Theories of Everything with Curt Jaimungal - Edward Frenkel: Revolutionary Math Proof No One Could Explain...Until Now

Episode Date: August 13, 2024

Ed Frenkel is a renowned mathematician and professor at the University of California, Berkeley, known for his work in representation theory, algebraic geometry, and mathematical physics. He is also the author of the bestselling book Love and Math: The Heart of Hidden Reality, which bridges the gap between mathematics and the broader public. YouTube Episode Link: https://youtu.be/RX1tZv_Nv4Y Become a YouTube Member Here: https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join Patreon: https://patreon.com/curtjaimungal (early access to ad-free audio episodes!) Join TOEmail at https://www.curtjaimungal.org LINKS: - Edward Frenke's book "Love & Math": https://amzn.to/4dB1URv - Edward Frenkel's book “Langlands Correspondence for Loop Groups”*: https://amzn.to/3Am99xX - Edward Frenkel's ebook (PDF with hyperlinks)*: [Langlands Correspondence for Loop Groups (free PDF)](https://math.berkeley.edu/~frenkel/loop.pdf) - Edward Frenkel's Official Website*: [edwardfrenkel.com](https://edwardfrenkel.com) - Edward Frenkel's Twitter*: [@edfrenkel](https://twitter.com/edfrenkel) - Edward Frenkel's YouTube*: [youtube.com/edfrenkel](https://www.youtube.com/edfrenkel) - Edward Frenkel's Instagram*: [@edfrenkel](https://www.instagram.com/edfrenkel) Timestamps: 00:00 - Intro 02:14 - Edward’s Background 07:04 - Robert Langlands 15:01 - Physics vs. Mathematics 34:14 - Unification in Math 45:48 - What Does Math Actually Describe? 01:02:57 - Langlands Program 01:22:08 - Counting Problem 01:25:55 - Harmonic Analysis 01:33:58 - “One Formula Rules Them All” 01:51:58 - The Shimura-Taniyama-Weil Conjecture 01:55:55 - Original Langlands Program 02:01:22 - A Twist: Langlands Dual Group 02:01:55 - Rosetta Stone of Math 02:11:33 - The Pleasure Comes From The Illusion 02:14:28 - Support TOE Support TOE: - Patreon: https://patreon.com/curtjaimungal (early access to ad-free audio episodes!) - Crypto: https://tinyurl.com/cryptoTOE - PayPal: https://tinyurl.com/paypalTOE - TOE Merch: https://tinyurl.com/TOEmerch Follow TOE: - NEW Get my 'Top 10 TOEs' PDF + Weekly Personal Updates: https://www.curtjaimungal.org - Instagram: https://www.instagram.com/theoriesofeverythingpod - TikTok: https://www.tiktok.com/@theoriesofeverything_ - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs - iTunes: https://podcasts.apple.com/ca/podcast/better-left-unsaid-with-curt-jaimungal/id1521758802 - Pandora: https://pdora.co/33b9lfP - Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e - Subreddit r/TheoriesOfEverything: https://reddit.com/r/theoriesofeverything Join this channel to get access to perks: https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join #science #physics #maths Learn more about your ad choices. Visit megaphone.fm/adchoices

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I'm doing great, Kurt. It's great to see you and it's great to be back on the Theories of Everything podcast. Yes. To give some context for people who are wondering what is this particular episode about, the reason we're here is that my good friend, professor of physics, Lucas Cardoso, WhatsApp messaged me a webpage titled, proof of the geometric Langlands conjecture. And then he asked me if this was legitimate so I then WhatsApp my other good friend Namely you professor of mathematics Edward Frankel to find out if it was indeed and you say yes

And moreover we talked about how you're like Forrest Gump and many mathematical achievements Have you peppered in either in the foreground or in the background. The first couple of mathematics. Yes, exactly. So Love and Math covers some of that. That's your popular book and the link to that will be on screen and in the description. And in that book, you provide an accessible introduction to what's one of the most abstract of all the mathematical topics, namely the Langlands conjectures. And since this is a recent monumental result, which is decades in the making, this channel is one of the places where we can go

into the technical details in podcast form instead of overviews. So you brought your trusty iPad and you're gonna walk us through what this result means, how it was achieved and what the future looks like. That's right, yes. So I'm happy to be back. And I also wanna say that from watching your videos

of your interviews, of your conversations on this podcast, Theories of Everything, I feel like your audience is more interested kind of in more in-depth discussion of some fairly sophisticated topics in the mathematics, quantum physics and other areas. So that's why I was actually happy to accept your invitation to talk about this subject, which is quite esoteric and quite sophisticated and quite technical, because I know that there

is a sufficient, at least sufficient number of your viewers, of your audience, who actually dig these kind of discussions, this kind of discussions which perhaps go more, go deeper into more technical aspects than discussions on other podcasts. Yes. So I just want to say from the beginning that this is really a very important achievement indeed.

And like you said, it caps several decades of work by a large group of mathematicians. I have been involved in this since about 1990, so I was very small. So I met actually at the time, I met Vladimir Drenfeld, who was one of the pivotal figures in the subject, a fields medalist and so on, and probably got all the major awards by now. He's a professor at University of Chicago. At that time, this is 1990, we were both at Harvard University visiting and he was actually very interested in my work because actually he was anticipating starting this new kind

of a new part of the Langlands program which became known as a geometric Langlands program. And he thought that my results, my ideas were useful for that. And so then we would meet every day and talk about it. And so then I was like, curious, what is this Langlands program? This is 1990, so there were not so many, it wasn't in the air. Like we weren't talking about this that much, especially it's new, but it was quite removed from my initial area of interest, which was representations of infinite dimensional algebra. And so I had the luck of actually having one of the main figures in the subject teach me about the basics.

And so this is 34 years ago. So that's how many years I have been involved in this. And indeed my work was important in Drinfeld's work with Bellenson, which launched this conjecture, geometric Langlands conjecture, that kind of like flowered recently in this series of works by, I think it's nine mathematicians, but led a team of mathematicians, teamwork, it's really like 500 pages or more, led by Dennis Gaizguri, who is an old friend of mine. We've written like, I counted today 11 papers together, eight of which are very closely related to the subject and in fact are used in the proof.

And there are many other great young mathematicians like Sam Ruskin and Dario Beraldo, whom I have known. I was basically his co-advisor when he was a graduate student here at Berkeley. So in other words, I'm kind of like well positioned to see the big picture and I'm happy to share it with you and with your audience because the subject is quite abstract, like you said, and so one needs some guidance to kind of penetrate through this abstraction, this kind of maybe somewhat obscure notions and concepts, but they are important. And I think that people don't necessarily have to understand the technical stuff precisely,

but just have an idea, have a gist of an idea of what these concepts are, how they relate to each other. Because after all, a lot of people know about quantum physics. They know about entanglement. They know about various other kind of weird aspects of quantum mechanics or quantum field theory. People know a little bit about string theory, for example, and so on, because physicists have done a great job kind of explaining these ideas in down to earth terms. Mathematicians haven't.

Mathematicians haven't as much. So you mentioned my book, Love and Math. One of the motivations for writing this book was exactly to present these ideas in an accessible form to for the general audience. It was published 2013, so about 10 years ago. And so I've always tried to do it. I always tried to share these ideas with the general audience. And I think this is a great opportunity to do this again,

because we are witnessing kind of a landmark achievement in the subject being done in this recent series of work. So I think it's very opportune time to revisit some of the aspects of the Langlands program, to look at the big picture, to talk about some of the concepts and ideas that go into this proof. Because these are concepts and ideas

not only important for the language program, but they are kind of bread and butter of modern mathematics. And so the more of us, the more people are aware of this, these things become coming to, you know, in the kind of in the air, in the conversation. I think the more we will benefit from it. Mathematicians benefit because it helps us also to leave our little office or our little desk and go talk to other people and kind of get maybe it helps us to also get kind of a bigger picture and but also helps other people to understand what these guys are doing,

what kind of ideas are being played with today in modern mathematics. Some of those people you mentioned that you collaborated with, did you ever collaborate with Langlands himself? Yes, I have. Just as an aside, for people who are listening, the name Langlands has been around for so long, decades now, the Langlands program, that it sounds, it would sound like he's no longer around because it's such an historic name, but he's alive and well at the Institute for Advanced Study. So I just want to show you Robert Langlitz here.

So here he is sitting at his office at the Institute for Advanced Study in Princeton. This picture was taken in 1999. Interesting fact, this is the office previously occupied by Albert Einstein. I'm serious. It's not a joke. That's the office occupied by Albert Einstein. I'm serious, it's not a joke. That's the office occupied by Albert Einstein. And in fact, if you look, if you compare the photographs of Langlands at this office, also he's given some talks, there are videos online and so on. With the picture of Einstein, you will see it's the same office. Now, as I mentioned, I have collaborated with

Langlands. We actually wrote a paper together with another mathematician, Baoqiang Guo, who is a Fields Medalist at the University of Chicago. That paper I think we published in 2010 or something. So I've known the English for many years and we talked a lot, but especially between 2008, 2010, we collaborated quite closely. I visited him a number of times at the Institute for Advanced Study. I spent a lot of time at this office, sometimes arguing with him. He's a tough client, you know, I have to say.

He doesn't take any bullshit. So sometimes we kind of went a little bit, you know, looked our horns. Can you give an example? Well, you know, he would be sometimes skeptical about things. And you know, I was kind of a, I was, I guess I was young, I was younger and I felt a little bit like I had to prove myself. Occasionally I would push back and say, no, that is correct. You don't understand.

And who ended up being correct? Well, I would be biased, but I think. Yes, two sides to every story. I think that I was, I held my own and well, we did complete the paper and I'm proud of it. It was published, like I said, about 2010. And, and, you know, so obviously he accepted some of it, but his comments were always on point. I have to say also, all jokes aside, you know, that even though he was already like in his seventies and so on, but he was still sharp and kind of like,

you know, quick on his toes and like, yeah, what is this about? He would pick up exactly the things where things were not quite fitting together, for example. So anyway, that was a lot of fun. So that's who he is. So he's a mathematician, Canadian born by the way, Canadian born from British Columbia.

And it's a beautiful story actually, because he was, you know, lived in a small town. His father was a carpenter. And for all he knew, he would just inherit family business and, you know, make the window seals and install window seals and stuff like that. But there was a teacher, there was a mathematics teacher at his school

who inspired him to go to university. He went to UBC, University of British Columbia. He wasn't planning to. This is after the war, you know, he was like, okay, well, this is my town. This is where I'm going to live here and, you know, do the stuff that my father is doing. But his teacher, this is all according to like, he wrote a couple of biographical sketches, notes for when he received some major awards, you know, so that's where I picked this up. He said something like, his teacher shamed him in front of the classroom full of kids,

his classmates saying that you have to go to university, you know, it would be unforgivable if you don't go to university in front of other kids. And he was like, okay, well, if he says so, maybe I should. And so he enrolled in UBC, University of British Columbia. And he describes how he felt early, you know, first year, second year, he felt kind of inferior to the kids from Europe who seem to know a lot more stuff than he did. He studied in this provincial town, small town, and he felt a little bit inferior to them. But I think it gave him a little bit of fuel like, I'm going to show them guys.

And so of course, years later, he comes up with these ideas, which became known as the Langlands program. So how did it happen? So this was in late 60s. In fact, 1967 is the year when he formulated his ideas for the first time in writing and it was in a letter. In those days there was no email obviously. So sometimes mail, obviously. So sometimes mathematicians would write things or type things and either send it by mail or if somebody was close enough, as was the case, Andre Wey was a great mathematician who was in Princeton at the same time as Langlands, and Langlands gave him this bunch of notes and they were actually handwritten. And there's a beautiful cover page which I'll show you later. But Andre Wey could not make you understand his handwriting.

So actually he asked him to type it, to type the letters. Eventually he typed it up and I think it's like 20 or 30 pages, still available, you can find it online, where he shared these ideas. It's not clear that Andre Wey, who was himself a towering figure in mathematics, a very important mathematician of the 20th century, a professor at the Institute for Advanced Study. At that time, Langlands was not a professor at the Institute for Advanced Study, he was at Princeton University, I believe. But they were in the same town, in the same place, in Princeton. I'm not sure Andrei actually understood what he wrote, but it was a great opportunity for Langlands to kind of like put these things organized on paper.

And then the whole thing just launched because this letter was shared. People made photocopies, Xerox copies at that time. And a lot of people got interested in this stuff. And that's how Lying Glass program was launched. So now, fast forward, 67, so fast forward, 2007, so 40 years, I wrote this book, I wrote this book before Love and Math, I wrote this book, Lying Glass Correspondence for Look Groups.

Correspondence doesn't mean like sending letters to each other, even though letters are very important in this story. Correspondence here means a kind of like one-to-one correspondence, a relation between two kinds of objects. And we will talk about this later on. And at very first sentence in the preface to this book, this was published by Cambridge university press in 2007. And the very first sentence I wrote is that the Langlands program has emerged in recent years as a blueprint for a grand unified theory of mathematics. I just wrote it, you know. So it was a bit of a showmanship, you know, like, okay, well, that's, you know, kind of like to present it in the most favorable light.

But interestingly enough, this expression caught on. And now I'm quoted in all kinds of places as one who said that this is a grand unified theory of mathematics. So now this gave me an idea for the opening. So because I know that you are interested, well, first of all, your podcast is called Theories of Everything. So obviously you're interested in this general idea. It's another question whether what we mean by a theory of everything and so on.

But obviously it's something that is very much on the mind of a lot of physicists these days, right? And actually not on these days, but for many decades. And so in physics, that's grand unified theory, theory of everything is something that people talk about all the time, right? So then the question then is, okay, so what about mathematics? Do we have a grand unified theory in any sense? And so in what sense the Langlands program is a grand unified theory? And so I thought I would just take a moment to discuss the difference between physics and mathematics.

Introducing Tim's new infuser energy beverages made with natural caffeine. They come in two refreshing flavors, blackberry yuzu and mango starfruit. Try them today only at Tim's. At participating Tim's restaurants in Canada for a limited time. I think it's a very essential point which is actually missed by a lot of people because somehow there is sort of like people kind of merge the two and kind of like don't differentiate them enough. People as in the general public or mathematicians? Even the ones, well I would say mathematicians faces do know, but in general public, it's this line between the two is not clearly marked somehow in the minds of most people.

I think it's important to talk about this, to understand. It's essential because this way we can understand all kinds of recent controversies about string theory. Because as you and I talked about about a year ago on your podcast, in my opinion, string theory has been great for mathematics, but it has not been great for physics. So see, there is a difference between the two because the original promise of string theory was to give us the unified theory of this universe and it failed spectacularly doing so. But it's not all bad. In other words, yes, it has helped mathematicians to come up with some

interesting ideas and it's pushed mathematics. So if we blur the lines between math and physics, we will say string theory is a success, right? So that's why it's very important to understand that mathematicians and physicists actually have different goals, different tools, and there are different expectations, different responsibilities if you will. So what is the difference? Let's talk about this. Yeah? Sure. Okay. So let me go back to the beginning of my slides. And so I want to talk about the unification because, you know, what does the idea of unification in mathematics actually mean? And so I want to start with a famous quote by Galileo.

At the times of Galileo, the two things were really close to each other, natural sciences and mathematics. And Galileo famously wrote, the book of nature is written in the language of mathematics. And he continued, its characters are the characters in the language are triangles, circles and other geometric figures without which it's impossible to understand a single word, without which one is left wandering in a dark labyrinth. So in other words, Mathematics is a language of nature. So now let's talk about physics.

In physics we have this nomenclature. We have Unified Field Theory, which actually started with early attempts by Albert Einstein to unify general relativity, which described his theory describing the force of gravity. And electromagnetism, there were only two forces of nature known at the time. Then eventually physicists found that there are two other forces, the weak and strong, the nuclear forces, subatomic forces. And we have now the standard model, which describes three out of four known forces of nature, electromagnetic, weak and strong with spectacular

experimental success. But of course, a lot of questions remain. For example, it has a Lagrangian of standard model has something like 19 parameters. A lot of people not satisfied with it. They feel that there are much deeper theories yet to be discovered, right? And so then grand unified theory in physics refers to an attempt to merge these forces into a single unified force, including gravity. So for now we don't have a

quantum field theory of gravity. String theory promised us to give one, but they failed in doing so during the period of 40 years. And it looks like, it doesn't look very good. If you had to bet, I think you shouldn't put your money on this. It's investment advice. Don't put money. Don't put any money on this. Maybe short it, you know, like in the markets, you could buy stock, we can short.

So shorting it probably still a good idea, but I think the stock has come down quite a lot because more and more people are kind of wake up from this dream and kind of like, look at the reality of the situation and say, look, you know, we have to move on to other things. All right. And so- Just a moment. Do you say that with any reservation

because you've published in string theory in the past or you've published with one of the most famous string theorists? That's so speaking of Forrest Gump, right? So not only did I publish with Langlands and other people that I've mentioned, I also published a paper with Ed Whitten, okay? And in 2007 2007 around the time

when this book that I mentioned was published. But see, this is the thing. With Whitten, we published a paper about mathematical ideas involved in string theory. And actually it was not even string theory, but more like quantum field theory in higher dimensions, you see. Well, in higher dimensions, it's not a proper term. You see, string theory kind of turns things High dimensions is not a proper term. You see string theory kind of turns things upside down because in string theory, the field theory which is involved is a two dimensional field theory because you're considering propagation of strings, right?

And what is a string is like a circle. And when it moves, it sweeps an area. So it sweeps something two dimensional. So in effect, you're studying embeddings of what we'll discuss later called, these objects called Riemann surfaces. Think of the surface of a donut or of a Danish pastry or a sphere. Their maps or embeddings into the space time.

So the space time is on the receiving end of these maps, but the theory itself is defined on the surface, you see, on the Riemann surface. That's weird because you're taking the two-dimensional field theory in which the target space is a space-time. Normally we take the theory is defined on the space-time. So the roles kind of switch, which is kind of like, it's really cool if you think about it. But string theory is not just a theory of quantum field theory on a Riemann surface, because in string theory, once you study these models, the quantum field theory is defined on surfaces. That's not enough. That's the first step.

The second step is to integrate over all possible Riemann surfaces. And that involves summation over all possible topological types like sphere or surface of a donut and so on. The parameter, the topological parameter is called a genus. So sphere is of genus zero. The torus is of genus one. The Danish pastry with two holes is of genus two. The pretzel is of genus three. So you have to sum up the results corresponding to each topological type of a Riemann surface. And then for each of them, you have to sum up over all possible complex structures, basically. In fact, you have to sum

up over all the metrics, but because the theory is what's called conformally invariant, you actually end up with a finite dimensional integral over what's called conformity invariant, you actually end up with a finite dimensional integral over what's called the modular space of complex structures on the Riemann surface. So string theory is not just a two-dimensional quantum field theory defined on a specific Riemann surface. So if you imagine some two-dimensional world, you know, like some bugs, let's say you have like a globe in your apartment or in your house and you have a little spider who lives on that globe. So that spider's space, well, it's more space than space time, but okay, let's forgive me for this, for this difference.

Let's assume that the space time of the spider is a sphere, is the surface of the globe. So then the quantum theory that the spider would observe if it were indeed his or her space time would be a two dimensional theory defined on a specific human surface, namely the surface of this globe. You see what I mean? Yes. But that's not string theory yet. String theory is when you can see the spiders living on all possible human surfaces and

you sum up the every calculation you sum up over what happens for across all of them. So that's string theory. So that's a very simple explanation of the difference between quantum field theory and string theory. So first of all, quantum field theory is not defined on the space time. Rather space time is on the receiving end of maps from a two-dimensional surface to the space time. So the theory effectively is two dimensional. Then the theory is what's called conformal invariant, which allows, which gives us a possibility to actually sum up over all possible choices over human surface.

And it's when you sum up, you get amplitudes from string theory, you see. So that's the, and you have to admit that it's a very cool idea. It's a very cool idea. You turn things upside down, kind of. That your space time becomes the target space of the theory.

It's not the space, the ground of the theory. It's a kind of additional ingredient of the theory. The theory itself is defined on the human surfaces, but it's sort of like, it's a cool idea, but at the same time, it's actually kind of like precipitates its own demise, because then the question is which space-time do you have to choose? You see? So it does not lock you because you don't define your theory on your space-time that is given to you, which is this our universe, which has

three spatial and one time dimension, but it becomes a parameter of your theory. But the point is that it's an essential condition is that the theory be conformally invariant on the Riemann surfaces, which means that if you rescale your metrics locally, then you will get the same result. And this enables you to go from an integration over some infinite dimensional spaces, which is basically intractable,

to actually finite dimensional spaces. Instead of integrating all metrics, what's called metrics on the Riemann surface, you go to the integral over all complex structures, which is a finite dimensional manifold, so that the integral in principle could be computed, even though there are all kinds of singularities that need to be regularized,

which no one, as far as I know, has been able to do in general, actually. But at least there is a path to doing it. But the problem is that your space-time is a kind of an external thing that you put into the theory. And the question is, how do you differentiate between them? How do you say, oh, our space-time has to be this?

And so one condition is that the theory has to be conformally invariant, and that means that it's Kalabiau. So you probably heard this said many times that the target space of string theory is a Kalabiau manifold. Kalabiau manifold is one for which the corresponding two-dimensional theory of maps from your Riemann surface, say a donut, surface of a donut, a sphere and so on, to your space-time is conformally invariant. But the problem is that there are too many Calabi-Yaus and the problem is the string theory is only when defined when this manifold is 10-dimensional.

Right. Super string theory to be precise. Super string theory. For string theory, it would have to be 26 dimensional. So it's a much bigger gap from what we observe to what string theory allows you to work with. So the first string revolution in the 1980s was going from bosonic string theory

to supersymmetric string theory, where you have a sort of a balance between bosons and fermions, which enable to bring the dimension down to 10. And 10 is much closer to 4 than 26. So you have six extra dimensions. And then physicists say, oh, okay, well, these six dimensions that curl onto this little Calabi-Yau manifold, but which one? And the problem is nobody has been able to find. And the problem is also that it's not just a single one, but

it's moving also with the dynamics of the theory, this Calabi-Yau manifold, this extra six dimensions, is also changing. And nobody was able to find, and those changes lead to some long-range forces which nobody has been able to observe. So therefore it's like immediately in contradiction with experiment, not to mention that it's supersymmetric. So you have to, and we don't observe supersymmetry in this universe. So anyway, that was a long kind of a side digression on string theory.

Just to show you what the problem is, if you're a physicist, this is a major issue because your job, your job, you only have one job if you're a physicist. Yes. Well, and I mean quantum physics. Obviously there are other areas of physics like solid state physics, statistical physics and so on, where models are much more diverse. But if you are quantum physicists who works, you know, on finding the theory of everything or you're grinding your fight theorem. If you're a high energy physicist in fundamental physics. If you're a high energy physicist, you only have one job, which is to describe this damn universe.

That's your job. You're not interested in describing all possible universes, 10 dimensional and so on. But mathematicians are interested in all of them. So that's the difference between us. That's the major difference. As a mathematician, I love 10 dimensional manifolds or 10 dimensional spaces, if you will. Yes, exactly.

They're just as good. They're my children, just like four dimensional space times. I don't differentiate between them. I love all of them, you see. Or even infinite dimensional spaces. So to put it briefly, it's mathematician is interested in a space of any dimension. A high energy physicist is only interested in four dimensional spaces. And not just some generic spaces, but the ones which are realized in this universe.

Now there may be some other universes which are ten dimensional. And maybe this theory describes those universes. Okay, first of all, how do you find out if it's true or not? Do you have like pick up a phone and they talk to some aliens who live in 10 dimensional and say, congratulations. You have found the grand unified theory of our universe. Congratulations. But what is good is it for us?

Now what, now I have to also add, there is also another aspect that by doing this more general theories, you can actually stumble upon some ideas which you may find it more difficult to understand in the realistic theory. And then you can try to adopt it, to tweak it, to apply to this universe. So for instance lower dimensional theories are usually simpler than higher dimensional theories. So for instance two dimensional theories have been a great playground for physicists to try to develop ideas or some three dimensional theories as well.

There's a famous work by Alexander Polikov, a brilliant Russian Soviet physicist about three dimensions, instantons and three dimensional gauge theories, which became classic because it gave you mechanism of how instantons contribute to creation functions, which we still don't understand how to apply in four dimensions. And if we could, we would be able to solve the confinement problem, trying to understand why quarks cannot be separated as you move them apart, which is a feature of four-dimensional gauge theory. This is just one example.

There's this phenomenon called instantons, which is understood much better in three dimensions and in two dimensions than it is in four dimensions. But this is that province, that area of research where you try different models in other dimensions is called mathematical physics proper, really. That's the overlap. If you think of a Venn diagram, I said Zen diagram, well, Venn diagram, okay. Something else. Venn diagram is something else, probably. So if you look at the Venn diagram, okay? Something else. Venn diagram is something else, probably. So if you look at the Venn diagram of mathematics and physics, yes, there is this overlap. And

that's called mathematical physics. So all my life I was very interested in that because, you know, I actually, as a kid, I was actually very interested in high energy physics and quarks and so on. And then I learned that actually you have to understand mathematics to even speak about those theories. And as I delve deeper and deeper into math, I realized that I actually love mathematics proper. But also occasionally- It was SU3 that got you inspired. SU3, that's right.

The quirks, the description of elementary particles in terms of quirks, which goes back to Mori Gelman and this Russian mathematician, Zweig, as well. So anyway, that's what my work with Whitten was kind of in mathematical physics because it was about understanding certain models which are closely related to quantum field theory and potentially string theory as well, but it doesn't, for example, those models are supersymmetric. The geometric Langlands conjecture that Witten worked on is in four dimensions though. Yes, in a sense, yes. But, yes, true. But supersymmetric theories, that's what I'm trying to say. People think the general public thinks of supersymmetry as just one supersymmetry, but there are extensions.

There are perversions of supersymmetry, n equals two, n equals three. That's right. So typically we study n equals one, n equals two, n equals four. And that's called maximal supersymmetry. So supersymmetry, by the way, is a very simple idea. It's just that you have one of the requirements of quantum field theory is the invariance under the Poincare group. The Poincare group combines the Lorentz group, which is kind of all rotations or pseudo-rotations in Minkowski space and translations. So, Poincare group is built as what we call semi-direct product of the Lorentz group and the group of translations. In supersymmetric and everything in your theory has to be invariant under this action. So for instance the space of fields, the space of states has to be what's called the representation

of this group, of the Poincare group. In a supersymmetric theory this representation has to be extended to a bigger group, but in fact not a group, it's called supergroup. Because it has both bosonic degrees of freedom and fermionic degrees of freedom. The bosonic degrees of freedom will stay the same, it will be the same Poincare group, but there will be some fermionic transformations, the kind of weird little transformations, they're kind of like little shifts, they are not even bona fide transformations in the ordinary way. And so this way you get an enlarged supergroup which, depending on its size, is classified as n equal 1, n equal 2, n equal 3, n equal 4. But the biggest one you can get, given all the requirements of quantum field theory,

is n equal 4. And that's the theory we're talking about. That's the theory which exhibits what's called electromagnetic duality, gauge theory, super Young-Mills or super gauge theory, n equal four in four dimensions exhibits something that's closely connected to the geometrical angle of correspondence, which is called electromagnetic duality. So anyway, so this was just to put things in perspective of how similar phenomena actually arise in both mathematics and physics, although today this phenomenon that we're talking about, the Langlands correspondence

or Langlands program or Langlands duality, Langlands conjecture, can use all these terms, they actually, so far we only know how to apply them to supersymmetric theories. So therefore, it's not physics proper, it's mathematical physics. You see what I mean? How do stop losses work on Kraken? Let's say I have a birthday party on Wednesday night, but an important meeting Thursday morning. So, sensible me, pre-books a taxi for 10 p.m. with alerts. Voila! I won't be getting carried away and staying out till 2. That's stop loss orders on Kraken, an easy way to plan ahead.

Go to kraken.com and see what crypto can be. Yes. Okay. But now let's go back to this idea of unification. So in physics, in theoretical physics, proper, not mathematical physics, like string theory, but real physics. Okay. So I'm not trying to offend anybody.

I'm just saying real physics means, and I'm not a physicist, so I'm kind of looking from outside. I have no horse in this race, to be honest with you. I'm just like, I'm kind of independent observer. And all I know is theoretical physics, you have to describe this universe. And I don't think anyone would argue with that. So then in the perspective of theoretical physics, there are these notions of standard model grand unified theory and theory of everything, TOE.

Okay. That's the one which unifies all forces. So I guess, yes, grand unified theory. That last one sounds familiar. It sounds familiar, right? I mean, so, but I have to say, GUT, grand unified theory, usually references kind of a better understanding of the standard model. So it's still three forces.

So TOE means that you include gravity. I think that's the nomenclature. Okay. And so now this is a beautiful quote from Einstein, which is something he said during his Nobel lecture when he received the Nobel Prize in 1923. He said, the intellect seeking after an integrated theory cannot rest content with the assumption that there exist two distinct fields totally independent from each other by their nature. So that's the impulse, which is a very natural human impulse. You want to simplify things, you want to reduce things to something, to believe that at a deeper level there is only one phenomenon that manifests itself in many different ways, right?

Well to be fair, he does say if you're seeking after an integrated theory, so by saying integrated theory, he already has the assumption that you can't have two distinct parts. That's right. So you could say that you could be like, because there's a philosopher named Nancy Cartwright who says there are different models of the world and one is the integrated approach. Another one is a patchwork approach, where it's akin to open sets that overlap and in the overlapping region you just have to have compatibility, but they could be incompatible outside of that.

And then furthermore, you could have regions where there is no law. No, I agree. I agree. Absolutely agree. And in fact, in mathematics, it's more like that. In mathematics, we do not seek. So that was my next point. In mathematics, we do not seek, so that was my next point.

In mathematics, we do not seek to have an integrated theory of everything. We do not actually. I didn't mean to preempt your point. So yes, so in a sense, what you described is more like how mathematicians approach it. But you see, a lot of it also has a social element. So when we brought up, and I can speak for myself because I studied mathematics since I was a kid, and I went through school, graduate school and so on. And so I can say that never ever at any moment were my teachers telling

me, Edward, find the grand unified theory of mathematics. It was never an issue. First of all, I think it's impossible. Mathematics is just so diverse. There's just so many different subjects. I like to talk about them. this is another analogy I used in love and math, the continents of mathematics. So there is like number theory, there is harmonic analysis, there's functional analysis, there's geometry, there is algebra and so on. And they are all connected somehow, but it's not like one field will subsume all other fields.

We don't have that idea. But physicists, I've spoken to a lot of my colleagues, physicists, oh yes, when they are studying, there is a lot of premium on this idea that maybe I'll be the one to come up with this theory of everything. So I'm hopeful that the next generation of physicists will see through this and understand that in some ways it gives you kind of a kick, it makes it more fun, it's more competitive, but also there is a great downside. And quite honestly, I think that the debacle with string theory is in many ways the result of this sort of attempt to subjugate everything to what you already know, you see. And then discard all other ideas

and say that they are not serious. Their ideas are propagated by amateurs and stuff like that. Instead of looking in the mirror and kind of like understanding what am I doing here? And on what ground am I to claim that the theory I'm working on is the only game in town, which was an expression used by many string theorists for a long time. Still is.

So, well, you know, one day, you know, it's like reality intrudes. So like you can pretend that it's not there but the others will look at you in the face every day and will say Okay, Edward. When are you going to? Accept reality and one day you will have to or you will die not accepting it Then other people will come and do it for you. Just a moment What why do you think because the string theorists will say that look in the past in physics when we've been on to something It did produce mathematical results as well. Now look in string theory while it's not producing physical results at least not experimentally testable in our range

currently it's producing plenty of mathematical results and we could use that as some proxy indication that we're on to the right physics. What do you say to that? I disagree I just explained the difference between mathematics and physics so what they're what string theory does is it where strength theory has been positive is in the area of mathematical physics because it points us to some very interesting phenomena in a whole range of theories which have nothing to do with our universe, you see. And that's useful for many reasons. Number one, it helps pure mathematics. And strength theory has been tremendously helpful to mathematicians.

Has led us to some paths which we probably would not have discovered in the 21st century. So in other words, it was some kind of mathematics just fell on mathematicians' lap in the 20th, 21st century because it was inspired by string theory. Yes. Also, these ideas, these theories with ten-dimensional spacetime, they are not useless at all. They are useful because studying those theories, you observe some phenomena which can help you eventually to come up with realistic theory

of this universe. There are these things like electromagnetic duality which we are going to talk about, which is connected to the Langlands program. There are things like ADS-CFT which are absolutely beautiful mathematically, but not only mathematically because they are the phenomena in theories which are in the same class as quantum field theory describing this universe. You see, they're different because they're different dimensions, they're supersymmetric and so on.

So they're not quite fitting the clothes that you have to put on. But you have to put on the clothes of your universe, otherwise it's not physics. You see, that's a very simple thing. Yes, it's true that you could say that in the first 10 years of development of string theory, that its success in mathematics gives you a kind of confidence that you should keep going and working on it because there must be something there.

But after 40 years, after 40, actually more than 40 now, because we're talking about 40, I would say, yeah, 85 would be the first string revolution. Since the first super string revolution. And then they saw the cancellation, the Green-Schwarz cancellation of anomalies. That was brilliant, a brilliant work that gave people, and I understand the human element of it. Of course you get carried away.

Of course you get excited. Of course you want to run around like, who was it? Archimedes who jumped out of a bathtub when they understood how the bodies float. Of course you want to jump and run around naked and say, guys, it's amazing. I have come up with this incredible discovery. I understand it. And of course, please do that, but not for 40 years. After 10, 20 years.

And I think the crucial moment was about 2012 when LHC was fully online and it was clear that there's no sight of supersymmetric, super partners. That was the time for reckoning in the field. That was the time for kind of a serious conversation where the adults, the elders of the theory, should have talked, should have come out and said, look, you know, it's not working out and allow other ideas to come in. That's the problem. The problem is not just, yeah, please, of course keep working on it. But when you deny resources to everybody else, then it becomes a bigger issue.

Now I don't want to spend the whole conversation talking about string theory. I think it's important to talk about it because I am on record on your podcast a year ago on the heels of my keynote talk, keynote lecture, what was called challenge talk at the big strength theory conference in at the perimeter Institute near Toronto, I guess near where you are in July of last year where I saw up close this community and I saw up close what's going on and I was very disturbed by it. So then I went on your podcast and I spoke about the failure, what I call the failure of the original promise of string theory.

So in today's conversation, I would like to bring up the connection between the Langlands correspondence, which is the main or Langlands program, which is the main topic of our conversation and electromagnetic dualities and supersymmetric Young-Mills theories, four-dimensional super Young-Mills theories. So some people, some viewers may be puzzled by this. So how can this guy be criticizing string theory and at the same time he's saying that, you know, that it's very interesting to study this supersymmetric theories. So that's why I took some time to explain this major difference

in the approaches of mathematicians and physicists as well as mathematical physicists. What is appropriate, what's not appropriate and where you have responsibility as a physicist to make sure that your results actually apply to this universe and not claim something that is not there. Because a lot of, you know, I'm not going to call names, but you can easily find quotes where people saying string theory has unified Einstein's theory of gravity with quantum field theory. Yes, it has in 10 dimensions. And even then it's still incomplete because of various issues which have not been addressed because it's only defined perturbatively and so on.

But let's just say that, yes, in 10 dimensions they have combined, but they don't put that in the sentence. They say we have combined, you see. And it's a lie if you just say it this way, quite frankly. And like the first time you say it, it's kind of cute, but like after 40 years, after 40 years, it's a lie if you keep saying that. And if you keep giving yourself a plus plus plus as one of the major

figures in string theory did in the recent public event, a plus plus plus is how he is a great, is a great. Yeah. Wow. Interesting. Wow. A plus plus plus, you know, I usually, when I

teach, I give a plus, you know, kind of like, okay, let's know be realistic. Like never give A++, especially for someone who actually couldn't solve any problem in the exam, you know, but that's just me. So I wanted to explain this difference. It's important in mathematics, we don't have this issue. Any consistent mathematical theory is valid. It doesn't have to be in four dimensional space time. It can be 10 dimensional space time.

That's why mathematicians love string theory. But it doesn't mean that physicists should love it. Understood. Okay. All right. Let's move on. So, there is another aspect of it, which is that physical theories get updated. And mathematical theories appear to be objective, necessary, and timeless. How do you explain that? That's a mystery to me. So what does mathematics actually describe?

And the point is that there are a lot of concepts and ideas in mathematics which a priori have nothing to do with the physical universe, with nature. In other words, they don't fit into the four-dimensional space-time, you see. So for instance, in this geometric languagelands correspondence, we talk about sheaves, not even functions, which are kind of like, okay, function is something that is rooted deeply in physical reality. For instance, think about temperature. At every point in the room, there is a certain number, which is a temperature at that room or barometric pressure. And so to every point, you assign a number and that's what mathematician call a function. So functions are very much, you know, bread and butter of physics, but sheaves, like who,

when was last time you saw a sheaf on the street? So that's just one example or infinite dimensional hybrid spaces or periodic numbers like this numerical systems, this is a theoretical numerical systems that are bread and butter of mathematics and yet we don't systems that are bread and butter of mathematics and yet we don't observe them in the physical reality. So then if so, what could unification mean for mathematics, you see? And so my favorite example is that I say if Leo Tolstoy did not write Anna Karenina, nobody else would have written exactly the same novel. But if Pythagoras did not live or did not discover his

theorem, we would still have Pythagoras' theorem. It would still be a squared plus b squared equals c squared and not equals c cubed. And it stays the same for the last 2,500 years. And by the way, it was actually, as it turns out, was discovered by others in other places like in Babylon, in Mesopotamia, in China, even before Pythagoras. And it hasn't changed. Whereas for instance, our theory of gravity got updated a hundred years ago, going from Newtonian theory of gravity to the Einstein theory of gravity. So there's something very special about mathematics. Some mathematicians believe that there is this platonic world of mathematical ideas named after the Greek philosopher Plato, who I must say was really following in the

footsteps of Pythagoras. These ideas really go back to Pythagoras in my opinion. And so the question then is usually posed as do we discover mathematics or do we invent it? And I used to be squarely on the side of like, it is discovered. In other words, there is this ideal world of mathematical objects and concepts and we just go there somehow, almost like ESP, like extra sensory perception, like mathematicians go there to this ideal world and discover and bring back the fruits from that world. Mind you, no smaller figure than Charles Darwin actually wrote that mathematicians are endowed with an extra sense. He actually wrote that.

Closer to the end of his life, he wrote that he regretted not to have studied mathematics more as a young adult or as a child because he wrote, and I think it's a direct quote, mathematicians are endowed with an extra sense. So there is this perception, and I have to say as a working mathematician, you do feel it. So a friend of mine recently asked me, like how do you experience it?

How do you experience the mathematical discovery? And so the best I could come up with is that, you know, imagine like there is this deep fog. So you know there's something there, but you don't know what. And then occasionally some part of it, the mist kind of starts dissolving and you see the contours, the contours of the trees or a castle or some exotic animals. And then maybe it will close again.

But that's the sense that you have. You don't feel that you invent something. you don't feel that you come up with something just on your own. You feel like it's always been there, but obscured by our inability to see it, was invisible. So there is this sense. But now I kind of also feel that Matmanx is a human activity. And therefore I kind of like, I'm reluctant these days to just say, yeah, it's discovered, it's just out there.

Because who discovers these mathematical ideas? Human mathematicians, right? And so, as far as we know, maybe some other animals too. Apparently, some animals can count up to a certain number, some can't and so on. But for now, all the mathematics that we know came from human beings. So there's something to be said about that. And so I'm kind of like, now I feel it's a mystery. It's one of those questions where it's neither and both.

Like an electron is both a particle and a wave. Neither and both. Kind of like that. But it's a very interesting thing to think about and contemplate. So, Kurt Gödel actually, the great logician, the greatest logician of all time, as far as I'm concerned, wrote that mathematical ideas form an objective reality of their own, which we cannot create or change, but only perceive and describe. Of course, this is very close to the conversation about artificial intelligence and so on, but I'm not going to go there because we want to get to the language program. So well,

I have this one example of the universality of mathematics. I don't know if we should go there or should move more faster to the... Do you want to say it quickly? Yeah, so there is this idea about numbers. So just to give you a sense of how weird mathematics is, weird in a good way. I want to pose this question. Is there only one mathematics or there are like different, imagine there is another universe, not another universe, like another exoplanet, like there is another civilization and they have their own mathematics. What if we meet and we start comparing our notes? Is it possible that their mathematics is totally different?

And so the traditional argument that it may be so goes like this. Imagine that this new civilization has like many different aliens and then you can imagine all kinds of pictures of these aliens. So then they are like us because we see other people kind of like similar to us. So naturally we start counting things. So numbers arise from the idea that there are many things which are kind of similar to each other.

But what if you have like a Solaris type intelligence? Solaris was a novel by Stanislaw Leom, a great Polish science fiction writer, and it's been made into film by Andrzej Tarkowski. And then also it was a Hollywood remake of this movie, which I highly recommend the Tarkowski version where the intelligence was this is one thing, was this whole planet was intelligent. So then the argument goes, okay, for that intelligence, there's no reason to come up with numbers because it's only one of it, they only observe one of it.

There's no need for it to develop the idea of numbers because it cannot count. There are no essential things to count. So, right? But I have an answer to that and I want to explain how Solaris-like intelligence would actually discover numbers. Not by counting the way we usually do or the way we teach our kids. Which by the way, I think it's a very, there are some subtle points there that we are not properly teaching additional multiplication, but that's another story. Let me not go there at the moment, but I can talk about it later.

So here is an alternative way to discover numbers. Here's an alternative way to discover numbers, which is more clean in some sense and kind of devoid of the deficiencies of the traditional way of teaching kids numbers of counting. Namely we can discover them through winding. You see, so this comes from what's called topology. So this I have a flaws here to demonstrate. So the point is that I can take, I put it on my finger and then I can go around.

I can go once, I can go twice, I can go three times, four times and so on. So if it's an infinitely long floss, in principle I can wind this string any number of times. So this way I represent all natural numbers, one, two, three, four and so on. But also observe that I can represent negative numbers if I go in the opposite direction. You see? Instead of going this way, I can go this way. And so this way I can actually introduce all integers without counting, by realizing that the topological structure, that if I cannot unwind things,

because to unwind, I would have to move the strings through my finger. So if I, if I'm not allowed to move my strings through my finger, then there is a distinction between winding the string once or twice, three times or a negative number. In the Solaris case, what would be the analogy? The solar prominences? So the analogy in the Solaris case is that instead of wrapping a circle onto itself, you can think of my finger. First of all, you don't need the whole finger, you can just look at the section of the finger. It's kind of the intersection of my finger with a plane. And then you can look at this picture that I put out on the slide,

where basically the circle is just wrapping onto itself, like many times, right? And so, but likewise, the sphere can wrap onto itself many times. This is much harder to imagine because you can easily see it from a four-dimensional perspective, but in 3D it's very hard to imagine a sphere wrapping into itself. But you kind of can grasp it by analogy that just like a circle wrapping on itself gives us an invariant, what might be called an invariant, topological invariant, which is the winding number. Likewise, there is an invariant of a map between a sphere, from a sphere to a sphere.

And the coverings of sphere by sphere is also labeled by all integers, because also there are two orientations possible, so you get both positive numbers, negative numbers. This is a rough rendering of what this might look like. This is an example of what's called a homotopy group. So in fact here this corresponds to the pi1, the fundamental group, the first homotopy group.

And here we're talking about pi2. Every time you have a sphere you're talking about pi2, the second homotopy group. When you have a circle you're talking about pi1 and that's the first homotopy group also known as the fundamental group. Anyway, this is just a kind is just something to realize how many interesting aspects of mathematics there are, which go far beyond what we use ordinarily, like counting. What it shows is that the same concept can actually arise from different continents of mathematics. In the first approach, you get to numbers through counting, so kind of from the point

of view of number theory, right? But in the second approach… It seems to be evidence for the discovery of mathematics. Right. Discovery, but this is a baby version of unification. That's what I mean by unification of mathematics, that numbers, natural numbers or whole numbers actually live on the intersection of two fields, number theory and topology, you see. You can discover the same numbers from these two different points of view.

And to me, that's an example of unification. In other words, you're not trying to say that topology is subsumed by number theory or the other way around, but you observe that there are certain phenomena which are reflected in both fields or which are manifested in both fields. And that's why it's prized in mathematics to have this type of situation, because it gives you a different perspective. It's like you're looking at the same object but from a different angle. And this is where a lot of great discoveries are made when you're able to realize the same thing from these two different angles or more, two or three or more different angles. So for me, mathematics is like a giant

jigsaw puzzle where you're kind of trying to build this picture from the small pieces without really knowing what this final picture is going to look like. And so when you solve a jigsaw puzzle with your friends, usually the strategy is to try to build like small islands. So you find a few pieces that fit together and you try to enlarge them and your friends do the same. And so then at some point you kind of build several islands in this picture, right? Which kind of like looks consistent. But the greatest

advance happens when you know how to fit different islands together when you play juke. And it is like that. That's what the Langlands program comes in. So Langlands program is like this, is a set of ideas which suggest how different continents could fit together. That's kind of like a brief summary of what it's about. Great. So that's like, so then what are the continents that we're talking about? And so you have number theory, which is obvious. You have harmonic analysis, which is this very important idea in mathematics, which

is that you can decompose different signals as a superposition, a signal that you can decompose different signals as a superposition, a signal, you can decompose a signal as a superposition of some basic signals and basic signals are given by what's called harmonics that they have frequencies which are multiples of each other. Oh okay so for people who are familiar with the terms Fourier series this would be an example of that. Fourier series, what I'm referencing right now is the idea of Fourier series where you can write

a given function as a combination, sometimes infinite, of sine and cosine functions. But not just two sine x cosine x, but sine nx and cosine nx, where n is a whole number, is an integer. So different sine functions have graphs which look like this, but as you increase n, it becomes more and more squished. And so that corresponds to a higher and higher pitch in the sound. And so we know that in the chromatic scale, essentially the frequencies are different by rational numbers.

Not quite, there is a gap, there is this wolf tone and so on. But roughly speaking, you can generate them from going up an octave where the frequency doubles and going to the Pythagorean fifth, where the frequency gets multiplied by approximately three halves. And by using this, you can then realize other nodes with a small gap at the end somewhere by rational number. But if it's by rational, it means you can always find kind of a smallest denominator frequency, so that every other frequency is a multiple of it.

So that brings you into that framework that I was talking about, where you can see the notes with frequencies which are multiples of each other, integer multiples of each other. And then if you think about an orchestra playing, then the sound of an orchestra is a combination of sounds of different instruments.

And each sound of an instrument at any moment is a note, is a particular note, and that note has a frequency. And so you can think of this decomposition of breaking into pieces of a sound as a decomposition of a signal into a combination of different notes, each of them with its own intensity. And that's the idea of Fourier analysis or Fourier series. Right, so that's the subject which studies this type of decomposition. So what do we need here? You need a particular space of functions and you need some preferred functions,

some special functions, a collection of functions like sine nx and cosine nx. And then they should be rich enough to give you every function as a superposition. It also has a continuous analog which is called a Fourier integral as opposed to Fourier series. These bubbles appear by the way. It liked what you were saying. These bubbles, it means that the AI overlords are liking what I'm saying, which is great. It did some harmonic analysis on that.

I think they like harmonic. It's kind of like a meta level realizing. It's harmonic, right? That you're in sync with it, yes. That's good. Okay, so you're outlining there's a harmonic analysis, there's topology, there's number theory or numbers. Geometry.

So you have this Riemann surface we talked about earlier when I talked about string theory. microfiber cushions engineered for comfort and a range of colors and finishes. Dyson On Track. Headphones remastered. Buy from DysonCanada.ca. With ANC on, performance may vary based on environmental conditions and usage. Accessories sold separately. For example, and so what is Langlands program? The Langlands program is this giant project aimed at finding common patterns in different fields of mathematics. And so the original formulation by Langlands in the 1960s, in 1967, in that letter to Andre Wey,

which I mentioned, was actually about connecting two specific fields, number theory on one side and harmonic analysis on the other side. But not harmonic analysis in the naive sense of just Fourier series on the circle, so to speak, or on the line, you see. That's just a kind of a baby version of harmonic analysis. You can have harmonic analysis for other spaces, multidimensional spaces, instead of the real

line, where the role of the sine functions and cosine functions will be played by some other functions. And the example which is relevant to this is the example of what's called modular forms. Modular forms on the upper half plane or on the complex disk. We'll talk about this in a moment. So now if it were just that, if the language program was just that, connecting questions in number theory to questions in harmonic analysis probably wouldn't get as much attention. But what happened is that over the years people in other fields of mathematics But what happened is that over the years, people in other fields of mathematics started discovering very similar patterns as well.

And so therefore, we actually have Langlands program developing not only in the original formulation but also in these other fields, which I'm going to talk about. And so now to circle back to this recent achievement in the geometric Langlands correspondence. What's the geometric Langlands correspondence? The geometric Langlands correspondence appeared as a particular way of generalizing these ideas of original ideas of Langlands of connecting number theory and harmonic analysis

and adopting them in the world of Riemann surfaces. Riemann surfaces like this. And the reason why it's connected to physics is because as I explained So there are these models of quantum physics where you also have Riemann surfaces You see so that's two-dimensional and then you have to make a leap You have to there is a way to connect four-dimensional young Mills to the theories defined on this Riemann surfaces So that's another step, but that's roughly why. So why the geometric Langlands is relevant to physics and not the original one.

The original one has to do with number theory. Whereas the geometric one has to do with geometry of Riemann surfaces. And that's much closer to the kind of stuff that physicists, high energy physicists are studying or quantum physicists are studying, you see. All right. So that's the picture of Langlands that I already showed before.

So I would say that Langlands program is kind of about building bridges between different continents of mathematics. And as a baby version of it, think about the example I gave with numbers. Natural numbers or whole numbers appear both from counting and from homotopy groups, the winding in topology. So that's a bridge between the two fields, right? It's a kind of rudimentary version of unification in mathematics. Langlands program is much more sophisticated program or set of ideas for connecting fields

at a much more abstract level. And so unification therefore consists of finding hidden connections, kind of invisible connections between areas of mathematics, which seem to be far apart. That's what unification is about in mathematics, not about finding one overarching theory which subsumes everything.

It's bridge making. It's bridge making. So if you are into building bridges, then mathematics is for you. Great. All right. So, now I can't resist showing this. So remember I mentioned how Langlands first summarized his ideas in a letter to Andre

Wey. And Andre Wey will play an important part later on in the story. He was a great mathematician, a French-born who moved to the United States during the war and has been a professor at the Institute for Advanced Study where Einstein was a professor as well as Langlands, Witten, and so on. So he was the luminary in this field and so therefore Langlands felt that he should run these ideas by him, by Andre Wey. W-E-I-L is how we spell his name.

It's very confusing because there is also Herman Weill, another great mathematician who was also at the Institute for Advanced Study at the same time. His name is spelled W-E-Y-L and is pronounced Weill, whereas Andre- Oh, is is it it's not vile so Andre way that we are talking about right now to whom language wrote his letter his last name is violence in ers like vile spinners

Weyl spinners, isn't that Herman? No, no the vile spinners go is due to is in honor of Herman while W E Y L. Yeah, that's right. But here we are talking about someone whose last name is spelled W E I L. Right, right, right. And it's pronounced very differently because he is from France. So it's pronounced in a distinctly French way without pronouncing the last letter, the last consonant. You say way, Andre way. Yes.

Whereas Hermann Weill was German. So his name is pronounced in a German fashion, Weill. Or Weill, I'm not quite sure. More like Weill. Yeah, I think it's vile for him. Vile, vile. Right, vile. So, but not to be confused also with Andrew Wiles. W-I-L-E-S. Or you see.

So, as if mathematics was not already confusing, these people then come up with these names that are really hard to tell apart, you know. So, that's why. Yeah, and so for people like myself and yourself, we see these all the time. It never even occurred to me that there was also Andrew Wiles that would be confused with Andre Wiles. And by the way, Andre Wiles is very close to the subject. It's not like they, if they all worked in areas which were far away from each other,

that would be one thing. But actually their works are overlap considerably. Like Herman Wiles definitely works, actually he was a kind of a polymath. He worked in so many different areas. And Andrei Wey algebra, geometry, number theory and so on. So it's very close to Andrew Weyles and the proof of Fermat's last theorem was actually based on the proof of what was called Shimur-Tanyama way conjecture where way actually is involved. So you have both way and while in the same sentence, you see. So, but you know, it's okay, bear with us. In the meanwhile.

Yes, in the meanwhile, we have this letter, we have this letter from Robert Langlands. So Robert Langlands, imagine he is 30 or 31, I think he's 30 years old. So this is like January of 67. He was born in 1936, like in October, I think. So he's a 30 year old man, very ambitious. He's a guy, he's a kind of a really tough guy, very full of energy. And he meets Andre Wey in the corridor before some seminar. And he hands him this letter, and this letter is not just like one page, it's like 30 pages, okay, handwritten. And what you're looking at right now is a cover page, which was preserved in the archive

of the Institute for Advanced Study, where Langlands worked for many years. Not yet at the time. At the time, Andre Wey was a professor at the Institute for Advanced Study. I think Langlands was not yet. He was a professor at Princeton University. And what does it say? So it's hard to make out, so I will read it for you.

He says, Professor Wey, in response to your invitation to come and talk, I wrote the enclosed letter. After I wrote it, I realized there was hardly a statement in it of which I was certain. If you're willing to read it as pure speculation, I would appreciate that. If not, I'm sure you have a wastebasket handy. That's what it says. Kind of show man a little bit, you know. you have a wastebasket handy. That's what it says. Uh-huh. It kind of showed me a little bit, you know.

Yes. Also, 30 pages. I would have so much anxiety sending that and thinking it may get lost in the mail. And then what happened, my understanding is I haven't looked into this in a while. I mean, I looked at it a lot when I was writing my book and it's quoted in the book. But my collection is that actually Andre Wey, there was like silence and after a week he sent a message through his secretary saying that, could you please type the letter because I cannot understand your handwriting.

Yeah, if you look at the handwriting, it's not exactly the easiest one to decipher. Anyway, so here's where we are. So we are now in January of 1967 and we have Robert Langlands, one of the greatest mathematicians of the 20th century. He doesn't know it yet. He's 30 years old. He is looking up to his... Andre Wey was not really his mentor, but he was a towering figure in the field.

And Langlands felt that he should be the first judge, the first one to judge his new ideas. So he, these days you would send an email to, with an attachment, you know, like PDF file to Andrevay. But in those days he actually wrote the letter by hand. And then the story went stratospheric. But there are several steps that we have to get to the recent work by Gates, Goury, Raskin, and others about the proof of the geometrical and conjecture. We have to make a number of steps. And as we make those steps, the story becomes more and more abstract

and more and more sophisticated. So I feel that I have to give an example at the outset of what this is about, because otherwise it feels like, kind of like, why are we doing this? You see, I want to give an example of what kind of questions Langlands actually was trying

to solve and in what way his ideas were so powerful in solving them. What do you think? Yeah, and the link to the recent work will be put on screen and in the description. Okay. And so to explain this, we have to recall what's called the clock arithmetic. Okay. So to explain this, we have to recall what's called the clock arithmetic. So to explain this we have to recall what's called the clock arithmetic. And the clock arithmetic is an arithmetic like on a clock where, let's say, in North America

as we go beyond 12 o'clock we don't say 13 o'clock what we say 1 1 p.m. We don't say 14, but we say 2 Right so we identify the numbers which differ by a multiple of 12 and We can do the same for a clock with any number of hours But it's especially nice to do it when the clock has P hours, where p is a prime number. Because then the addition and multiplication of numbers with this identification satisfies all the usual rules of what we call a field. So for instance, on this picture you have a clock with seven hours, seven is a prime

number. So effectively the only numbers you're focusing on are 0, 1, 2, 3, 4, 5, 6 because seven brings you back to zero. You can add two numbers like this. If the result is in the same range, then that's your answer. If the result is out of this range, you replace it by a number in this range, which differs from it by 7, and likewise with multiplication. So that's what we call arithmetic modulo, a prime number. In this case, modulo number 7.

And so the next notion we have to introduce is a notion of an elliptic curve. So an elliptic curve, for our purposes right right now is an equation of the kind, a cubic equation of the kind that is written on the slide. Namely, you have basically two variables, y and x, and the highest power that you raise y to is 2 and the highest power you raise x to is 3. And so that's like a normal equation if you think about it. So what is the solution of this equation? Solution is a pair of numbers x and y such that the left hand side when you substitute that number y will be equal to what you get from the right hand side by substituting number x.

But the question is what do you mean by number? And so typically we could say, okay, well, real numbers, that's a good choice. Or it could be complex numbers. If we do, to kind of jump ahead, if we do that, if we consider solutions in complex numbers, what we're going to get is precisely a Riemann surface. The set of solutions is more or less the surface of a donut. It's an elliptical. So a particular genus.

It's a curve of genus one, specifically. I see. And this curve of genus one is called elliptic curve. Is there a straightforward way of seeing that or is that some result that would take us off course? Not immediately. Yeah, you have to do a little work. And honestly, you have to get the entire torus.

You have to also allow solutions at infinity. So without that, it will be the torus without a point actually. But this is what gives the name elliptic curve is traditionally was used as a name for this complex torus. But since we look at the same equation and we look at solutions in another numerical system, we also call it an elliptic curve within that context you see. Over a certain field.

Over a certain field, exactly. It's the nomenclatures over a field. That's right. So in other words let's say you have a basket you have a basket here it has tiny balls inside you pull out one of the balls it's a ball number two pull out another one it's ball number ten and another one is ball number twelve you're like okay this is you look at the label it says this is the integer basket. That's right. So now you're thinking okay so, so if I was to take one of these integers, put it into my equation and it comes out with a solution that's like saying this is the elliptic curve

over the integers in that case. Now if you have another basket and it's modulo primes, you pull it out, it's 0, 1, 2, 3, 4, and then goes back to 0, 1, 2, 3, 4 over and over and you have the modulo prime 5, then that solution's over that finite field. That's the term. The phraseology is you have solutions over a field, meaning what are the people, what are the balls that I'm putting into my equation?

What basket are they coming from? That's the field that it's over. Exactly. Or we now have our new numerical system, right? This modulo p. So we can actually look at this equation because it has coefficients which are 1 and minus 1. 1 and minus 1 makes sense modulo p as well.

So therefore the equation also makes sense modulo p, where p is a prime. So now what would be a solution, say, modulo 7 or modulo 5? It would be a solution in which the left-hand side is not necessarily equal to the right-hand side on the nose, but they differ by a multiple of p. Like if p is 7, it's a multiple of 7. If p is 5, it's a multiple of 5. You see? And so here's a problem.

Find the number of solutions moduloulo prime p, for every prime. And so here is an example. If you take p equals, so that's the equation, right? And so take p equals five. So what are the solutions? For example, x equals zero, y equals zero is a solution. It's actually a solution on the nose. Left-hand side is zero, right-hand side is zero. Or x equals one, y equals zero, also on the nose, because this is zero and this is zero. But there are two more solutions. What do you mean on the nose?

Do you mean like it's quite obvious that it's zero? No, they're actually equal to each other. Not only modular five, but just as- Oh, I see what you're saying. Okay. As integers, you see? But here's an example of something which is not on the nose,

which is X equals zero and Y equals four. Let's calculate. Y squared is going to be 16, right? Plus four, 20. Left-hand side is 20 for this choice, right? And the right-hand side is going to be zero still. So the left-hand side is zero, and the right-hand side is zero. If we wanted to consider solutions in real numbers or in integers, then this would not be a solution because they don't agree with each other, left-hand side and

right-hand side. But in our new world of clock arithmetic, modular five, they do agree with each other. Because in this new numerical system, number 20 is the same as number zero. But modular seven, it would not be, because modular seven, 21 is zero, but 20 is not zero, it's minus one. Right, right. You see? So that's how you see that it is very subtle question,

because let's suppose you calculate how many solutions you get in a module of 5. In fact, it's easy by inspection to just plug in all possible values for x and y, namely 0, 1, 2, 3, 4, into the left-hand side and right-hand side and see when the results differ by a multiple of 5. If you do that, you will find that these are the only solutions. There are four solutions.

But when you move from 5 to 7, which is an X prime number, and you address the same question, find the solutions, module 7, you see that all the previous calculations are completely useless. Well, except maybe this first two will still give you solutions, module every prime. But like last two solutions, solutions are not going to be helpful because the fact that the left hand side and the right hand side differ by a multiple of five is not going to help you to find a solution where they differ by multiple seven. You see, so it's a very strange kind of question where

for every prime number, every prime number has its own quirks. And for every prime number you get a particular number of solutions. And so the question is, can you actually describe all of them in one stroke somehow? Obviously, for very small p's, you can just do it on a calculator. And for large p's, you can easily write a computer program, say to do it up to 1000, 10,000 and so on. But try to do it for all p. And so this is where Langlands program comes in. Langlands program gives you a solution to, a completely unexpected solution to this problem, all in one stroke. And that's the, it's a really good example because it shows you the power of this idea

because absolutely out of the blue, out of the left field, you get the solution. So what is it? What is the solution? So let's arrange this as a, as it in the table. So for every prime number, right? So for every prime number, like two, three, five, seven, 11, 13, we have the number of solutions and I've calculated them for you. For five, we found them on the previous slide, four, and these are the numbers for other

primes between two and 13. But it turns out that it's better to consider not the number of solutions itself, but rather this number, which is kind of the difference between the number of solutions and the prime itself. So P minus the number of solutions. And the thing is that the number-

And how is one supposed to get this idea? If you look at these numbers, you will see that these numbers grow almost linearly with p. So let's just put it this way. It's a kind of an error because naively you could say, it's probably about p solutions. Because after all, think about it this way. You have two free variables, right?

And each of them takes p values. For example, if p is five, you have zero, one, two, three, four, and then five is not an extra element because five is zero. So you have five values for X and you have five values for Y. So altogether you have five times five, 25. But you have one equation, which means one degree of freedom drops. So you have five squared

because there are two degrees of freedom X and Y, but there is one equation. So kind of like as a rough estimate, you could say the number of solutions should be close to five. And number of solutions close to dimension drops by one if you have an equation, right? So for instance, if you have a circle is defined by one equation or a line, let's say a line on the plane. The plane has X and Y, two coordinates, so it's two-dimensional. But if you write the equation X equals Y, you get a diagonal. So the equation drops the dimension by one. One equation drops dimension by one, two equations usually drop by two and so on. That's kind of like a rough back of the

envelope calculation. But one-dimensional in this new world means p possibilities, like for 5 it means 5 possibilities. Two dimensional means 25. Three dimensional means 125. So that's how you come up with this rough estimate that two variables with one equation should give you approximately p solutions. And then you say, okay, well, let's calculate what is the error. So the actual numbers, the P minus the actual number of solutions. Then of course, once you understand the Langlands program, there is another explanation why this is a good number to consider.

There is another way to explain it, but let's just stick with that. Okay, so now, so here is a miracle. This number is AP, so AP is what's in the last slide, colon. So it's not exactly the number of solutions, but it's the difference between P and the number of solutions. But of course, if you know AP, then you know the number of solutions because you simply take, you know, the number of solutions can be found as P minus AP. So the two problems are equivalent to each other, right?

So what we're going to describe is not numbers of solutions for every prime, but we're going to describe these numbers AP, but that's equivalent to the original problem. And so it turns out that these numbers can be described all at once in the language of harmonic analysis. So remember I said the original formulation of the Langlands program was in relating or connecting number theory and harmonic analysis. The problem we have discussed with counting numbers of solutions is squarely in the field of number theory, right, because we are, well, all we are doing

is arithmetic with numbers, right? And comparing like left-hand side, right-hand side, modular prime and so on. And harmonic analysis is about functions of some special kind. And so here is how the two fields come together, number theory and harmonic analysis. Consider the following infinite product. And at first it looks intimidating, but if you look closely, you will see that there is a system here. So Q is a variable. In high school, we usually denote variable by X. That's the traditional notation.

But in this subject, this particular variable is traditionally denoted by Q. Don't ask why, nobody knows. It's just like... In every field... He's like position is usually X and then it gets mapped to Q in Hamiltonian mechanics. I don't know why. It's weird because Q is also used in the field which is called quantum groups, quantum algebra, so for quantum.

And interestingly enough, there are many results which were obtained in arithmetic a long time ago or in theory of modular forms where for some reason people chose the variable Q, they connect to results in quantum groups, for example, in quantum algebra. And this is like, how did they know? Because when you translate them, you don't even have to make a change of variables. So it's a mystery. But anyway, somehow people did it with Q and it caught on.

And it's a traditional notation in this theory for this variable, okay? It's Q, it's called Q, not X. Just for the sake of explaining or perhaps not explaining, this function here, Q times 1 minus Q squared times 1, blah, blah, blah, that also looks like it's dropped from the sky. So are we going to explain that or are you just going to say that this... I will explain.

Yes, I will explain. But first I want to give the answer. Okay? Sure. So first I have to explain what it is so that you're not intimidated by it. So first of all, there is this Q. You just write it once and you forget about it.

Then you have this guy, this guy, this guy, and this guy, and it goes on. What do they look like? It's one minus Q squared, right? This is Q to the first power basically, but then squared. This is one minus Q squared, squared, right? This is one minus Q to the third power squared. So each of these terms is a square of something

which looks like one minus Q, one minus Q squared, one minus Q cubed, one minus Q to the fourth. So you can easily guess the next one will be one minus Q to the fifth squared and so on, right? That's clear. Then in addition, you have these guys, Q to the 11 squared, Q to the 22 squared, Q to the 33 squared. So what are these numbers? 11, 22, 33, right? These are multiples of 11. So you have one progression where you have Q, Q squared, Q cubed, and Q to the fourth.

And every time you put a square. And then you have a second progression. The first one is underlined with red. Second one is underlined with blue. And you got here Q to the 11th, Q to the 22th, Q to the 33th. And each time you square it. So you can see what the next two terms are. First you will have 1 minus q to the 44 squared, and then you have 1 minus q to the 5 squared. So after that we open the brackets. Now, in principle you could say, okay, well, how can we possibly open the brackets? There are infinitely many terms here.

So this dot dot dot means that we continue ad infinitum. But the point is that the degrees grow, these powers grow. So if you're interested in just the coefficients in front of Q or Q squared and so on, there will only be finitely many terms which will contribute. For example, Q is already there, so Q times 1 in each of these factors will give you Q. But there is no other way to get Q because every other term will have Q times Q to some other power, some positive power, right?

So that's how you know for sure that q will appear with coefficient 1. What about q squared? So for q squared we write actually this is 1 minus 2q plus q squared. That's this term, right? So you see there is minus 2q and this minus 2q will conspire with q to produce minus 2q squared. This. And there is no other way in which you can get q squared out of this product. And so on. So what I'm trying to say is that this is well defined.

The coefficient in front of every finite power like q to the fifth, q to the sixth and so on is well defined. Is a combination of things that come from just finite sums and products. Even though the whole thing is infinite, you see? In other words, only finitely many terms will affect a particular power of q when we open the brackets. And so now we get this expression

where you have each power of q has a particular coefficient. Like we have just calculated that in front of q you have one, in front of Q squared you have minus 2. And then if you continue along this path, you will see that the coefficient in front of Q to Q cubed is minus 1, in front of Q to the fourth is 2, and so on. So what does this have to do with the original problem? And the amazing thing is that you recover all these numbers as the coefficient in front of Q to the P. Can you believe this? So for example, in front of Q cubed,

you have minus one, right? Okay, and so this doesn't work for the non-prime numbers. Because there's no data. For what? For the non-prime numbers, it doesn't work. So in fact, this series has more information that we needed, right? Because it also has coefficients in front of non-prime powers. And there's an interesting question of what that corresponds to, which can be answered

as well. But let's just focus. Is that information about what it corresponds to new work that should be covered today or no? There's some more intricate properties of these equations, but it's not really relevant. It's not really necessary to understand the original question. So that's why we'll ignore it. We will just take the coefficients in front of prime powers. And I claim that in fact, they match perfectly. These numbers AP that we have in this. Let's do it. So for P equal 2 is minus 2. And that's the coefficient in front of Q squared, right?

Yes. For P equals 3 is minus 1. And that's the coefficient in front of Q cubed. For Q equals 5, remember we're not considering 4 because we only want prime powers. So we don't care about the coefficient in front of q to the 4th. But we do care about the coefficient in front of q to the 5th. Right? And this coefficient is 1, which is the number AP for p equals 5. Which by the way is what we found, because we found 4 solutions.

But remember AP is not the number of solutions, it's P minus the number of solutions. So P here is 5, and so P minus 4 is 1. And that's exactly the coefficient in front of Q to the 5. In front of Q to the 7, we have minus 2, and that's exactly what we have here. In front of, for P equal 11, we should get one and that's exactly what we get. And for 13, we should get four and that's what we get. And it is a theorem, it is a mathematical theorem. This is an ad by BetterHelp. What are your self-care non-negotiables? It's hard to make time for the things that keep you healthy, but being consistent with self-care non-negotiables? It's hard to make time for the things that keep you healthy, but being consistent with

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This is a well-known fact. It's a well-known theorem, right? So no human being can actually behold all these numbers at once in some sense. However, they are all defined. And you can check for any finite element of this infinite sequence. You can check that the left-hand side corresponds to the right-hand side. But imagine how much stronger this result is. It doesn't tell you that it works for p from 1 to 13, from 2 to 13, or from 2 to the smallest prime less than 10,000.

It actually tells you that it's true for every p. And that statement contains within an infinite sequence of statements. It's amazing if you think about it. So one formula rules them all. And so we call it the finding order in seeming chaos. If you take the coefficient of this power of Q in this infinite series, then it will give you exactly this AP, which is P minus number of solutions for all primes. And so what happens is a kind of a colossal compression of information. Because just one, think about just one line of code and I explained how this, how this to produce this code, right? So it's, it's, there's some regularity here. You don't

need infinite amount of information. It's finite amount of information. You just say 1 minus q to the i, where i is equal to 1, and then repeat for i plus 1. So, you have q squared times q squared square, and each time square it, right? So, these are this term, this term, this term, this term. And then do the same with q to the 11 instead of q. So Q to the 11, Q to the 22, Q to the 33. It's easy to program. It's extremely easy to program it on the computer in such a way that for any prime number that is accessible, that you have access to on your computer, on your hard drive or your memory that you can actually store it, you can find infinite amount of

time the coefficient in front of Q to the P. And guess what? It's going to be exactly the number of solutions, well, more precisely it's going to be P minus the number of solutions for that cubic equation. That's an example of the Lengence program. And that's the most beautiful, it's the simplest, most beautiful example. I have to say I learned it from Richard Taylor. He's a great mathematician, he's the simplest, most beautiful example. I have to say, I learned it from Richard Taylor. He's a great mathematician, he's at the Institute for Advanced Study. He was a co-author of Andrew Wiles

in their famous paper, Solving Fermat's Last Theorem. So I learned an example from him actually. And it's in my book as well. It's in Love and Math, in chapter seven, I think. So if you want to know more, to learn more about this, kind of slow reading, that's where you can find it. So- I also recommend that book just for people who are

listening or read it approximately a year ago or so. And in one of the early chapters, you cover braid, braid groups and knot theory, which is something you learn in third year in university, but it's covered in one of the early chapters. That was my first mathematical, That was my first mathematical work. That's why I talk about it, right? So that was my first excitement of solving something

which nobody knew, making discovery. We talked about whether mathematics discovered it or invented. Certainly in that moment, I felt like I discovered something about break groups. But anyway, so what I'm trying to say is that this just one line of code kind of gives us a simple rule for solving this infinite counting problem for all prime numbers at once. That's what I mean by finding hidden

connections. Remember I said to me, unification in mathematics is about finding hidden connections between different fields, which seem to be far apart. Here it's between the field of number theory and the field of harmonic analysis. All right, so now let's talk about what is this product? In what sense does it reside in harmonic analysis. Okay? And so the point is that when we talked about it first, we talked about it as a kind of a formal expression. Right? So it's like a product Q. So Q was just a variable. But actually it turns out that we can assign a numerical value to Q and this product will converge.

So even though it is a product of infinitely many numbers, it turns out that this product actually makes sense. And not in a kind of esoteric sense as like 1 plus 2 plus 3 plus 4, but actually like the rigid, precise, rigorous sense of limits that we study in calculus. The product of finitely many terms of this from the first to the nth of this expression is actually going to have a limit as n goes to infinity. And that therefore, but only if q is a number between minus 1 and 1, a real number between minus 1 and 1. Or we could also work with a complex unit disk.

You know, the complex numbers live on a plane because they have both real and imaginary part. And every complex number has a norm, which is the distance from this, from a geometric representation of this number as a point on a complex plane and the point zero. So for example, number i has has normal one, because that's the distance from i to zero. If you take all numbers whose norm is less than 1 you obtain the interior of a disk, of a unit disk, right? And for every point in that disk, if you view it as a complex number and substitute in this infinite expression, we will get a well-defined number. It will have a limit. This product will have a limit, you see. So therefore we get a function on the unit disk.

For every Q, we get a specific value, which is encoded by this infinite product. So we get a function on the unit disk, and it is called a modular form. And the point is that it has very special transformation properties under the group of symmetries of the unit disk, which is denoted PSL2Z. So instead of explaining exactly how this group acts on the unit disk, I'm going to show you how a similar group acts by showing you what's called the fundamental domains. So the action of a group like this,

it's actually going to be a subgroup of PSL2Z. of a group like this. It's actually going to be a subgroup of P, C, L, to Z. And this is not exactly the same subgroup, but it will do as an illustration. So here's what I'm talking about. Remember our harmonic analysis on the circle, which we- Can you go back just a moment? Can you go back to slide 40?

Okay. So this function is defined on the complex unit disk. Yes, which is think about this. That's it's like this. Right. So two complex numbers to all of complex. That's right. So not just upper half.

So look, here's a complex. So here's a complex plane, which has two axes, X and Y, where if you have a point here, a point with coordinates XY, we assign to it number X plus YI. That's a complex number assigned to a point, right? And so you have number one here, you have number I here, you have number minus one here,

you have number minus 1 here, you have number minus i here. And then you can draw this circle of radius 1. And so your number q is going to be some number inside this circle, you see. So that's your q. The condition is that q is less than 1, which means that X squared plus Y squared is less than 1. Does it make sense? My question is, where is it going to?

What's the target space? Oh yeah, so for every Q, which is like this, this function has a value, which is a complex number without any restrictions. Okay. Some complex, some complex number without any restrictions. Okay. Some complex, some complex number. And it's unique? It's unique, yes.

It's unique, absolutely. Yes, otherwise it's not a function, right? So when we say function, we mean single-valued function, which means a rule which assigns to every Q some complex number. But every Q, not everywhere, Q is constrained by the property that it is within the unit disk. Okay. The value is, the only constraint is that it's unique, is finite, it's well defined and it's uniquely defined.

So now explain where SL2Z comes in. So SL2Z, yes, so there is a SL2Z. Before I explain this, let's look at a simpler situation. Okay? So, at the simpler example, namely, the harmonic analysis that we discussed earlier when we talked about sounds of music, right? So, we have these basic harmonics, sine, nx, and cosine nx, where n is an integer. And we discussed the fact that every function on the circle..., so they are periodic functions, because if you send X plus X two pi,

the value will be the same. So they are periodic with period two pi, right? And the point is, so the invariant under the shift X plus two pi. And the point is that harmonic analysis allows us to write any function on the circle, essentially, any continuous function, let's say, as a linear combination of these guys. That's called Fourier series.

And you can think about it as decomposing the sound of a symphony, of an orchestra, into the notes of different instruments in time. So that in this case, the X is a time because it plays music in time. And each note, roughly speaking, is a wave like sine of an X or cosine of an X. And as I said, we can arrange things in such a way that these notes have approximately,

proportional to a specific note, to a specific frequency. The frequencies of these notes are proportional to a specific note, to a specific frequency. The frequencies of these notes are proportional to a specific frequency with integer multiple, with rational multiples more precisely, but we can always find a kind of a common denominator so that it will be just integer multiples.

So that's the setup of harmonic analysis. But what I want to focus our attention on is the fact that actually there is this invariance. What's special about these functions, sine and X and cosine and X, is that they are invariant under the shift X goes to X plus two pi. But if so, it's also invariant under the shift X plus two pi times some integer, for example, four pi, six pi, eight pi, minus two pi and so on. So you see what happens that actually there is a group of symmetries that is lurking in the background when you talk about these functions.

This function really is defined, think about it, this function is defined on the real line. So that's your real line, right? And so your function is going to be a cosine function or sine function. So it's going to look something like this. Right? It's a wave like this. But now the point is that if you shift it, you see, if you shift it by this amount, if you shift it by this amount, it will stay the same. Right?

So if you imagine shifting this whole curve from here to here. here so this point goes to here this point goes to here, so it will go to itself That's what I mean by being invariant Right I think that's the best approach for this part And then in part two we can go to the graduate student level yeah I do I do I do I want people to understand it because it's really basic in beautiful stuff So so what I'm trying to say is that there is a shift under which the whole thing goes to itself. And so this shift is a shift by 2 pi.

Let's say that I'm writing the basic one which corresponds to n equal 1. And then the sine 2x say they will have, they will oscillate more frequently, but they will still be invariant under the shift but it's also invariant under the shift by 4 pi It's also invariant under shift by 4 pi which would be like twice this twice you put this and one more time, right? So as a result It's invariant under all of the shifts and all of the shifts gives you an action of the group of integers on the real line And what I would like to focus on is what's called the fundamental domain of this section. And the fundamental domain of this section

is just this interval. Because it's like, remember like, when we talked about arithmetic module of five, then we identify things which are related by a multiple of five. So the fundamental domain consists of just zero, one, two, three, four. Every other thing you can get by adding to one of those, a multiple of 5. So the fundamental domain consists of just 0, 1, 2, 3, 4.

Every other thing you can get by adding to one of those a multiple of 5. And likewise now, on this picture, every real number can be obtained from a number just on this interval plus a multiple of 2 pi. So that's called a fundamental domain. Fundamental domain. But actually it's not the only fundamental domain. The whole real line breaks into fundamental domains which are going to be from this point to this point, from this point to the next and so on. They all have lengths 2 pi. And now let's go back to this. So here on the unit disk, there is another group which is called PSL2C, which plays the role of Z, 2

pi Z in our baby version, which is here, is 2 pi M. And the analogs of these intervals are precisely this kind of hyperbolic triangles, both red and white. So this gives you a sense of how this group acts. It acts by hyperbolic transformations. And this visualizes how, what it does, it exchanges different triangles. It exchanges points in red triangles with points in other red triangles or white triangles.

You see? So this gives you a rough idea of the special properties of this function that we are considering. This function, which is obtained from our infinite series that solved for us all the counting problems for all prime numbers, all at once. This function that we obtained from this series by evaluating it at points in the unit disk actually has special symmetry properties with respect to the group which acts like so.

Which is similar to how sine and cosine functions act with respect to the action of a much simpler group, namely the group 2 pi z. Z being the integers. You see? So that's a rough explanation. And again, if you want to know more, for instance, you can read about this in my, in the chapter seven of love and math and so on. But that's a kind of a rough explanation of what is so special about this function. Now the weight in this one is what is two, you said here, or is it one?

The weight, because modular forms have a weight. Weight is two. Weight is two because roughly speaking because well it's elliptical so it's like two-dimensional. It has to do something two-dimensional in the background. Sure. Now this is actually, so now this is one example remember it's what we're considering is just one example. There is one counting problem. There is a specific cubic equation, right? It was y squared plus y equals x cubed minus x. And we are looking for every prime number, we get a number of solutions, then we slightly

modify it by subtracting it from p. Those are coefficients in front of prime powers of q in this infinite series. Can we generalize it? It turns out that this has a vast generalization. In other words, you were just given an elliptic equation, so that y cubed minus y squared equals x cubed or whatever it was before. That's right. Okay, so you were given an equation. Then you from that asked, well, what are the number of solutions in modulo prime or the clock arithmetic prime number? That's right. And then actually you took a modified form of that with some error term the clock arithmetic prime number. That's right.

And then actually you took a modified form of that with some error term, but it doesn't matter. That's right. Found a correspondence between that and a modular form. That's right. Now you're wondering... A specific modular form, right? Which we could actually write out explicitly. Right. And now you're wondering, okay, well, that was for one elliptic curve. Can I be greedy? Is there a class of elliptic curves that this may work on? Or does it work for every single type of elliptic curve, can I be greedy? Is there a class of elliptic curves that this may work on?

Or does it work for every single type of elliptic curve? Exactly. Every, every smooth, every smooth. Is there something special here? It's, it is actually works for all smooth elliptic curves. So there is some kind of condition, not the generic kind of equations. Okay.

So every, so there are lots of equations you can write of this nature that we wrote, right? Infinitely many, in fact. And for all of them, there will be its own, for each of them, there will be its own modular form which will encode numbers of solutions for all primes. Save maybe finitely many. In this case, we lucked out actually every single prime for every single prime we got matching perfect matching with the coefficients in general there will be finitely many prime numbers for which it will not work

But okay, so the big deal compared to infinity of other primes for which it will work you see so and that is called the Shimura-Tanayama-Wei conjecture. And it's a beautiful story. So this is the 1950s. There were two Japanese mathematicians, Yutaka Tanayama and Goro Shimura, and Andre Wey, the recipient of that infamous letter from Robert Langlands. They were actually at a conference in Japan. There was a famous Tokyo Niko symposium in 1955. Remember, this is after World War II. And obviously, Japanese mathematicians were isolated from American mathematicians. This was essentially the first meeting. Of the mathematicians from the two nations. And it was extremely influential. So there's a picture, the talks were held in two cities, in Tokyo and Nikko, and they are on the train here,

going from Tokyo to Nikko, or maybe the other way around. And who do you see here? So you have, this is Andre Vain, this is Yutaka Atanyama, who's this brilliant Japanese mathematician, committed suicide at the age of 31. I talk about this more in love and math. And so if you're interested, it's a very interesting story.

This is Goro Shimura, another Japanese mathematician who actually worked most of his life at Princeton University. And this is Jean-Pierre Serre, a legend from Paris. You see a beautiful picture, huh? And so the Shimura-Tanyama conjecture, Tanyama-Vey conjecture is, what is it again? Let me say it one more time. So it's a statement that for every cubic equation, what we call the elliptic curve over finite fields or elliptic curve of the rationals and then we specialize modular prime numbers. Every cubic equation of the kind we considered will have its partner in a different

world in the world of modular forms. So they're going to be of weight two and also satisfy some additional condition. They're going to be what's called a new form, which I'm not going to get into it. It's a technical but not very complicated condition. And it turns out that there's a bijection. And it goes in both directions. One to one correspondence. Isn't it amazing? One to one correspondence.

That for every modular form, there is its own cubic equation waiting in the wings, whose counting problem will be given by the coefficients of this modular form at prime powers, save maybe for finitely many of those. And conversely, for every cubic equation, there is a modular form waiting to give the solution to the counting problem. So that's the kind of things we are talking about. It's really mind-boggling. And how did they come up with this? It's absolutely genius.

Like now we look back after so many years, like what, 70 years, it looks kind of, yeah, sure, it makes sense. But it absolutely did not fall from anything. Just the fact that there were several examples known, it doesn't mean that there should be really one-to-one correspondence. And yet there is, you see. That's how groundbreaking this is. And to add to the kind of the glory of this result, I want to mention that Fermat's last theorem actually follows from it. In 1996, my colleague here at UC Berkeley, Ken Ribbett, proved that if you can show Srimotanyama wave conjecture, then you get Fermat's last theorem.

And Fermat's last theorem, of course, I imagine most of the viewers have heard about. It's an impossibility of solving this equation with positive integer numbers. You see, so Fermat's last theorem was proposed by Pierre Fermat on the margin of D'Aphantus's book on arithmetic 350 years ago. And he wrote famously on the margin that he found a beautiful proof of it but the margin is too small to contain it. So for the next 350 years people tried to prove it, professionals and amateurs, only to have their hopes dashed, people would find mistakes and so on. Until finally in 1996 Ken Ribbett, my colleague here at UC Berkeley was able to relate it

to something, to a different conjecture, which is the Schmurtanian M of A conjecture we just talked about. In fact, we talked about one specific example of this conjecture, but the Schmurtanian M of A conjecture is much vaster than that. It services every cubic equation and every modular form, which is of weight two and a new form. So finally, what remains to prove is the Schmurtanium of A conjecture, and that was done by Andrew Wiles and Richard Taylor in 1995, I believe.

In other words, Fermat's Lysterium follows from Schmurtanium of A. But Schmurtanium of A is a special case of the Langlands program. So now, already, the Schmurt-Anhem-Wei is a vast generalization of this one example we considered, right? Because our one example referenced specifically a particular cubic equation and a particular modular form, which was given by the infant product. So Schmurt-Anhem-Wei congestion is a vast generalization of that to arbitrary cubic equations, satisfying some conditions and modular forms. And that's just a very special case of the Langlands program. So now Langlands program is a vast, you see,

that's what it's about. Now, the original program, Lang's program, was about this type of questions in number theory and a possibility of solving them in terms of much more easily tractable questions in harmonic analysis. Now, I have a feeling that we're approaching kind of a nice middle point in this conversation. I feel like we won't be able to get to the end of it today, unless we talk all day, you know? So do you think it would make sense to have a second installment where I will pick up on this. We have now a very nice kind of like,

we build the foundation. We see now an example of what this is about, a very concrete example of what this is about. So this will give us motivation to study this further and to think about generalizations of this Langlands program, moving it away from questions in numbers theory to the questions in geometry that are related to Riemann surfaces and things like that. And that will bring us to up to date to the most recent achievement of this recent papers that we talked about.

Sure. So here's what we'll do. People who are watching, if you have any questions, leave them in the comments below, because then this way we can pull from them to ask next time if you're confused at any point. And a quick question I have is, the Langlands program is usually formulated as, if I have a question in number theory, I can more easily answer it in harmonic analysis. Right. But I don't hear the opposite, that if there's a question in harmonic analysis, can I throw

it over to number theory, it's much more easily solved there and throw it back to harmonic analysis. Interesting enough, such questions did emerge. Because obviously if you have a bijection, then initially it looked like harmonic analysis is much simpler. And so this is kind of an advantageous approach is to reformulate number theoretic questions. Not all number theoretic questions, mind you, I want to make sure that there is no misunderstanding.

I'm not saying every single question in number theory can be formulated in these terms. No, very specific questions like the questions that we discussed. Right? So it's less like a bridge between two continents and more like a bridge between a city in one continent To see it's more than the city. It's more like a State like a big state like California's state of California Yeah, the point of it be it's a subset proper subset It's a proper subset, but it's a very big subset.

And the question is, and one could hope that maybe for now we think it's a subset, but maybe eventually we'll see. In a sense it is much more than just a subset because, let me comment on this because it's important. So the proper formulation of this correspondence is not in terms of equations that we did on the side of number theory. It's in terms of the Galois group. It's in terms of equations that we did on the side of number theory. It's in terms of the Galois group. It's in terms of the Galois group.

So Galois group is a very important concept in mathematics. Galois group is roughly speaking describes symmetries of numerical systems that you can get out of rational numbers. By rational numbers I mean fractions. A divided by B, like one half or three fifths. They form a field of its own, and we can throw in solutions of various polynomial equations into it,

like square root of two or I. As a result, we get what's called algebraic closure, a much vaster field. And one could argue, and I think this is a very good argument, that every problem in number theory boils down to a problem about Galois group, this Galois group. Now, if you accept that, the Langlands program actually gives you something more or less gives you an answer to most reasonable questions about Galois groups because it describes describes n dimensional representations of the Galois group for all dimensions. What we are discussing now has to do with two dimensional representations that have to do with the slip decurves. But in fact,

the conjectures cover much a wider spectrum of questions about Galois groups. So in some sense, it's not a big exaggeration is that we are actually covering a very big territory. It's not a city for sure. It's not even like a state. It's much more. It's like a coast. It's a coast, half of the country maybe.

At least half of the country, let's just say. So it is really serious. It's not like a small province. Understood, understood. I didn't mean to diminish or support it. No, no, I'm not trying to. I'm not trying to. By the way, it's not really my field, so I'm not, I don't have a horse in this.

So because people are going to be championing at the bit thinking like, oh, I wish that I had more to chew on when it comes to what are the recent results about. Why don't you just give a flyover? Yeah, let me give a teaser. Let me give a teaser for what's coming next. So first of all, well, that's the kind of a summary of what we talked about. The cubic equations are connected to modular forms and the general correspondence is about representation of Galois group is what's called automorphic functions, which generalize these

and these generalize this. So when I say it's not really my field, I mean, I come into Langlands program more from the side of geometry. So this is something that I don't necessarily work on on a daily basis. So it's kind of like a more of a hobby, this area of the Langlands program for me.

But now we're going to move closer to what my field is, the field in which I have worked for the last four years. So- And if you'd like to save time, address a graduate student audience in math or physics. Okay. So first there is one twist, which I haven't mentioned, which is what's called the Langlands dual group,

because in fact, there is a group, what's called a group appearing on both sides, but these two groups are not the same. One of them is a dual, so-called dual. And why it is so is a big mystery, but it's one of the indications that there's something very non-trivial. There's Lie groups described in terms of the Dynkin diagram.

So there's some beautiful combinatorics. There's a beautiful story underneath. And so finally we get to the crucial point, which is that in fact, we talked about number theory and about how the questions in number theory can be related to some questions in harmonic analysis, right? But there is a separate idea in mathematics which is called the Rosetta Stone of math, which is due to Andre Wey. That's the guy to whom Robert Langlands wrote his initial letter in 1967. Andre Wey in turn wrote a letter to his sister, Simone Wey, who was a great philosopher and mystic and humanist from prison in 1940, he was jailed because he refused to serve in the army during the war. And André Wey

in this letter formulated an analogy between three different areas of mathematics, number theory, curves over finite fields and Riemann surfaces. So that's not the same kind of list as what we have here. So Langlands correspondence is from number theory to harmonic analysis. But number theory has analogues to other fields, this and this. And so the question arose as to whether there are analogs of harmonic analysis for this guy and for this guy. And did we not just do curves over finite fields? That's right. So curves over finite fields have already occurred,

but in fact, for a different reason, and this is a bit confusing, but bear with me. The reason why, what occurred here is not really a curve over a specific finite field, but remember we had one equation which actually made sense, modulo every prime. So in fact, it was an equation over the integers because the coefficients were integers. That's what enabled us to relate it to an equation over modular prime. Right? So in other words here you actually have the truth is that what we have here is an elliptic curve, elliptic curve over the integers or the field of rational numbers.

And if you have such a curve, you can consider you attach to it a curve over all primes for all modular all primes. You see? All primes. But here we consider finite field with a specific, with a fixed prime. So it's a different setup. Even though the same objects appear here and here, but they appear in a different way, in a different, they have different meanings.

So, even though they appear in both, they actually kind of have different interpretation. Okay, so there are three relationships here. There's number theory to harmonic and back, and then there's curves over finite field to something else not defined yet and remont surfaces to something else. So we have to find, and this is a question, what are these connections in these two other realms? Okay, so it's a conjecture about conjectures. That's right. Interesting. And so you see the point is actually to honest, this connection was kind of obvious,

because these two areas are so close to each other that it was very easy to find an analog of the harmonic analysis that is necessary for these guys. There are already connections going downward from number theory to curves over finite fields and then from number theory to Riemann surfaces. These are not connections. So you see, horizontally, it's really like a correspondence, right? Because we saw that for that cubic equation, that corresponds to that cubic equation, that corresponds a particular modular form, right? So that numbers of solutions of the equation

modular primes can be expressed as coefficients of this modular form, right? So it is not just an analogy, it's actually bona fide one-to-one correspondence. So horizontal is a correspondence where one thing on the left, object on the left corresponds to object on the right. But the vertical is not a correspondence, it's an analogy. You're saying whatever happens for number theory questions like that should also be

true for questions, similar questions for cores over finite field. And something like that should also be true for human surfaces. This is much more vague, much less defined. You see? Andre vague. Why vague? Oh vague. I got you.

That's a good one. Yes. Andre vague. That's it. So Andre vague's genius was to see the analogy between these three fields and not just to see them kind of like as a dream, but in fact come up with some tangible conjectures, the so-called, you know, the vague conjectures, which is very closely related to this stuff, which is kind of Riemann hypothesis for this middle field, which was eventually proved by various people including Alexander Grothendieck and Pierre Deligne. But here we're not talking about that.

We're talking about now a hypothetical generalization of this one-to-one correspondence called Langlands correspondence in the original setup to a similar correspondence for this field and for this field. You see, that's the idea. And the point is that go from here to here is relatively easy, and it was actually clear from the outset,

but to go here is complicated. Okay. Okay, so we've lingered on slide 50 for too long, so for the synoptical glimpse that we want to give rapidly, can you go over just for the graduate students as a tease? Yeah, okay. So look Rosetta Stone is going like this analogies between these three fields Number theory here. I have a nice picture of Ivaris Galois because it's all about Galois groups

Like I said curves over finite fields Equations like this but for a fixed prime and we don't care about and it's not about Galois groups, like I said. Curves over finite fields, equations like this, but for a fixed prime. And we don't care about... And it's not just about solutions of this, but about some more concrete questions, and only for a fixed prime. And then there is a theory of Riemann surfaces and kind of geometric objects that are associated to them. So here I want to... And perhaps this is where we will stop,

so because I think that otherwise it's kind of going to be a little bit overwhelming. But I want to, I want to quote Andre Wey from that letter. Andre Wey wrote, my work consists in deciphering and trilingual text. I remember the actual Rosetta Stone. Rosetta Stone was a stone with three texts, text in three different languages about the same thing. And archaeologists were able to decipher them because they knew that these three texts referenced the same thing. So that's an apt analogy that he was talking about this trilingual text.

Trilingual because of these three fields that he's discussing. Number theory, curves over finite fields, and Riemann surfaces. And of each of the three colons, I only have disparate fragments. I have some ideas about each of the three languages, but I also know that there are great differences in meaning from one colon to another. In the several years I have worked at it, I have found little pieces of the dictionary. And I want to quote another place from that letter where he talks about how the process of mathematicians, how at first you come up with some conjectures, with

some analogies, and it's all very vague at that point. It's all like a dream. But eventually some of it may work out and kind of solidify and crystallize into something which is a rigorous bona fide theory. And in this passage, Andriy Vey talks about not only what is gained when you go from this dream to its realization, but also what is lost. He says, when this happens, gone are the two theories that you tried to, of which you saw some kind of nebulous analogies, but which you have now connected to each other. Gone their troubles and delicious reflections in one another, their furtive caresses, remember he's French, their inexplicable quarrels. Alas, we have but one theory whose majestic beauty can no longer excite us. Nothing is more fertile than this illicit

liaisons. Nothing gives more pleasure to the connoisseur. The pleasure comes from the illusion and the kindling of the senses. Once the illusion disappears and knowledge is acquired, we attain indifference." In the Gita, he's referencing Bhagavad Gita, the sacred text of Hinduism. He actually spent a lot of time in India. He learned Sanskrit. He met personally Gandhi. So he was a very deep guy. And he's referencing Gita here.

In Gita, there are some lucid verses to that effect. And then he goes, that's my favorite part. But let's go back to algebraic functions. So I think it's a good place to, to, to take a break. And, uh, uh, given all the knowledge we have acquired, I think we'll be, this will be a good stepping stone for us to explore the Langlands correspondence in this three, in these two other columns of the Rosetta stone until eventually we'll arrive at the geometric Langlands correspondence in these two other colons of the Rosetta stone until eventually we'll arrive at the geometric Langlands correspondence and some other versions which I will also

talk about the so-called analytic Langlands correspondence which are the versions of the Langlands program for human surfaces. Wonderful. I have to get you to expand on the pleasure comes from the illusion. So that to me sounds like a difference between Buddhism and Hinduism, because in Buddhism, they would say that the suffering comes from the illusion, or more specifically attachment, though still there's a heavy emphasis on removing an illusion. And here it says the pleasure comes from it.

Yeah, well, suffering comes from attachment, I would say, in Buddhism, since that's the Buddhist idea, from attachment. So the question is can you have an illusion without an attachment? I think so, I think so. An illusion is something, you know how in in both of these traditions and other Eastern traditions there is this concept of Maya. The world is an illusion, is a play of Maya. And so in a sense we have our own Maya in mathematics as well. So I think that I kind of see it a little bit as a comment on that.

In other words, there is a play. And this is important, I think, also, by the way, in terms of discussion about what is creativity and what is the difference between the human consciousness and artificial intelligence, human intelligence and artificial intelligence. And so there is a temptation, given how powerful our current models, large language models and so on, of artificial intelligence. It's very tempting to say that they can do everything human beings can do. And I disagree.

I think that the work of the great mathematicians such as Andre Wey, Robert Langlands, Alexander Grothendieck and others shows that true discoveries in mathematics are really points of departure from what is known. It's very hard to imagine that these discoveries can be made by simply reshuffling and correlating and interpolating known data. It comes from somewhere else. It comes from this inspiration, which is very hard to quantify. But all of us, mathematicians, scientists in general, in fact, all of us, I think, who do what we love, we all know this feeling of inspiration, this feeling when you start flying, when you are not in control anymore, when you are propelled.

It's almost like you're possessed by something, by an idea perhaps, by beauty, by love. And so to me, the story, that's what it's all about. It's about human creativity and how much we can do as humans when we let ourselves to do it and not to block ourselves. Thank you, professor. I'm glad I WhatsApp messaged you. Okay, well, I'm enjoying it as you can see.

It's my favorite subject, honestly. I can talk about this all day, but I also feel that at some point it becomes a little bit too much overwhelming and so I think that it's a good place to break, but I'm looking forward to the continuation. Firstly, thank you for watching. Thank you for listening. There's now a website, KurtJaymungal.org and that has a mailing list. The reason being that large platforms like YouTube, like Patreon,

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