
},
{
"end_time": 6895.725,
"index": 282,
"start_time": 6874.224,
"text": " So in twenty seventeen i was playing around with this and the reason is very why i was playing around with this was very simple is because my son was born. And i had infinite sleepless nights and i couldn't do anything right i had like you know there's the kid and then you know there's the kid there's the kid and you know and you wake you up at two."
},
{
"end_time": 6921.067,
"index": 283,
"start_time": 6896.391,
"text": " Put him to bed and I was bottle feeding him and I had a daughter at the time so my wife's taking care of the daughter. They're passed out. And then I got this kid, I passed him out, put them into bed and I'm wide awake at this point. It's like 2 a.m. It's like I can't fall asleep anymore and I can't do real, you know, serious computation anymore because I'm just too tired."
},
{
"end_time": 6934.428,
"index": 284,
"start_time": 6921.425,
"text": " So let's just play around with data. At least I can let the computer help me to do something. And that's what I learned. What's this thing that everybody's talking about? Well, you know, it's machine learning. Right."
},
{
"end_time": 6959.548,
"index": 285,
"start_time": 6935.282,
"text": " So that's why I got through this. It's a very simple biological reason why I was trying to learn machine learning. So then I think I was hallucinating at some point, right? I was like, well, if you look at pictures like matrices, we're talking about 500 million of these things, right? Yes. Certainly I wasn't going through all of them. And they're being labeled by topological invariants."
},
{
"end_time": 6988.524,
"index": 286,
"start_time": 6959.667,
"text": " How different is it if I just saw a pixelated one of these and label them by this? And all of a sudden this began to look like a problem in hand digit recognition, right? This is like, how different is this or image recognition? So, and I just literally started feeding in, I took 500, I mean, 500 million is too much, right? So I took like 5,000 of these, 10,000 of these, and then trained them to look and recognize this."
},
{
"end_time": 7010.708,
"index": 287,
"start_time": 6989.053,
"text": " to"
},
{
"end_time": 7040.384,
"index": 288,
"start_time": 7011.408,
"text": " And it was recognizing it to great accuracy. And now, I mean, people have improved this, like loads of people, like in Affinatello, there's a group there that did some serious work on just trying to this problem. But this idea suddenly didn't seem so crazy anymore. The idea seemed completely crazy to me because I was hallucinating at 2 a.m. And it was so. But what's the upshot of this? The upshot is somehow the neural network was doing algebraic geometry like this kind of algebraic geometry."
},
{
"end_time": 7069.872,
"index": 289,
"start_time": 7040.879,
"text": " Really sequence chasing very complicated bobacki style stuff without knowing anything about our drug geometry It somehow was just doing pattern recognition and somehow it's beating us because you know if you do this computation Seriously, it's it's it's double exponential complexity But it's just now but pattern recognition is bypassing all of that So then I became a fanatic right then I said well all of algebraic geometry is image processing"
},
{
"end_time": 7077.415,
"index": 290,
"start_time": 7071.032,
"text": " So far I have not been shot by the Algebraic Geometers because it's actually true if you really think about it."
},
{
"end_time": 7103.49,
"index": 291,
"start_time": 7078.012,
"text": " You know, any algebraic, the point of algebraic geometry, the reason I like algebraic more than differential is because there's a very nice way to represent manifolds in this way. Manifolds in algebraic geometry. So in differential geometry, manifolds are defined in terms of Euclidean patches. Then you do, you know, transition functions, you know, which are differentiable, C infinity, you know, blah, blah, blah. But in algebraic geometry, they're just vanishing low-side polynomials."
},
{
"end_time": 7130.145,
"index": 292,
"start_time": 7104.889,
"text": " and then once you have systems of polynomials you have a very good representation so for example here this is i'm just recording the list of polynomials the degrees of polynomials that are that are you know embedded in some space and and that really is algebraic geometry so basically any algebraic variety so that's fancy way of saying the this polynomial representation of a manifold"
},
{
"end_time": 7156.869,
"index": 293,
"start_time": 7130.384,
"text": " which is called an algebraic variety. This thing is representable as in terms of a matrix or a tensor, sometimes even an integer tensor. And then could be computation of invariance or topological invariance is the recognition problem of such tensor. But once you have a tensor, you can always pixelate it and, you know, picturize it. You know, at the end of the day is doing this because it's just image process and algebraic geometry."
},
{
"end_time": 7185.998,
"index": 294,
"start_time": 7158.217,
"text": " Do you mean to say every problem in algebraic geometry is an image process or is an image processing problem or just problems involving invariance or image processing or even broader than that? I think it is really more broad. I think, you know, at some level, you know, I think in my view, I try to say, you know, bottom up mathematics is language processing."
},
{
"end_time": 7215.981,
"index": 295,
"start_time": 7186.476,
"text": " And top-down mathematics is image processing. Interesting. Of course, this is, I mean, take with a caveat, but of course, at some level, there is truth in what I say. Of course, it's an extreme thing to say. But, you know, in terms of what mathematical discovery is, is that you're trying to take a pattern in mathematics. So in algebra, if you use perfect example, you can pixelate everything and you can just try to see certain images have certain properties. And so your image processing mathematics,"
},
{
"end_time": 7242.756,
"index": 296,
"start_time": 7216.34,
"text": " Where's bottom up you're building up like mathematics as a language. So it's like processing. Of course. All of this will be useless if you can't actually get human readable mathematics out of it, right? So this is the first surprise. The fact that it's even doing it at all to a certain degree of accuracy. Now we're talking about courses like now it's been improved to like 99.99% accuracy in these databases."
},
{
"end_time": 7271.032,
"index": 297,
"start_time": 7243.302,
"text": " But that's the first level. That's the first surprise. The second surprise is that you can actually extract human understandable mathematics from it. And I think that's the next level surprise. So the memorization conjectures, this beautiful work in DeepMind that Jody Williamson's involved in, in this human guided intuition, you can actually get human mathematics out of it. And that's really quite something."
},
{
"end_time": 7299.292,
"index": 298,
"start_time": 7271.63,
"text": " So that's maybe that's a good point to break for part two, which is an advertisement of, you know, here is like we've gone through many, many things about what mathematics is and to, you know, how it got this through doing, you know, this interaction between algebra, geometry and string theory. And then a second part would be how you can actually extrapolate and extract"
},
{
"end_time": 7328.029,
"index": 299,
"start_time": 7300.179,
"text": " mathematics, actual conjectures, things to prove from doing this kind of experimentation, which are summarized in these books. I keep on advertising my books because I get £50 for a year of, what do they call it, royalties, you know, so I don't have to sell my liver for my kids. But it's actually, it's kind of fun. It's a complete, I mean, academic publish is a joke, right? You get like, I don't know, like £100 a year."
},
{
"end_time": 7350.896,
"index": 300,
"start_time": 7328.268,
"text": " because you don't actually make money out of it. But maybe that's a good place to break. And then for part two, how we try to formulate what the Birch test is for AI, which is sort of the Turing test plus. Because the Birch test is how to get actual meaningful human mathematics out of this kind of playing around with mathematical data."
},
{
"end_time": 7371.323,
"index": 301,
"start_time": 7351.237,
"text": " I see two of your sentence that will be these maxims for the future will be that machine learning is the 22nd centuries math that fell into the 21st. So this machine learning assisted mathematics or that the bottom up is language processing and then the bottom the top down is image processing. Yeah, I like those two. Yeah."
},
{
"end_time": 7398.968,
"index": 302,
"start_time": 7371.493,
"text": " Anyone who's watching, if you have questions for Yang Hui for part two, please leave them in the comments. Do you want to give just a brief overview? Oh, yeah, sure. So just so I'm going to talk about what the birch test is and what which which papers so far have have gone have come close. They've gone to the birch test. And then I'm going to talk about something more experiments. Number three. And the one that I really enjoyed doing with my collaborators, Lee Oliver, Postnikov,"
},
{
"end_time": 7425.316,
"index": 303,
"start_time": 7399.411,
"text": " which is to actually make something meaningful that's related to the Bletch-Thunwind-Dyer Conjecture just by just letting machine go crazy and finding a new pattern in elliptic curves which is fundamentally a new pattern in the prime numbers which is completely amazing. You mentioned quanta earlier so this quanta feature that featured this one considered this as one of the"
},
{
"end_time": 7446.647,
"index": 304,
"start_time": 7425.316,
"text": " The breakthroughs of of twenty twenty four great and that word murmuration which was used repeatedly throughout it was never defined but it will be in the part two. I'm looking forward to it me too we took okay thank you so much thank you this has been wonderful i could continue speaking to you for four hours both of us have to get going but. That's so much fun pleasure."
},
{
"end_time": 7470.009,
"index": 305,
"start_time": 7447.517,
"text": " Don't go anywhere just yet. Now I have a recap of today's episode brought to you by The Economist. Just as The Economist brings clarity to complex concepts, we're doing the same with our new AI-powered episode recap. Here's a concise summary of the key insights from today's podcast. All right, let's dive in. We're talking about Kurt J. Mungle and his deep dives into all things mind-bending."
},
{
"end_time": 7486.51,
"index": 306,
"start_time": 7470.657,
"text": " You know this guy puts in the hours like weeks prepping to grill guests like Roger Penrose on some wild topics. Yeah, it's amazing using his own background to dig in really challenging guests with his knowledge of mathematical physics pushes them beyond the usual."
},
{
"end_time": 7502.858,
"index": 307,
"start_time": 7486.852,
"text": " Definitely and today we're focusing on his chat with mathematician yang we he they're getting into AI math Where those two worlds collide and it's fascinating because it really makes you think differently about how math works How we do math and where AI might fit into the picture?"
},
{
"end_time": 7532.79,
"index": 308,
"start_time": 7503.131,
"text": " You might think a mathematician's life is all formulas and proofs, but Yanqui, he actually started exploring AI-assisted math while dealing with sleepless nights with his newborn son. It's such a cool example of finding inspiration when you least expect it. Tired but inspired, he started messing around with machine learning in those quiet early morning hours. So let's break down this whole AI and math thing. Yanqui, he talks about three levels of math. Bottom-up, top-down, and meta. Bottom-up is like building with Legos. Very structured, rigorous proofs."
},
{
"end_time": 7561.886,
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"start_time": 7533.046,
"text": " That's the foundation. But here's where things get really interesting. It has limitations. Right. And those limitations are highlighted by Gödel's incompleteness theorems. Basically, Gödel showed us that even in perfectly logical systems, there will always be true statements that can't be proven within that system. It's mind blowing. So if even our most rigorous math has these inherent limitations, it makes you think. Could AI discover truths that we as humans bound by our formal systems might miss? Could it explore uncharted territory?"
},
{
"end_time": 7583.473,
"index": 310,
"start_time": 7562.176,
"text": " That's a really deep thought, and it's really at the core of what makes this conversation revolutionary. It's not about AI just helping us with math faster. It's about AI possibly changing how we think about math altogether. So how is this all playing out? We've had computers in math for ages, from early theorem provers to AI assistants like Lean. But where are we now with AI actually doing math?"
},
{
"end_time": 7605.674,
"index": 311,
"start_time": 7583.729,
"text": " Well, AI is already making some big strides. It's tackling Olympiad-level problems and doing it well, which makes you ask, can AI really unlock the secrets of math? And that leads us to the big philosophical questions. Is AI really understanding these mathematical ideas, or is it just incredibly good at spotting patterns? It's like that famous Chinese room thought experiment."
},
{
"end_time": 7631.561,
"index": 312,
"start_time": 7606.049,
"text": " You could follow rules to manipulate Chinese symbols without truly understanding the language. Yang Hui, he shared a story about Andrew Wiles, the guy who proved Fermat's last theorem, trying to challenge GPT-3 with some basic math problems. It highlights how early AI models, while excelling in tasks with clear rules and plenty of examples, struggled with things that needed real deep understanding. It seems like AI's strength right now is in pattern recognition."
},
{
"end_time": 7657.346,
"index": 313,
"start_time": 7631.869,
"text": " And that ties into what Yan Kuo he calls top-down mathematics. It's where intuition and seeing connections between different parts of math are king. Like Gauss. He figured out the prime number theorem way before we had the tools to prove it. It shows how a knack for patterns can lead to big breakthroughs even before we have the rigorous structure. It's like AI is taking that intuitive leap, seeing connections that might have taken us humans years, even decades to figure out."
},
{
"end_time": 7687.671,
"index": 314,
"start_time": 7657.961,
"text": " And it's all because AI can deal with such massive amounts of data. Which brings us back to Yang Hui. He's sleepless nights. He started thinking about Calabiao manifolds, super complex mathematical things key to string theory, as image processing problems. Wait, Calabiao manifolds? Those sound like something straight out of science fiction. They're pretty wild. Think six dimensions all curled up, nearly impossible to picture. They're vital to string theory, which tries to bring all the forces of nature together."
},
{
"end_time": 7712.858,
"index": 315,
"start_time": 7688.404,
"text": " Now mathematicians typically use these really abstract algebraic geometry techniques for this. But Yang Wei? He had a different thought. So instead of equations and formulas, he starts thinking about pixels. Yeah. Like taking a Klabi-Yau manifold, breaking it down into a pixel grid, like you do with an image. He's taking abstract geometry and turning it into something a neural network built for image recognition can handle. That is a radical change in how we think about this."
},
{
"end_time": 7743.319,
"index": 316,
"start_time": 7713.439,
"text": " It's like he's making something incredibly abstract, tangible, translating it for AI. Did it even work? The results blew people away. He fed these pixelated manifolds into a neural network and it predicted their topological properties really accurately. He basically showed AI could do algebraic geometry in a whole new way. So it's not just speeding up calculations. It's uncovering hidden patterns and connections that might've stayed hidden, like opening a new way of seeing math."
},
{
"end_time": 7753.08,
"index": 317,
"start_time": 7743.626,
"text": " And that leads us to the big question. If AI can crack open complex math like this, what other secrets could it unlock? We're back."
},
{
"end_time": 7782.654,
"index": 318,
"start_time": 7754.411,
"text": " Last time we were talking about AI, not just helping us with math, but actually coming up with new mathematical insights, which is where the Birch test comes in. It's like, can AI go from being a super calculator to actually being a math partner? Exactly. And now we'll look at how researchers like Yang Hui He are trying to answer that. Remember, the Turing test was about a machine being able to hold a conversation like a human. The Birch test is a whole other level. It's not about imitation. It's about creating completely new mathematical ideas. Think about Brian Birch back in the 60s."
},
{
"end_time": 7812.022,
"index": 319,
"start_time": 7783.029,
"text": " He came up with this bowl conjecture about elliptic curves just from looking at patterns and numbers. So this test wants AI to do similar leaps to go through tons of data, find patterns and come up with conjectures that push math forward. Exactly. Can AI like Birch show us new mathematical landscapes? That's asking a lot. So how are we doing? Are there any signs AI might be on the right track? There have been some promising developments like in 2021 Davies and his team used AI to explore not theory,"
},
{
"end_time": 7841.715,
"index": 320,
"start_time": 7812.483,
"text": " knots like tying your shoelaces. What's that got to do with advanced math? It's more complex than you think. Knot theory is about how you can embed a loop in three-dimensional space and it actually connects to things like topology and even quantum physics. Okay, that's interesting. So how does AI come in? Well, every knot has certain mathematical properties called invariants. It's kind of like its fingerprint. Davey's team used machine learning to analyze a massive amount of these invariants. So was the AI just crunching numbers or was it doing something more?"
},
{
"end_time": 7870.35,
"index": 321,
"start_time": 7842.073,
"text": " What's amazing is the AI didn't just process the data, it actually found hidden relationships between these invariants, which led to new conjectures that mathematicians hadn't even considered before, like the AI was pointing the way to new mathematical truths. That's wild. Sounds like AI is becoming a powerful tool to spot patterns our human minds might miss. Absolutely. Another cool example is Lample and Charton's work in 2019. They trained AI on a massive data set of math formulas. And what did they find?"
},
{
"end_time": 7887.637,
"index": 322,
"start_time": 7870.452,
"text": " Well, this AI could accurately predict the next formula in a sequence, even for really complex ones. It was like the AI was learning the grammar of math and could guess what might come next. So we might not have AI writing full-blown proofs yet, but it's getting really good at understanding the structure of math and suggesting new directions."
},
{
"end_time": 7908.968,
"index": 323,
"start_time": 7887.841,
"text": " And that brings us back to Yang Kuhi. His work with those Calabiao manifolds, analyzing them as pixelated forms, that was a huge breakthrough. Showed that AI could take on algebraic geometry problems in a totally new way. Like bridging abstract math in the world of data and algorithms. Exactly. And that bridge leads to some really mind-bending possibilities."
},
{
"end_time": 7929.821,
"index": 324,
"start_time": 7909.462,
"text": " Yang Hui and his colleagues started exploring something they call murmuration. It's a great analogy. Think of a flock of birds moving together like one. Each bird reacts to the ones around it, and you get these complex, beautiful patterns. Well, Yang Hui"
},
{
"end_time": 7958.865,
"index": 325,
"start_time": 7930.23,
"text": " He sees a parallel between how birds navigate together in a murmuration and how AI can guide mathematicians towards new insights by sifting through tons of math data. So the AI is like the flock, exploring math and showing us where things get interesting. Yeah, and they've actually used this murmuration idea to look into a famous problem in number theory, the Birch and Swinerton-Dyer conjecture. That name sounds a bit intimidating. What's it all about? Imagine a donut shape, but in the world of numbers, these are called elliptic curves."
},
{
"end_time": 7974.292,
"index": 326,
"start_time": 7959.326,
"text": " Maths"
},
{
"end_time": 7999.821,
"index": 327,
"start_time": 7974.565,
"text": " and a specific math function, like linking the geometry of these curves to number theory. I think these are definitely getting complex now. And it's a big deal in math. It's actually one of the Clay Mathematics Institute's millennium price problems. Solve it, you win a million bucks. Now that's some serious math street cred. So how did Yang Hu, his team, use AI for this? They trained an AI on this massive data set of elliptic curves and their functions."
},
{
"end_time": 8027.875,
"index": 328,
"start_time": 8000.282,
"text": " The AI didn't actually solve the whole conjecture, but it found this new pattern, this correlation that mathematicians hadn't noticed before. So the AI was like a digital explorer mapping out this math territory and showing mathematicians what to look at more closely. Exactly. This discovery, while not a complete proof, gives more support to the conjecture and opens up some exciting new areas for research. It shows how AI can help with even the hardest problems in mathematics. It feels like we're on the edge of something new in math."
},
{
"end_time": 8056.766,
"index": 329,
"start_time": 8028.319,
"text": " AI is not just a tool, it's a partner in figuring out the truth. What does all this mean for math in the future? That's a great question and it's something we'll dig into in the final part of this deep dive. We'll look at the philosophical and ethical stuff around AI in math. We'll ask if AI is really understanding the math it's working with or if it's just manipulating symbols in a really fancy way. See you there. Welcome back to our deep dive. We've been exploring how AI is changing the game in math."
},
{
"end_time": 8083.541,
"index": 330,
"start_time": 8057.108,
"text": " from solving tough problems to finding hidden patterns in complex structures. But what does it all mean? What are the implications of all of this? We've touched on this question of understanding. Does AI really understand the math it's dealing with, or is it just a master of pattern matching? Yeah, we can get caught up in the cool stuff AI is doing, but we can't forget about those implications. If AI is going to be a real collaborator in mathematics, this whole understanding question is huge."
},
{
"end_time": 8097.108,
"index": 331,
"start_time": 8083.848,
"text": " It goes way back to the Chinese room thought experiment. Imagine someone who doesn't speak Chinese has this rule book for moving Chinese symbols around. They can follow the rules to make grammatically correct sentences, but do they actually get the meaning?"
},
{
"end_time": 8114.394,
"index": 332,
"start_time": 8097.551,
"text": " So is AI like that, just manipulating symbols and math without grasping the deeper concepts? That's the big question, and there's no easy answer. Some people say that because AI gets meaningful results, like we've talked about, it shows some kind of understanding, even if it's different from how we understand things."
},
{
"end_time": 8141.323,
"index": 333,
"start_time": 8114.394,
"text": " Others say AI doesn't have that intuitive grasp of math concepts that we humans have. It's a debate that's probably going to keep going as AI gets better and better at math. Makes you wonder how it's going to affect the foundations of mathematics itself. That's a key point. Traditionally, mathematical proof has been all about logic, building arguments step by step using established axioms and theorems. But AI brings something new, inductive reasoning, finding patterns and extrapolating from those patterns."
},
{
"end_time": 8167.841,
"index": 334,
"start_time": 8141.766,
"text": " So could we see a change in how mathematicians approach proof? Could we move toward a way of doing math that's driven by data? It's possible. Some mathematicians are already using AI as a partner in the proving process. AI can help generate potential theorems or find good strategies for tackling conjectures. But others are more cautious, worried that relying too much on AI could make math less rigorous, more prone to errors. It's like with any new tool. There's good and bad."
},
{
"end_time": 8186.391,
"index": 335,
"start_time": 8168.234,
"text": " Finding that balance is important. We need to be aware of the limitations and not rely on AI too much. Right. And as AI becomes more important in math, it's crucial to have open and honest conversations. We need to talk about what AI means, not just for math, but for everything we do. It's not just about the tech. It's about how we choose to use it."
},
{
"end_time": 8216.374,
"index": 336,
"start_time": 8186.596,
"text": " We need to make sure AI helps humanity and the benefits are shared. That's everyone's responsibility. A responsibility that goes way beyond just mathematicians and computer scientists. We need philosophers, ethicists, social scientists, and most importantly, the public. We need all sorts of voices and perspectives to guide us as we go into this uncharted territory. This has been an amazing journey into the world of AI and math. From sleepless nights to those mind-bending manifolds, we've seen how AI is pushing the boundaries of what's possible."
},
{
"end_time": 8241.834,
"index": 337,
"start_time": 8216.374,
"text": " And as we wrap up, we encourage you to keep thinking about these things. What does it really mean for a machine to understand math? How will AI change the way we prove things and make discoveries in math? How can we make sure we're using AI responsibly and ethically in our search for knowledge? These are tough questions, but they're worth asking. The future of mathematics is being shaped right now, and AI is a major player. Thanks for joining us on this deep dive."
},
{
"end_time": 8246.596,
"index": 338,
"start_time": 8242.346,
"text": " We'll catch you next time, ready to explore some other fascinating corner of the universe of knowledge."
},
{
"end_time": 8275.247,
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"start_time": 8248.08,
"text": " New update! Started a substack. 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": 8287.312,
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"start_time": 8275.247,
"text": " Also, thank you to our partner, The Economist."
},
{
"end_time": 8311.954,
"index": 341,
"start_time": 8289.565,
"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": 8336.578,
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"text": " which means that whenever you share on Twitter, say on Facebook or even on Reddit, etc., it shows YouTube, hey, people are talking about this content outside of YouTube, which in turn greatly aids the distribution on YouTube. Thirdly, there's a remarkably active Discord and subreddit for theories of everything where people explicate toes, they disagree respectfully about theories, and build as a community our own toe."
},
{
"end_time": 8359.991,
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"text": " Links to both are in the description. Fourthly, 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. I also read in the comments that, hey, toll listeners also gain from replaying. So how about instead you re-listen on those platforms like iTunes, Spotify, Google Podcasts?"
},
{
"end_time": 8383.677,
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"text": " I'm"
},
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"text": " 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. Either way, your viewership is generosity enough. Thank you so much."
}
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}No transcript available.