Theories of Everything with Curt Jaimungal - Monumental Breakthrough in Mathematics (Part 2) | Edward Frenkel

Episode Date: October 2, 2024

Edward Frenkel is a renowned mathematician, professor of University of California, Berkeley, member of the American Academy of Arts and Sciences, and winner of the Herman Weyl Prize in Mathematical Physics. In this episode, Edward Frenkel discusses the recent monumental proof in the Langlands program, explaining its significance and how it advances understanding in modern mathematics. SPONSOR (THE ECONOMIST): As a listener of TOE you can get a special 20% off discount to The Economist and all it has to offer! Visit https://www.economist.com/toe Check out Edward Frenkel's New York Times Bestselling book "Love and Math" at https://amzn.to/4gPAVn1. Also, please consider following Edward on Linkedin at https://www.linkedin.com/in/edfrenkel/ LINKS: •⁠ Edward Frenkel's Twitter: https://x.com/edfrenkel •⁠ ⁠Edward Frenkel's Official Website: https://edwardfrenkel.com •⁠ Edward Frenkel's YouTube: https://youtube.com/@edfrenkel •⁠ Edward Frenkel's Instagram: https://www.instagram.com/edfrenkel •⁠ Edward Frenkel’s SoundCloud (DJ Moonstein): https://soundcloud.com/moonstein •⁠ ⁠Edward Frenkel's 1st TOE Episode: https://www.youtube.com/watch?v=n_oPMcvHbAc •⁠ Andre Weil’s letter on “Rosetta Stone” of Math: https://www.ams.org/notices/200503/fea-weil.pdf •⁠ ⁠"Proof of the Geometric Langlands Conjecture" (Papers): https://people.mpim-bonn.mpg.de/gaitsgde/GLC/ •⁠ Etingof-Frenkel-Kazhdan, “A general framework for the Analytic Langlands Correspondence” https://arxiv.org/abs/2311.03743 •⁠ Yuri Manin’s book “Mathematics and Physics”: https://www.amazon.com/Mathematics-Physics-Progress-Mathematical/dp/1489967842 •⁠ Edward Frenkel’s papers: https://edwardfrenkel.com/frenkel-biblio.pdf •⁠ Edward Frenkel’s previous lecture on TOE (Part 1): https://www.youtube.com/watch?v=RX1tZv_Nv4Y •⁠ Mathematics and Physics (book): https://www.amazon.com/Mathematics-Physics-Progress-English-Russian/dp/3764330279 •⁠ Richard Borcherds on TOE: https://www.youtube.com/watch?v=U3pQWkE2KqM TOE'S TOP LINKS: - Support TOE on Patreon: https://patreon.com/curtjaimungal (early access to ad-free audio episodes!) - Listen to TOE on Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e - Become a YouTube Member Here: https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join - Join TOE's Newsletter 'TOEmail' at https://www.curtjaimungal.org TIMESTAMPS: 00:00 - Edward’s Previous Appearance on TOE 01:15 - Discoveries in Mathematics 04:31 - Langland’s Program 11:02 - Counting Problem 14:58 - Symmetries of the Unit Disc 26:55 - Part 1 of Edward’s Talk 30:20 - Shimura-Taniyama-Weil Conjecture 40:02 - Quick Recap 42:38 - Langlands Dual Group 51:50 - Rosetta Stone of Math 01:00:10 - Riemann Surfaces 01:10:20 - Proof of the Geometric Langlands Conjecture 01:21:42 - Tribute to Legends 01:26:02 - Langlands Correspondence for Riemann Surface 01:43:30 - Galois Groups 01:53:33 - Other Objects Involved 02:10:40 - Outro / Support TOE SPONSORS (please check them out to support TOE): - THE ECONOMIST: As a listener of TOE, you can now enjoy full digital access to The Economist. Get a 20% off discount by visiting: https://www.economist.com/toe - INDEED: Get your jobs more visibility at https://indeed.com/theories ($75 credit to book your job visibility) - HELLOFRESH: For FREE breakfast for life go to https://www.HelloFresh.com/freetheoriesofeverything - PLANET WILD: Want to restore the planet's ecosystems and see your impact in monthly videos? The first 150 people to join Planet Wild will get the first month for free at https://planetwild.com/r/theoriesofeverything/join or use my code EVERYTHING9 later. Other Links: - 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 - Subreddit r/TheoriesOfEverything: https://reddit.com/r/theoriesofeverything #science #physics #math #podcast #sciencepodcast #maths Learn more about your ad choices. Visit megaphone.fm/adchoices

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kraken.com slash legal slash ca dash pru dash disclaimer for info on Kraken's undertaking to register in Canada. Professor Edward Frankel, thank you for coming on the podcast again. Welcome. The previous part one did fantastic and it's an exposition into the geometric Langlands recent proof. That's right. Well, good to be back with you and with you, Kurt, and your dedicated audience of your podcast The Theories of Everything, I was very pleased

by the response to our first part on the Langley's program. There was a lot of interest, it seems, and so it confirmed what I expected, that you have some of the most sophisticated people watching this podcast, people who do not shy away from technicalities, from the details of some mathematical theories. So I'm kind of, I'm really excited about continuing along these lines and revealing more secrets of the Langlands program. Oh, wonderful.

So one of the questions, and by the way, I was happily surprised about the reception. I knew it would be successful, but I didn't think the previous episode would be that successful. So that was a pleasant surprise. Definitely for me too. It feels like people are digging this stuff and there was a lot of comments and you know, a good response to our first part. So let's continue. Okay, so professor, many people are tuning in and they're hearing these terms like geometric Langlands correspondence, many people are tuning in and they're hearing these terms like geometric Langlands correspondence, many unfamiliar terms, analytic Langlands, number fields, Galois groups. Explain why this recent proof is monumental.

The way mathematics develops is that it's like you're searching for something in the dark room or in the dark space. You may be searching for a key. The way you would proceed usually is that we're trying to imagine how things could work out. This way we come up with some analogies first and then maybe with some more specific conjectures. In today's conversation, we'll have both. We will have conjectures. We will have conversation, we'll have both. We will have conjectures. We will have analogies. And we will also have theorems. A theorem is something which is

proved, which is valid universally. There is no argument in mathematics that something is proved or not. There is an argument initially when somebody writes a paper and then submits it to peer review. But these days it's more than that because most people actually post their papers on what's called archive, which is a kind of depository for all the new preprints so that people working in the field already have access to it before it's published, before it's peer reviewed. And therefore everybody can evaluate it for themselves.

And this way a mistake could be found by some graduate student, for example, not necessarily who is an official referee of this paper. And then they would make it known to others and so on. So there is a very... There is a scrutiny for all these papers. So they are all subjected to scrutiny by peers, not necessarily through the peer review process, which is how it used to be before the internet, but now actually by a lot more people. The point is that people who are interested, so it has to be something that enough people

are interested in so that they would actually spend time reading it and evaluating it. So once this is done, we say, okay, it's a theorem, so then the expectation is it's something unassailable. If it's true today, it will be true in 100 years or a thousand years. Same way as Pythagoras' theorem was true 2,500 years ago and still true today, and still we believe it will be true tomorrow in 100 years and so on. That's mathematical rigor. So you see you have this process of progress in mathematics.

You don't immediately come with theorems. You usually come with certain visions, with some intuitions, with some analogies, with some conjectures, and then you try to prove them. But the point is that on the path to conjectures, you kind of veer around, you're searching. So it's very important to check ourselves. So every once in a while, we actually have to come up with some hard proof where these visions, these intuitions become concrete, you see. So it's not just some wishy-washy stuff. But actually we say this is a theorem, this is proof, this is correct, this is known,

this is valid. And that's an example. An example of that is what we are talking about here. So this series of papers by Dennis Geisgurye, Sam Ruskin and others actually establishes an important conjecture in the Langlands program. And from my perspective, it's a very interesting result. But in addition to that, it also validates in some sense the whole project, or it's one

of the publications where you say, okay, so we must be on the right track if actually the conjectures we come up with can eventually be proved. Yeah, plenty of questions from the audience. So one of them that I recall was, what is the difference between geometric Langlands and then Langlands proper? Right, that's what I'm going to address right away

as we jump into the topic. Okay, so why don't we go straight into the slides? Okay. So I have some slides prepared into the topic. Okay, so why don't we go straight into the slides? Okay. So I have some slides prepared and also I will make some, do some writing actually, like in real time. So new developments in the language program. Okay.

So, but I want to start by giving a little recap of what we discussed last time. So the Langlands program, as I mentioned, is a kind of a giant project aimed at finding common patterns in different fields of mathematics. The original formulation by Robert Langlands, a Canadian-born mathematician who worked most of his life at the Institute for Advanced Study in Princeton, which was home to such luminaries as Albert Einstein, Kurt Gödel, John von Neumann, and others. So Langlands formulated these ideas first in 1967 in a letter that he wrote to his colleague, a senior colleague, a great mathematician on his own, right, Henri Wey. And the original idea was to link number theory, some specific questions in number theory,

to harmonic analysis, which is another field of mathematics. And the idea was that some of these questions are quite complicated if you look at them within number theory. But when you translate them to this other field, harmonic analysis, they become more easily solvable, more tractable. And I talked about some specific problems actually, right? So well, here's a picture of Robert Langlands sitting at his office at the Institute for Advanced Study in 1999. Well here's a picture of Robert Langlands sitting at his office at the Institute for

Advanced Study in 1999. As I mentioned in our conversation, in our first conversation, this is actually the office which used to be occupied by Albert Einstein. So Langlands had Einstein's office for many years. And the example that I gave was that of what we call the elliptic curves over modular prime numbers. So here we consider what I explained last time called as arithmetic modular prime numbers, the kind of what is often called clock arithmetic, just like our clock has 12 hours, and when we talk about time, we usually don't say, at least in North America, don't say

14 o'clock, we say 2 o'clock. So in other words, we take numbers, module 12. 13 means 1, 14 means 2, 15 means 3, and so on. And one can do the same thing if we replace 12 by any other number. So for instance, on this slide, you can see arithmetic with a clock that has seven hours. So then the possible hours are zero, one, two, three, four, five, and six, and seven brings us back to zero, eight is same as one, and so on. So seven is a prime number. It's not divisible by any other whole number other than itself and number one.

And we can likewise consider clocks with P numbernumber, p-hours, where p is one of those primes, 2, 3, 5, 7, 11, 13. And then we get a new numerical system, what mathematicians call a number field, corresponding to each prime number p and it is the arithmetic of this field that we are interested in when we consider what's called an elliptic curve. So now over here there was a point of confusion last time because this is an equation with a 2 and a 3 and some people were thinking wait a moment I thought Fermat's last theorem had to do with any n greater than 3, and so they were thinking this itself had something to do with

a to the n plus b to the n equals c to the n. The problem that I'm talking about, this counting problem, counting the number of solutions of an equation like this, of a cubic equation like this, modular prime numbers, is indeed closely connected to Fermat's Last Theorem. But indirectly, Fermat's Last Theorem is about a totally different equation, which will show up at a later slide. And it's not that we're solving that equation. We are trying to count the number of solutions of this equation at the moment.

I will explain more precisely what the link to the to the Fermat's Last Theorem is later. Right. So last time we had this equation, right? So by the way, feel free to jump back to part one of this conversation where you can find more details about this introductory part. I'm not going to go over the same details again, obviously, right? So I'm just going to give a quick recap. So the equation is like this, y squared plus y equals x cubed minus x squared is just one

example of the cubic equations that give us what's called elliptic curves. But the idea is that we are considering the same equation, but we are looking for solutions, modular, every prime number, you see. Because we can think of this equation as an equation where x and y take values in a finite field that is defined by a given prime number p. And what we want to do is count the number of solutions, right? And so here's a little table which shows you how many solutions this equation has Modulo prime number 2 3 5 7 11 and 13 which are the first few prime numbers you see so for instance last time we discussed in detail solutions module of 5 in this case There are four solutions there are four possibilities for X and Y such that the left hand side is equal to write to the right hand Side not exactly as numb as natural numbers, but as natural numbers module 5.

So that the difference is divisible by 5, right? And likewise for the prime number 7, we have 9 solutions and so on. So the third colon of this table is obtained by taking the difference between the first and the second value. For instance, here is minus 2, it's 2 minus 4 and so on. It's p minus the number of solutions. A general expectation is that the number of solutions is going to grow the same way as p. So there will be roughly p solutions on average, but there is a certain error, there is a certain deviation from p, and that's what this number a of p measures.

Of course, if you know a of p, then you know the number of solutions. You simply take, you have to simply take this number, you know, you take P minus this number and then you'll get number of solutions. So in other words, we're just relabeling things. But it turns out that those numbers AP, which appear in the third colon, are the ones which will be more useful in what follows. So the question is to find all of these numbers at once. So here I just calculate them. It basically can just do it with a pencil and paper

for the first few primes. Then eventually you can program this on a computer and you can find these numbers for prime numbers. Let's say less than 1000 or less than 10,000, less than 100,000. But we know that there are infinitely many primes and you would like to have an answer for every prime out of the infinite set of all prime numbers.

Is it possible? Right? So it seems like a daunting task. And the miracle is that you can solve it in one line. And that's the miracle of harmonic analysis. Remember, as I said, the Langlands program in this original formulation is about translating complicated questions of number theory, such as finding these numbers a of p, counting solutions of a cubic equation, modular prime

numbers, in terms of harmonic analysis. So this is what the solution looks like in harmonic analysis. We consider this infinite series, which we talked about in part one in great detail. Here you have this variable q, and you consider q multiplied by a product of factors. So there are two types of factors. There is like 1 minus q squared, there's 1 minus q squared squared, minus 1 minus q cubed squared, q to the fourth squared. Then there will also be 1 minus q cubed squared, q to the fourth squared, then there will also be 1 minus q to the fifth squared and so on. And in addition, you also have things where the powers are divisible by 11.

So you have 1 minus q to the 11th squared, 1 minus q to the 22th squared, 1 minus q to the 33th squared and so on. So it's very easy to write down these factors, right, up to any power of q, and then you can open the brackets and you find what we call a q-series. It's an infinite series where each term is a power of q, like here, q to the fourth with a coefficient 2, or q to the fifth with a coefficient 1, or q to the sixth with a one, or q to the sixth with coefficient two, and so on. And the statement is that you can find the numbers that you wanted, the numbers of, essentially

numbers of solutions of this cubic equation, module prime number p, as coefficients in front of the pth power of q, you see. So for instance, if you look at this term, it corresponds to P equal five, because we are talking about Q to the five, right? And the coefficient here is equal to one. And lo and behold, that's exactly the number that you have in the third colon, which is associated to five,

which is a prime number, which is one of the prime numbers, right? Yes, yes. There were two quick questions that the audience had here, if you don't mind. One was that we made plenty of references to sines and cosines when talking about harmonic analysis and then we go to this Q polynomial and there are no sines or cosines. So someone wanted to know what is the connection between this and harmonic analysis precisely. Right.

So I explained in part one that a prototype, trigonometric functions give us a nice prototype for the way harmonic analysis develops. It is a very special case and one could say one of the most basic cases of harmonic analysis, one could say the whole subject originated from the study of trigonometric functions, right? So in this case, the space on which functions are defined is just the real line. And we are considering functions which are invariant under the shift by 2 pi, or translation by 2 pi along the real line.

We know that the cosine of x and the sine of x, both functions are invariant under shifts by 2 pi along the real line. We know that the cosine of x and the sine of x, both functions are invariant under shifts by 2 pi. And there are more, namely you can say it can take cosine of 2x or cosine of 3x, cosine of any integer multiple of x, or sine of any integer multiple of x. They will still be invariant under shifts by 2 pi. And so the idea of harmonic analysis is to use these trigonometric functions as a basis for the space of all functions and try to decompose general functions as linear combinations of these basic harmonics they are called.

And so it is like decomposing the sound of a symphony into the notes of specific instruments, which each note being a kind of sine or cosine function, right? It was a particular frequency. Great. Responding to a particular frequency. So this is a basic example where your space is the real line. But a similar idea then was applied by mathematicians to other spaces. In this specific example, we're applying the idea of harmonic analysis to a unit disk on

a complex plane. Let me explain this. So but before we get to this, I just want to emphasize one more time what are we talking about. So we have this infinite series, right? And we have coefficients in front of prime powers of Q here after we open the brackets, like here, right? And basically the statement is that these coefficients are precisely the sought out

numbers AP, which are essentially the numbers of solutions of the cubic equation, modulo, the corresponding prime, right? And so you see, just to kind of appreciate the power of this result, one line of code gives you a simple rule for solving the counting problem for all primes at once. And there are infinitely many of them. So it's one formula to rule them all, you see. And kind of a colossal compression of information, or we could say finding order and seeming chaos.

That's one of the examples of what the Langlands program is about. But now let's go back to this idea. In what sense this is an object of harmonic analysis. So far it is just, it is just infinite series. Looks like an algebraic, like an arbitrary algebraic equation. It's a kind of algebraic equation where Q is actually just some formal variable, right? But actually it turns out that you can actually substitute for Q any number which is less

than, say, positive real number which is less than one. And moreover, you can substitute any real number which is between negative one and one, so its absolute value is less than one. But then you can be even more ambitious and say, I want to substitute a complex number. It turns out that this infinite product will converge for any Q which is a complex number whose absolute value is less than 1. So, here I want to remind you that complex numbers can be represented as points on a plane, right?

So each number has a real part and an imaginary part, and the real part will correspond to its x-coordinate and its imaginary part will correspond to its y-coordinate, right? And the absolute value of a complex number, let's say, let's call it q, the absolute value is just this distance between the origin on the plane, the zero point on the plane, and this point corresponding to complex number q. So if we want to consider all complex numbers whose absolute value is less than 1, geometrically it just means taking a disk of radius one, right? So these are exactly all the points such that the distance from the point

to the origin is less than one. In other words, it's a disk, but without the boundary, without the circle. Okay, so now what I'm saying is that if you have a point in this unit disk, open disk, in other words, without the boundary. Any point which is a complex number corresponding to this inequality or satisfying this inequality, then we can substitute it into this infinite series for Q and the series will actually

converge. And it will converge for real, not like 1 plus two plus three plus four and so on, where it can give you minus one over 12 in a certain esoteric sense, not on the nose, so to speak. Definitely one plus two plus three plus four and so on goes to infinity. But in this case, it actually converges to a specific complex number, you see.

Provided that Q is absolute value is less than 1, you see. If you take Q with absolute value greater than 1, it will diverge, same way as 1 plus 2 plus 3 plus 4 and so on. But we don't want to do that. We just want to take substitute Q, which satisfies this inequality. And so as a result, you get something which is a function on the unit disk, you see. And so it is this unit disk which is a playground for the harmonic analysis which is relevant to this function.

In the same way as the real line is relevant to the harmonic analysis of the standard harmonics, the trigonometric functions, cosine of nx and sine of nx. I explained this in more detail last time. What I want to say here is that this picture is meant to illustrate the kind of fundamental domains of the group of symmetries of this disk the same way as intervals from 0 to 2 pi, from 2 pi to 4 pi and so on are the fundamental domains of the action of translations by integer multiples of 2 pi on the real line. That group is relevant to the harmonic analysis on the real line.

Here we have what's called the group PSL2Z, the modular group, and its subgroups. In fact, in this example, these are the fundamental domains for a particular subgroup. I just want to give you a general idea how these things work. The function that we have when we look at it as a function on a unit disk turns out to be what's called a modular form. It has special properties with respect to the action of this group PSL2Z or perhaps its subgroup. This group has a family of subgroups which appear naturally in this setting.

So that is this notion of a modular form. And what I'm trying to say is this infinite series which contains the answer to all of the counting problems for this specific cubic equation, the Synthin series actually gives rise to a modular form. And this modular form is an object of harmonic analysis on the unit disk on the complex plane. So what we have done, therefore, is we have translated what seemed like an intractable problem in number theory, that is to say, counting the numbers of solutions of this specific cubic equation, modular all prime numbers, we have

translated this problem to a much more tractable problem of finding the coefficients of this modular form, which can be easily programmed much more easily than counting the numbers of solutions. You see? Yes. So let me see if I could do a 20 the numbers of solutions. You see? Yes. So let me see if I could do a 20 second recap of that. This is called a Q series.

This is a generalization of sine, cosine, which is a Fourier series. So that analogy is like Fourier series is the real number line as modular forms are to the complex unit disk. So this Q series is an example of a modular form. That's right. Or it has a corresponding modular form. What is the correct way of saying that? You could say that this Q series represents this modular form. You can write, you know how people are familiar with Taylor series expansion, right? Yes. So usually we talk about Taylor series in the context of real analysis. So let's say you have a single variable calculus, you have function on the real line and you

have a point and then you can write an expansion of this function in Taylor series in the neighborhood of this point. Let's suppose this point is zero, then this Taylor series is going to be an X series because usually we use the X coordinate in single variable calculus. So the difference is that in this case, first of all, we use the variable Q instead, which is a tradition in the subject. Don't ask me why, but it kind of fits with quantum also and so on, even though people who invented this, they did not have this in mind.

So it's very interesting how this labeling or this coordinates, coordinate naming appears in mathematics. Anyway, so we use this symbol Q instead of X, the more traditional in single variable calculus. And second, we are doing an expansion not on the real line in the neighborhood of zero, but on a complex plane in the neighborhood of zero. And there is the range of convergence which is also familiar in single variable calculus. For example, we know that one divided by one minus x, maybe I should give this example, you can have this one divided by one minus x, which you can write as a sum one plus x plus x squared plus x cubed and so on.

Very similar to this, much simpler of course, because here all the coefficients are equal to one. So this formula is true if x is a real number with absolute value less than one. So this is a Taylor expansion here. It's a Taylor expansion on the right-hand side. The Taylor expansion of this function, which is a bona fide function of one real argument on the interval from negative one to one. So we're doing something very similar, except we are now working with a complex plane, with

complex numbers. Q is a complex number now and not a real number, but the idea is very similar. Just like this series converges when X is a real number whose absolute value is less than 1, actually it also converges when x is a complex number whose absolute value is less than 1. So it's actually very similar. You can actually allow to expand from real numbers to complex numbers in this equation as well, and then the analogy becomes even more precise.

Just like this series converges for all complex x, which satisfy this inequality. So is this series, which is another way, it's just expanding this product, is converging when Q has the same property, satisfies the same inequality. Does it make sense? Yes. Wonderful. And the quick question that will probably get to the heart of it that I assume you're going to answer. We had an elliptic curve, which seemed to be an arbitrary elliptic curve.

And then we had some other Q-Series, and these are from different fields in math. Then the question is, how do we find, given a Q-Series, a corresponding elliptic curve and also backward? How do you find the Q-Series? So you're jumping ahead, right? So far, so far, so far, there is only one cubic equation, there is only one counting problem. And it looks like we have lucked out. So we found this thing, but it's not clear at all that this is a general phenomenon,

right? So of course, you're right, that's how mathematicians usually approach this, that if you observe something like this, which seems like a freaky coincidence, you say, okay, well, can this happen for more general cubic equations? And it turns out that yes, it does. And this is a subject of what is called the Shimura-Taniyama-Wei conjecture. In our first conversation, I talked about these three people

and I showed you a photograph of them at the colloquium at the conference in Japan in 1955. And this talk that Edward keeps referencing is called part one with the revolutionary proof that no one could explain until now. Until now. That's I believe the current working title. It's a fantastic title by the way. And that link is on screen. So if. That's, I believe, the current working title. It's a fantastic title, by the way.

And that link is on screen. So if you haven't watched that one, that's a prerequisite. Even though this is for if you're an undergrad in math, this one should be followable. This is largely independent because what we're going to focus on is on how this idea is play out in geometry. So in principle, if you're most interested in that subject, you can kind of just watch this short recap and then the rest of today's conversation.

But if you're more interested in the details, then part one is a place. And of course, there's also a book by Edward Frankel, a book I've read and recommend called Love and Math. It's a fantastic book and there are so many advanced math concepts that you encounter in third year, fourth year, some even in graduate school that were covered in this book in an elementary fashion. Thank you, Kurt.

I recommend you check it out. This book, I wrote this book about 12 years ago, 11, 12 years ago, it was published in 2013, precisely to explain the ideas of the Langley's program. So in fact, almost everything we talked about in part one, and most of what we will talk about today is in this book. So if you would like to really go deeper and to really understand these concepts and ideas more precisely, then this is one source for you. We will put also some other survey articles I have written about the subject in the description

of the video so that you have several resources for that. And in Love and Math, I try to explain it for the general audience, in other words, for non-mathematicians, which is the idea today as well, people interested in mathematics, but not necessarily specialists, not necessarily experts. But we will include also some sources for more advanced viewers, for most advanced audience, more advanced audience members. All right, so let's move on because we have a lot of stuff to cover. So I explained that it is a modular form and now it turns out that this is not a freaky

coincidence, it's not just a one-off thing, but in fact this link between a specific cubic equation and a specific modular form that we have discussed does have a vast generalization. What does it look like? It looks like a one-to-one correspondence between more general elliptic curves over the field of rational numbers. And more concretely, you can think of those elliptic curves as given by cubic equations very similar to the one we have discussed up to now.

That is to say an equation with two variables X and Y where one of the variables appears in degree two and the other one appears in degree three with coefficients in rational numbers. If so, we can by multiplying by the common denominator of these numbers, we obtain an equation with integer coefficients. And once we have that, we can consider it modular every prime, you see, so that we have an analog of the counting problem for any such equation.

And then it turns out that this counting problem for all primes can be solved by a particular modular form that is associated to this cubic equation or this elliptic curve over the field of rational numbers, you see. And this way one obtains a one-to-one correspondence between those elliptic curves over q and what's called modular forms of weight two with integer coefficients. There is one more technical thing that one has to say is that this is what's called modular forms of weight 2 with integer coefficients. There is one more technical thing that one has to say is that this is what's called normalized new forms. I'm not going to get into these details, but normalized means that it starts out with Q, doesn't have a constant term, starts out with Q and then there is some term, some coefficient times Q squared, etc. All of these coefficients are integers, and those coefficients in front

of prime powers of q will give you the number of solutions of the corresponding cubic equation or number of points on the corresponding elliptic curve, save perhaps finitely many primes in general. We were lucky in our basic example that in fact it covered all primes, but in general it could be maybe it covers all primes except number p equal 11 or something like that, you see. So almost all. But that's not such a big deal given that it covers it for all the rest of them, of which there are infinitely many, you see. I also want to mention one other thing, which is that in the basic example that we discussed, which by the way, as I mentioned in part one, I learned from Richard Taylor, a great mathematician

in Princeton, it's a very special case in that the modular form can be written as a product. In general, we do not expect modular form can be written as a product. In general, we do not expect that it can be written as a product. You see, so in general, the explicit formulas are much more complicated. That's why I want to present this case where it's easier to write down a formula, but I don't want to create the impression that it is always so. Okay. All right. So now I want to mention the link between the Shimur-Tanayama-Wei conjecture.

I should say this conjecture was formulated initially in 1955 by this Japanese mathematician Yutaka Tanayama, interesting human story. So the original formulation was actually incorrect. And then these two other mathematicians, Goro Shimura, who was actually his friend, also Japanese born mathematician who has worked in Princeton most of his life. And Andre Wei, whom I already mentioned,

the senior colleague of Robert Langlands, with whom Langlands first shared his ideas in 1967. So Shimura and Wei made some contribution. They kind of corrected the initial formulation by Tanyama. But what Tanyama did was kind of like a quantum leap, if you will. So he really came up with something which nobody expected. And Shimura, so Tanyama unfortunately took his life and committed suicide at the age of 31. He was this brilliant mathematician who died very young.

And there is a beautiful tribute to him written by his friend and colleague, Gor Shemura. And I remember one quote from there that Shemura says that Tanyama has this unique ability to make good mistakes. Right. And he said, it's not easy. He said, I tried to make good mistakes, but. And he said, it's not easy. He said, I tried to make good mistakes, but I couldn't, I failed. It is a very unique talent.

So what does it mean, good mistakes? It means that you reveal something important and maybe you mess up the details a little bit, but you kind of are on the right track. Other people maybe try to formulate it perfectly. And so they never kind of come up with a good idea. It's of course, ideally you would like to get the right formulation from the first goal. But in life sometimes you start out with the first draft is kind of maybe flawed in some

ways, but it can still inspire you to go deeper. I like this story because it shows you the drama of mathematical ideas, the drama of mathematicians trying to come up with these ideas, to come up with these conjectures, trying to prove them. Oftentimes, these conjectures may be wrong, so then other mathematicians come along who disprove them. But at the end of the day, it is collective enterprise of sharing these ideas and trying

to push the subject further to make progress. Yes. In the screenwriting world, there's an adage about you come up with your horrible first draft. Don't try to perfect it because you'll just write two sentences a day at most if you're trying to get pristine dialogue out the gate. But rather what you should do is just almost stream of consciousness, right? But it doesn't sound like what Taniyama was doing was stream of consciousness.

No, no, not at all. It was, you know, he was 90% correct. Let's just say. Imagine that, imagine a 90% correct screenplay on the first. Maybe 80%, okay, it's hard to measure. Still even half. Even half, okay, but he did something that,

he said he proposed something that nobody else had the insight or the courage, or perhaps perhaps both to propose, you see. And so his colleagues appreciated that in the fact that for instance his name is in the name of the conjecture, Smoltanian Way conjecture, also known as modularity conjecture or today is called, is known as modularity theorem because it has actually been proved. It was proved in 1995 by Andrew Wiles and Richard Taylor. Not in the most general case but in so-called semi-stable case but later on this proof was extended to cover the general case.

And this semi-stable case was already enough to prove Fermat's last theorem. You see, so the link between the two, Shmur-Tanyama-Wei conjecture and Fermat's last theorem is highly non-trivial. It was actually established by my colleague here at UC Berkeley, Ken Ribbett, in 1986. So what Andrew Wiles and Richard Taylor actually proved was the the Shimura-Tanyama-Wei conjecture in the semi-stable case. But because of the work by Ken Ribbett, this implied Fermat's Last Theorem, and here is the formulation of Fermat's Last Theorem.

So you see the equations are very different. The connection between the two is indirect and complicated, and we're not going to talk about it. But this is just to show how important this result is. That Fermat's last theorem was the most important, one could say, problem in all of mathematics for about 350 years. Many mathematicians have tried to prove it, but without success.

And so finally Ken Ribbett was able to connect it to the Schumann-Tany-Movec conjecture and then Andre Weiss and Richard Taylor proved Schumann-Tany-Movec conjecture. So that's how we finally got to the proof of Fermat's Last Theorem. There is still, as far as we don't have any kind of elementary proof of Fermat's last theorem itself that doesn't go through the intricacies of Schmurtanian-Movei. So it's not enough to take x to the n, y to the n. It's not enough to take y to the n to the right-hand side, so to speak. It's much more subtle.

All right. And so now, what does it tell us about the Langlands program? Well, guess what? This Shimur-Tanyama-Wei conjecture, the link between the counting problem for cubic equations and modular forms of way two, is a very special case of the Langlands program. So Shimur-Tanyama-Wei by itself is a vast generalization of this one example that we started with for a specific cubic equation and a specific modular form. But the Langlands correspondence or the Langlands program, the Langer's conjectures, are vast

generalization of this Shmurtanyama-Vey conjecture, you see. All right. So now, what do I mean by this? You see, let's recap. The original Langlands program was about difficult equations in number theory, such as counting numbers of solutions of algebraic equations, such as cubic equations. It can be reformulated in terms of more easily tractable equations in harmonic analysis, like finding coefficients of modular forms.

So Schimortani MFA could be represented schematically by this diagram. You have two types of objects on the left-hand side and on the right-hand side. On the left-hand side, you have a cubic equation like the one we considered. And we have numbers of solutions, or more precisely this number AP, which is, I remind you, is not exactly the number of solutions, but it's P minus the number of solutions, but it's P minus the number of solutions. Right, the error. The difference between the prime number itself and the number of solutions.

And on the right-hand side, you have objects called modular forms, more specifically, normalized new forms of weight two with integer coefficients. And it turns out that there is a bijection or as we say one-to-one correspondence between objects on the left and objects on the right under which this number of solutions or error of a number of solutions AP match with coefficients of the modular form in front of... So you have here it's BPQ to the P. That's the term in the modular form corresponding to the peace power where P is a prime. And Langlands correspondence, which is a giant generalization of this one-to-one correspondence is about more general representations of what's called the Galois group on the left.

In this special case that gives us this, these representations correspond to cubic equations, and certain numbers associated with them correspond to these numbers of solutions that we discussed. And on the right-hand side, modular forms get replaced by the so-called automorphic functions. So modular forms are special cases of more general automorphic functions arising in the Lagrange correspondence or Lagrange conjecture in the original formulation. It's just so remarkable that firstly the top even has an arrow in any direction and Then it's remarkable that it's in both directions It's remarkable that you can generalize that and that has two arrows right by way. There is a lot more

This is only the beginning okay Even more is going to become even more mind-boggling. Okay? So, and so first, the first twist that happens is the appearance of what's called the Langlands dual group, which is one of the biggest mystery of the whole subject. So the point is that on both sides of this correspondence, of this Langlands correspondence, we have a Lie group, what's called a Lie group, or one could call it reductive algebraic group. And on one side it's a group G, but on the other side it's not the group G, it's another one. And you see there's a very nice notation for it.

We put L as a kind of an upper index on the left. Yeah, why is it on the left? Usually indices are on the right. Langlands introduced this notation. And in fairness, he did not do it for his first initial, but it just happened to be his first initial. He used, he, it is named after what's called L function, L function. And so, but also happens to be, happens to be his, the first initial of his last name.

Yeah, it would be like, imagine if I said, this is a CJ space. And then someone's like, oh no, why are you naming it after yourself? No, no, no, that's a Kojois Manifold. As you know, on Theories of Everything, we delve into some of the most reality-spiraling concepts from theoretical physics and consciousness to AI and emerging technologies, to stay informed in an ever-evolving landscape, I see The Economist as a wellspring of insightful analysis and in-depth reporting on the exact topics explored here and even more. The Economist's commitment to rigorous journalism means you get a clear

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One that you won't regret. As a listener of Toe, you get a special 20% off discount. Now you can enjoy The Economist and all it has to offer for less. Head over to their website www.economist.com slash totoe to get started. Thanks for tuning in and now back to our explorations of the mysteries of the universe. And the notation kind of caught on. So people like this and we've been using this for more than 50 years. It's called the Langlands dual group.

He definitely introduced this idea. He definitely came up with this idea of the Langlands dual group. And this was one of the most revolutionary aspects of his theory. And why this group appears is still a big mystery. One of the goals of, you know, of people working on this subject, myself included, is to understand why the Lenglis-Duel group appears. There are some explanations. I came up with an explanation with my co-author and former advisor Boris Fegin many years

ago which uses what's called conformal field theory. But I'm still not satisfied with it. I think there is still a deeper reason. Other people have come up with some explanations, but in my opinion, the jury is still out. We still have not found the right explanation at the deepest level. In any case, there is this phenomenon. So you can ask, what about the Shmur-Tanyamovay conjecture? I said this is a special case of the Langlands correspondence.

So what are the groups which appear on both sides in this case? In this case, the groups that appear are GL2. GL2 is the group of two by two matrices with non-zero determinant. And let's say with complex coefficients, but in fact, you can consider with coefficients in other algebraically closed fields. And it's one of the simplest groups that mathematicians study. And in this case it just so happens that the dual group is the same. So at the level of the Schmurt-Tanyamovet conjecture you cannot actually see this phenomenon.

But if you start generalizing it to go from GL2 to more general groups, orthogonal, symplectic, or E8 is a famous group which appears in the study, in some physical theories and so on, then you will discover that there is this duality appearing, two different groups appearing on the two sides of the language correspondence. So here is one rabbit hole that we could go into, which is classifying all possible groups that appear here. So here, they are what's called reductively groups or reductive algebraic groups. And those are essentially products of Abelian groups and what's called semi-simple groups.

And so the semi-simple groups in turn are kind of like products of simple Lie groups or simple algebraic groups. And finally, those are classified by what's called Dynkin diagrams, more precisely, the so-called simply connected simple algebraic groups are classified by these Dynkin diagrams where you have four infinite series of diagrams like so, and then there are five exceptional ones. Why this is so is also a big mystery, but at least we know the classification, you see. And so it's a very interesting question to understand what this are. And the point that I'm trying to make is that under this language correspondence, a group

of G gets replaced by the language dual. And so for example, what this results to, for instance, that B and C and get replaced, the arrow has to be reversed. So if you reverse the arrow, BN becomes CN. And then there are more subtle transformations for groups of a different, of a given type, but with different, what's called a center. Each of them has a finite center. And there is also switching going on at the level of the finite center.

But I'm not going to get into details, but there is a very precise sort of combinatorial description of what this duality does to algebraic groups vis-a-vis this classification in terms of the Dynkin diagrams and the centers, the finite centers of the corresponding algebraic groups. It's not always the case that a reductive group's dual or Langlands dual, if that's what it's called, is also a reductive group? It's that a reductive group's duo or Langlands duo if that's what is called is also a reductive group. It's always a reductive group. But in particular-

So they're both always. Oh yes, that's right. Reductive, so Langlands duality is an involution on the set of all reductive groups. Each reductive group goes to another one. What we can focus on is the first approximation is what this does to simple algebraic groups. And then each simple algebraic group has what's called the Lie algebra, which kind of captures the local neighborhood of the group around the identity element. And it is this Lie algebras that are classified by this.

But the groups themselves, the connected groups which have a particular Lie algebra, there are only finitely many of them if this Lie group is simple. So they share this Lie algebra, but they have this additional parameter which is finite, an element of a finite set. Understood. This duality sends the Lie algebra to its dual, which only affects b and c, because it basically reverses the arrows. You can see here there is also an arrow, but if you reverse it, you get the same diagram.

Likewise here. But here, if n is greater than 2, this will really change the type of the Lie algebra. And in addition to that, there is a switch at the level of groups, which switches this finite parameter that I mentioned. Anyway, this perhaps is a bit too technical. I'm not even sure you should keep this part about the exact, exact, how the dual is. That's all right. Nothing exact, exact how the duel is. That's all right. Nothing is too technical for the toe audience. They crave and they appreciate the details.

Well, it's apparently nothing is too technical for people in your audience. Okay. I feel comfortable doing that. All right. So anyway, this was to say that there is this phenomenon, there is this twist, the group that goes to the Langlands duel, which kind of indicates that this connection cannot be trivial, because how can you possibly go from one group to another? They have nothing to do with it.

This group of type B is actually an orthogonal group in the odd dimensional vector space, and the group of type C is a symplectic group. A priori have nothing to do with it. So the fact that this duality appears, going from Langlands group to the Langlands dual group, as we go from one side to the other side of Langlands correspondence, suggests that this correspondence

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So now, so here's and 9th. All right. So now, so here's what we're going to do. We are going to discuss a generalization of this original formulation of the Langlands Program. So you could say, okay, wait a minute. You said that Shimur-Tanyama has been proved, but Shimur-Tanyama is a tiny part of the general Langlands correspondence. So why don't we focus and try to prove it in other cases?

And there has been a lot of work in the last 50 plus years on that. But it's extremely complicated. So we are like a guy who is looking for his key under a streetlight. And people ask him, why are you looking under the streetlight? Didn't you drop the keys over there? And he says, yes, but here at least I have a chance to find it. So we kind of follow the guy under the streetlight and say, okay, well, the original formulation maybe is too hard.

Let's find out where else, in what other areas of mathematics can these patterns be observed? And maybe this will teach us some lessons, maybe we can learn some insights by kind of shaking it up a little bit and going outside of the original realm of the Langlands program. So that explains why we're interested in the reformulations of these original ideas, but in other domains, in other areas of mathematics. And so what helps us here is this Rosetta Stone of mathematics, which I also talked about last time, which was proposed by André Wey, whose name has already been mentioned several times, in a letter to his sister, Simone Wey, from prison actually in France in 1940.

He wrote about the analogies between these three areas of mathematics. One of them is number theory, and that's what appears in the original formulation of the Langlands program, right? But then there are two other areas. One is called curves over finite fields, and the other one is called Riemann surfaces. And he showed that in fact these three areas are, there are many analogies between them. And so you can sort of move back and forth and translate various statements between these

three areas of mathematics. And Revé wrote in this letter, my work consists in deciphering a trilingual text. So it is indeed just like Rosetta Stone in some sense. Of each of the three colons, he says, or these three areas, I have only 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 of them to another. In the several years I have worked at it, I have found little pieces of the dictionary. So now you see we have this additional input. The Langlands program is about connecting number theory to harmonic analysis.

But guess what? Number theory has these two other areas which are analogous to it. So the question that we can ask is how does Langlands correspondence play out in those other areas? You see? So we're kind of starting to play a three-dimensional chess. The original chess game was trying to relate number theory and these automorph functions, right?

That's the original version. But because we now know that these two other areas are analogous to number theory, it is natural to ask whether one can observe a similar correspondence in those areas as well. And so in fact, this area, which I will explain in more detail in a moment, curves over finite fields, is actually very closely connected to number theory. And the objects which arise here on the other side are very similar to the objects which arise in the number theory version of the Langlands correspondence. In fact, when Langlands wrote his original proposal, he talked about both of these areas,

both of these areas and the other sides. Because for these two areas, the formulation is very similar. So in fact, both of these are in the original formulation. It's just that up to now, I only focused on the number theory version. But this is totally- From left field? Totally from the left field. And the only reason why we're even there to believe that there is some kind of language correspondence for Riemann surfaces is because of Andre Wey, because Andre Wey has taught us that Riemann surfaces are in many ways

analogous to these two areas. So therefore, the most audacious of us asked, well, if so, could there be a Langlands correspondence here at the level of Riemann surfaces, you see? So now we are getting closer to what's called the geometric Langlands correspondence. In fact, it turns out that the Langlands program patterns or this Langlands correspondence can indeed be observed in each of the three areas of the Andre Weiss Rosetta stone that I have listed in the previous slide. And the idea is that we want to study how this Langlands correspondence is realized

in each of those areas because we believe that this will help us to better understand what is this all about. The best understood, in fact, is the middle area. Middle area meaning curves over finite fields. That's a kind of a sweet spot, which kind of is a turntable, as Andre Wey called it, is a kind of a bridge between these two. In the case of curves over a finite field, on the one hand, you don't have some of the difficult aspects of number theory, but it's close enough to the objects

of number theory to kind of have a similar formulation. And at the same time, we can use geometry because we're talking about curves. We are talking about some algebra geometric objects here. So in fact, one of the biggest developments in the Langlands program across all three domains or all three areas of mathematics has been the proof of the Langlands correspondence in the basic case of the group GLN. We have talked earlier about the group GL2, which appears in the Schimur-Tanyama-Wei conjecture from the perspective of the Langlands correspondence. GL2 is the group of 2 by 2 matrices with non-zero determinant with respect to the usual matrix multiplication. Likewise GLN is the group of n by n matrices

where n is an arbitrary positive integer which have non-zero determinant. It's important to have non nonzero determinant because then you have the inverse, the multiplicative inverse. So if you consider all n by n matrices with nonzero determinant with respect to multiplication of matrices, you actually get a group. This group is called GLN. And this is the first group in which you want to try all of these correspondences, all of these conjectures. And in fact, in this case, it's no longer a conjecture, it is a theorem.

First Vladimir Dreamfeld, a brilliant Soviet American mathematician, proved it in the 1980s for N equal two, for GL2. And then a French mathematician, Laurent Laforgue, found a way to generalize this, which was actually very hard, from n equal to arbitrary n. In the early 2000s, both of them received Fields Medals for their works. So this is a highly-priced achievement in mathematics of the last 40 years, perhaps. Okay? Many interesting results have also been obtained in the number theory setting proper in recent years.

But since we are more interested in the geometric correspondence, geometric conjecture, which arises in this area of Riemann surfaces, I'm not going to talk about this today. All right? So finally, let's talk about the case of Riemann surfaces, which was our original goal, right? That's where the geometric Langliss correspondence is. That's where, that's what this recent series of works by Gates, Goury, Raskin, and others is about. So what are the Riemann surfaces?

First of all, these are the Riemann These are examples of human surfaces. The sphere, the surface of a donut, surface of you could say Danish pastry or something. Yes. So I guess this one is Homer Simpson's favorite. Sure. Now these are compactann surfaces. Is that important? Yes. Well, there is a more general formulation where we consider non-compact ones, where we remove points. You may see sometimes in papers written about the subject the term unramified or ramified.

So we are going to specifically talk today about the unramified case. And the unramified case means that we are considering compact human surfaces. The ramified case corresponds to having finitely many marked points and kind of removing those points so that you allow some sort of singularities at those points. Marked points. Marked means that you marked some sort of singularities at those points. Marked points. Marked means that you marked them, you chose them. Marked is a word mathematicians use when they want to say that they specified something. It's like you marked it, you marked it.

Here's the first one, here's the second, and the third one. That's just the way people say it. You can say chosen or specified. It's the same. All right, so these are the Riemann surfaces. And the question we're going to discuss now is how to formulate the Langlis program for these objects, for these Riemann surfaces. As if the Langlis program was not complicated enough already. We are now going to move to a different domain.

We are going to move under the street light, if you will. And try to find the key there. So for human surfaces, here is one more twist. In fact, for human surfaces, there are two versions. The first one was initiated by great mathematicians, Pierre Deligne, Vladimir Dreinfeldt, whom I have already mentioned, and Gerard Lamont in the 80s. And it is called the Geometrical Anglans Program. I should also mention another mathematician who is very heavily involved in this, Alexander

Bellinson. Bellinson and Dreinfeldt actually made perhaps the biggest contribution to this Geometrical Anglans correspondence for human surfaces, in the dream field in the in the 1990s and 2000s and it is it forms the kind of the cornerstone the foundation of the recent work by gaze Guri Ruskin and others But one thing to note is that this formulation called geometric Langlands program or geometric Langlands conjecture or geometric Langlands Correspondence is very different from the number theory version, which is where the Langlands program was started, initiated, originated. Instead of functions, one considers kind of esoteric objects called

sheaves. And I'll try to give some pointers, some ideas what these objects are. But this is what makes this formulations really sophisticated Now for a very long time up until about five years ago The prevailing prevailing wisdom in the subject was that this is the best that you can do in other words There is no formulation in terms of functions for human surfaces That was a general belief that the only thing one could do for human surfaces was the theory in which the traditional role of functions is taken over by Sheeves. But interesting enough, in the last five years, a new version of the Langlands correspondence

for human surfaces was proposed, and I've been involved in this actually in a series of works by my two co-authors, Pavel Ettinghoff and David Cashdon. We have found a function theoretic version, you see, which is much closer to the original formulation in number theory and for curves over finite fields. This version is called Analytic Langlands Program. So today I'm going to talk about both of these versions. They are not in contradiction with each other.

They actually complement each other. They are both interesting, you see. And if you asked me five years ago, if we were doing this conversation five years ago or perhaps six years ago, I would just talk about the shift theoretic version because that was the only thing available. In fact, one of the things that prompted me and my co-authors to do our work and to develop this new analytic Langlands correspondence, this function theoretic version, was the work of Robert Langlands himself. In 2018, he was awarded the Apple Prize, one of the most prestigious prizes in all of mathematics. So I actually traveled to Oslo, Norway to give a public lecture after his award ceremony

where the King of Norway awarded Robert Langlands this prize. So around that time, Langlands actually published a paper in which he argued that there must be a function theoretic version He did not like the shift version and in some sense I feel I feel guilty because I spent a lot of times trying to explain it to him And I guess I didn't do a good job Why didn't he like the chief version? He's more traditional mathematician for for mathematicians of his generation sheaves are kind of an of not exactly anathema, but it's

like bells and whistles that you don't need. But he's of the generation of Grotendieck, no? Yeah, but he comes, he's sort of on the other side of the spectrum. Oh, the Serres side? Well, no, Serres was closer to Grotendieck. The division between Serres and Grotendieck is not that big. Grotendieck, Alexander Grotendieck, one of the most brilliant mathematician of the 20th century, is someone who brought the ideas of sheaves and categories into mathematics,

as well as others like Jean-Pierre Serre, who was his contemporary and still alive, Grottinger died in 2014. But at the same time, you also had a development in parallel, which was kind of more concrete mathematics, where people did not work with categories or sheaves. They worked with representation theory proper, where you have bona fide functions and you have Hilbert spaces and you have operators. Which by the way is much closer to physics, to quantum physics because the kind of problems

that people considered was actually very much influenced and inspired by developments in quantum physics. Among them were Langlands, Harish Chandra was a great Indian born mathematician who was a professor at the Institute for Advanced Study. And I also want to mention another great mathematician, Israel Gelfand, who was another important figure in the subject. So it's not that they were against the sheaves, it's just that their style was much more concrete and much more rooted in classical mathematics, classical mathematics of functions, of spaces, operators and so on, as opposed to sheaves, categories and functors.

Did you and your collaborators coin the analytic Langlands program and create it or did you just develop it and popularize it? We did, we started it. So six years ago there was no analytic Langlands program. No. If I was to search that. The first time I talked about it was actually, so in 2018, soon after Langlands received the prize, so he got Apple prize in May, received Apple prize in May of 2018. And he gave a

talk about his ideas. The main point of his talk was there has to be a function theoretic version. Now he himself was not able to do it, and he and I had a little tension about this, but he had absolutely correct idea that this version has to exist. In many ways, that inspired me to look closer into this. There was also another mathematician, mathematical physicist, Jörg Teschner, who works in Hamburg, who also worked along these lines, and he actually published a paper around the same time.

So then, maybe a year earlier, about some special cases of this, how this could work out. In November of 2018, there was an Abel Symposium organized by the Abel Prize in Minnesota, in the University of Minnesota. And this is where I gave my first talk about these ideas. And so then with Pavel Ettingov, who by the way was my classmate in Moscow when we were kids basically, when we were in college. And David Cashdon, who is one of the luminaries in representation theory, he was the favorite

student of Israel Gelfand. So in some sense, he's kind of one who has continued the ideas of Gelfand more than anyone else perhaps. He used to be at Harvard University. In fact, when I first came to Harvard, he was one of my mentors and he moved to Jerusalem about 20 years ago where he is now. And so with Pavel Tengof and David Kazinian, we have worked on this for the last five years. We've published four papers on this.

So there is this now new formulation of the language correspondence for Riemann surfaces in the language of functions and Hilbert spaces and operators and spectral problems for those operators. In addition to what was known since the 1980s due to the work of these brilliant mathematicians, which came to be known as the Geometric Langlands Program, in which instead of functions one considers sheaves, instead of Hilbert spaces one considers categories Instead of operators, one considers functors, and so on. And the recent work by Gates, Curry, Ruskin, and so on is about that formulation.

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So let me actually show you. This is the webpage of Dennis Gaysgury, who by the way has been my co-author, my collaborator for many years. And a lot of this of our work is actually used in this recent work by him and his colleagues, his co-authors. So Dennis has this webpage about the proof of the geometric Langlands conjecture. Here's a list of his co-authors. So they made this sort of final push to actually give a proof of this statement for arbitrary

groups, which is a really big deal. It's a really wonderful achievement. So there are five different papers listed on this website. Haven't been published yet, haven't been refreed yet, but they're still in a preprint form. But I don't expect that there are any issues that one finds. In some sense, it's a culmination of an effort by many mathematicians. So all five of these are new papers that came out simultaneously? I think Dennis has maintained this page for years.

And in fact, one could say already about 10 years ago, the outline was kind of, he wrote a paper which was called the Outline of the Proof in 2014, I believe, where kind of the main, the foundation was laid out already. But there were so many technical issues that one needed to deal with. And remarkably they are addressed in this new series of papers. But those papers I think were written over the years. They didn't just dump them all at once. So here's what the first page of one of them looks like. It's a second one, second paper in the series.

And so I want to show you something which is kind of to give you an indication of what a grand project that was and how many people have been involved. Just to give you an idea how much goes into this. It really takes a village, if you will, as the proverb says. So the first thing they mention here, this is the acknowledgments to one of the papers, one of the papers, in which they give tribute to some of the mathematicians who came before them whose results they're using, which by the way, I'm impressed by the way they are kind of acknowledging the contributions of other mathematicians.

They're showing a great example for young mathematicians. This is how it's done. And so the first two people they mentioned, Alexander Belenson and Vladimir Dreamfeld, whom I have already mentioned, so they kind of initiated the whole thing in the early 90s. They started developing the geometric Langlands theory using the kind of tools that these new papers are using as well. They say countless ideas can be traced back to BD, which is a paper by Bell and Sandrinfield. So then they give a shout out to me and my co-author and my former advisor Boris Fagan. So indeed we contributed on the side of representation theory of what's called cut smoothie algebras.

And this gives a kind of the main technical tool for establishing the geometric language correspondence. But you see how many things. I'm actually looking at it as like, oh my gosh, you know, like looking back and they, they, they even shout out to semi-infinite flags. Yes. We, we, we, we developed that the offers actually offers were developed by balance and reflux but me or offers. Yes.

BRST functors. Yes. Making more the modules generalization of you helped develop BRST functors in this particular context. Yes. No, the original BRST. But we use BRST in a more generalized sense these days. And then there is this Fagen-Frankel Center, which plays the central role as they say.

And so then of course, then there is another deep idea of factorization. And so they mentioned- There's also Jacob Lurie down there. And Jacob Lurie. Yeah, Jacob Lurie, a tremendous, works of tremendous importance, which kind of like gives you proper language for what's called higher algebra. So just want to show you because, you know, people who are not mathematicians, it's very hard to imagine what goes into kind of like a project like this.

When somebody says, says okay we have proved this conjecture and this subject has been developing for like 40 years you know so like what kind of just to show what amount of work goes into this and how we always when we make these contributions how we always stand on the shoulders of others who came before us in this case actually Dennis and I Dennis Gait-Gurion and I actually wrote eight papers together on this subject closely related to this Katz-Muhl-Lübsberg at the critical level, which gives the tool for connecting, for the establishing of geometrical and its correspondence.

So in fact, some of those are important in this series of papers. I have been involved in this project and up until a few years ago when I switched to this new version which is the function theoretic version called the Analytic Languages Correspondence. But I'm very pleased and impressed that my friends and colleagues have been able to push this through, bring it to completion. Now this hasn't been published yet so in principle in mathematics we say, okay well, until it has been published in the journal, properly reviewed and so on, we cannot say that it's finished. But I have all the confidence that the proof is correct. And it is indeed a very essential achievement because as you see in the subject, there are

a lot of statements which are conjectural. And so every time you have somebody actually proving something, giving a precise, giving a proof, it's important. It kind of lifts all boats in some sense because it gives us confidence that what we're doing is right, that our general sense, our general understanding, our general intuition are correct. So that's one of the consequences of this series of works by Denis Gayskory, Sam Raskin and others. All right? So then I want to...

Well, I also want to show you the first page of my recent paper with Pavel Ettingov and David Cashdon about the analytical English correspondence, you see, which was recently published in this journal. It was actually a special issue in honor of a wonderful Italian mathematician, Corrado de Concini. You mentioned writing screenplays earlier. Fun fact, Corrado de Concini's father,

Ennio de Concini, was a famous Italian filmmaker. He actually got an Oscar. He co-wrote a screenplay of the film Divorce Italian Style, 1961 with Marcello Mostroiani. And he received an Academy Award with his co-authors for the screenplay. Anyway, it's just kind of an interesting connection to film. And what was his son's contribution to this? So there was a... This paper appeared in a volume, in an issue of a journal,

Pure and Applied Mathematics, in honor of his 70th birthday. So we dedicate the paper to him with admiration. He's a great mathematician, great human being as well. So there is this now, this other thread. In addition to geometric Lenglans correspondence, there is now this analytical Lenglans correspondence. It doesn't mean that it supersedes, just to be clear, it doesn't mean that it's better than the other one. It's different and the two are connected,

so they complement each other. I will comment more on that as we proceed, all right? Sure, the Langlands would probably say this one's better. So the Langlands would say it's better. Yes, so this is what happened. It's like, I mentioned in part one that I collaborated with Langlands. We wrote a paper together.

This was about from 2008 to 2010. I spent a lot of time talking to him. Our work was not in the geometrical Langlands, Chris, but more like a traditional one, but in the curves over finite fields mostly, as well as number field case. With another mathematician, by the way, Ngo Bao Chau, who is a professor at University of Chicago. But during this time, I spent a lot of time talking to Langlands, and he kept asking me,

what is this geometric thing it's about? What is this about? What are these shapes? How do they fit? And so on. And I tried to explain. And the result of it was that he hated it. Okay, so sorry to say.

But it wasn't that he couldn't understand it. It was that he understood it but didn't like it. He didn't like it. He said, this is totally extraneous to the subject. The shifts have no place. There has to be a functional theoretic version. He even went as far as saying during his talk at the Abel prize symposium, not symposium, but it was kind of acceptance speech after that,

he said that they might as well remove my name from geometric correspondence. I don't want it to be at that. I want to be at the context. Which I think is a little too much because it is really, to me, geometric linear corresponds is right. It has a place in this general overarching theory. But different matrices have different tastes, and they have different styles. This is the reason why I'm sharing with you these stories, is to show you what a robust dialogue we have with each other.

That even though we respect each other tremendously as practitioners and as human beings, there is a lot of disagreement from time to time about which direction to take, which subject is more interesting than other, and so on. So in this case, here is Langlands. He's in his 80s, right? But he's still very full of energy. And he's like, no, I don't like this shift theory formulation. There has to be a function theoretic formulation.

Which by the way, at that time, in the subject, pretty much nobody believed it. And he tried to do it by himself. He couldn't. But the fact that he pushed for it, it really inspired me and my co-authors to try to do it. And it turns out indeed that it was possible to do. I should mention, as I mentioned, there was a York Teshner, was another mathematician in Hamburg, a metaphysicist who tried to do something similar. There was another Russian born mathematician who works in France, Maxim Konsevich, who

considered that in special case and so on. So it's not like we were the only ones who did it, but we did it systematically as a kind of a really a branch of the language program, which is what we called the analytical language program. Now is that the same Konsevich who generalized mirror symmetry to homological mirror symmetry? So Konsevich is a very brilliant mathematician who has worked in many different areas and made very important contributions.

At some point he got interested in the Langlands program and so he tried to do kind of what Langlands was suggesting. It didn't go very far, but he did come up with a few calculations in some special cases which we kind of see now from our perspective where we have this general theory that this is a special case of what we are considering as the analytical language correspondence. Got it. Yeah. Okay, so let's move on.

And let's talk about, since we're talking about people, so I wanted to actually pay tribute to two giants who have influenced the subject. One of them is Israel Gelfand, who died in 2009. He was a patriarch of the Moscow Mathematical School, world famous. And the other one is Yuri Manin, who actually died last year, in January of last year. And he had his own mathematical school, it was a little bit smaller than the Gelfand's, but they were closely connected and they were extremely influential, not only in the Soviet Union but all over the world. I want to mention them because in many ways all of these ideas that I'm going to talk

about that have to do with the Halangansk correspondence for Riemann surfaces, both the geometric formulation and analytic formulation, in some ways can be traced to these two giants, Kelfand and Mannion. So for example, Bellinson and Dreinfeld, these two mathematicians who are perhaps most responsible for these developments, were his students, you see, of Yurimania. They were also very closely connected to Gelfand. So Gelfand had this famous seminar. By the way, in Love and Math, I tell the whole chapter about his seminar and what it felt

like to be a young student attending the seminar. It's really hard to capture it in a few words, but I really wanted to mention our debt to these two people because I was not a student of either of them, but my teachers, Boris Fagan, whom I have mentioned, and Dmitry Fuchs, were students of Gelfand. So I was Gelfand's grand student and I attended his seminar. And when he came to America, it was the same time as when I came in 1989, we spent a lot of time talking to each other and so on. So I have been greatly influenced by his, not only his ideas, but his general approach to mathematics. One of the things that he taught his students

is to pay attention to physics, to pay attention to quantum physics. He was always attuned to the relationship between representation theory specifically and functional analysis and quantum physics. And he obtained some absolutely foundational results in this area. So it's kind of some of them at the level of von Neumann, you know. So there is this famous Guelph-Neimark theorem about C-star algebras and so on. But he worked in so many different fields. He also worked with computer scientists. He worked with biologists.

He was really a renaissance man. And all of the people that I have mentioned who have worked in the subject, in this geometrical language and analytical language, have been greatly influenced by Gelfad. For instance, David Cashter, my co-author and collaborator on the analytic language response was one could say his favorite student of Gelfand. In some ways, he's perhaps the most connected to this whole circle of ideas. My other co-author Pavel Ettingov was a student of Gelfand as well to some extent. But also Yuri Manin played a great role.

So I think his ideas also kind of like between the lines, between the lines, between the formulas, between the ink of all of these papers. Interesting. And now I wanted to also cite this quote from a book by Yuri Manin. This book is called Mathematics and Physics. He was also someone who has contributed to not only to mathematics but also to quantum physics, quantum field theory, gauge theories and so on. A very beautiful quote. What binds us to space-time is our rest mass, which prevents us from flying at the speed

of light. When time stops and space loses meaning. In a world of light, there are neither points nor moments of time." Right? Because light travels with the speed of light. So light is everywhere and everywhere. Being woven from light would live nowhere and no when. And then he says, only poetry and mathematics are capable of speaking meaningfully about such things.

Interesting. All right, but now let's get down to it. So what are we doing? We are now discussing the Langlands correspondence for Riemann surfaces. And what I want to explain is how actually we come up with a formulation for Riemann surfaces. In the first place, in what sense is it connected actually to the original formulation of the Langlands correspondence or Langlands program in number theory?

And the first thing to observe is that what is the most important object in number theory? It is the field of rational numbers, most important number field. So everybody's familiar with integers, right? Integers are whole numbers, both positive and negative. And then we have fractions. A over B, where A and B are relatively prime integers and B is non-zero. So these fractions are called rational numbers and they form what mathematicians call a field, which is denoted by Q.

To say that it's a field is simply to say that there are two operations that we can do with rational numbers, addition and multiplication. And these two operations satisfy certain properties. Right? Right. So, now that's what happens in the number field. Remember in the, I'm talking here about the Andre Wey, Rosetta Stone.

Right? So in Rosetta Stone that I mentioned earlier, there are three different areas of mathematics. One is number theory, the other one is so-called curves over finite field, and the third one is Riemann surfaces. So I'm starting by giving a more precise explanation of what the analogy is between them. To explain this analogy, we have to look in number theory at the field of rational numbers. The analogous object in the second area is a function field. And here's what it looks like.

We have to fix a finite field. Remember we talked about arithmetic modular prime number where we have only numbers from zero to p minus one. With p taking us back to zero, then p plus one is one and so on, right? We have addition and multiplication here, modular p, which also satisfies the same properties as additional multiplication in rational numbers. But the analogy is not between q and fp. The analogy is between q and what's called fp of t, where we are considering fractions

like so, where f and g are polynomials with coefficients in fp. Okay. You see? So, let me explain this a little bit more precisely. So rational functions valued in FP? That's right. So you see, let me do kind of a, divided this into two parts and explain it as a kind of a translation between two languages.

The highlighting the objects which are analogous to each other. The first thing to start with on the left hand side, left hand side has to do with number theory. We have the set of integers, right? So you have like, you know, minus one, t, which is where t is with square brackets, which is all polynomials, polynomials over FP. So what does it look like? So for example, you can have a polynomial, so you have a0 plus a1t plus a2t squared and so on, plus some an t to the n, where each

ai is an element of fp. You see, so it's a very similar concept to what we study, I suppose we all study in school, polynomials with real coefficients, right? So a typical such polynomial would have this form where a0, a1, and so on is a real number. But now we're considering these expressions where each coefficient is an element of this finite field fp. You see? So now, why are the two things connected to each other or similar to each other? What's the analogy? To see the analogy, let's consider a special case when p is equal to 2.

Let's consider a special case when p is equal to 2. In this case, ai is either 0 or 1 because the field f2 actually has only two elements, 0 and 1, because 2 is already back to, takes us back to 0, right? So then a polynomial would look like this, for example, it would be like 1 plus 0 times t plus 1 times t squared, and so on. For example, here's an example. 1 plus t squared is an example of a polynomial over f2, right? So what you see is that you have a sequence of 0s and 1s, really. The information that it contains is a collection of coefficients.

And the coefficients is a finite string of numbers, but these numbers are only zero and one, so the binary. Yes. So it's a collection of binaries. There are n of them, where n is a degree of the polynomial. But if you think about natural numbers, which are positive integers, you can also represent them in binary form, right? And you also get a string of zeros and ones you see so this is one way to see

The link between the two that you see it is size-wise. They kind of look the same For instance number five can be written as One plus zero times two plus one times two squared So you see it? You kind of say, all right, well, here you have 1 and here you have 1, here you have 0, and here you have 1. The difference is that here we take 2 and then 2 squared and 2 cubed and so on. And here we have this additional variable t. And we take t, t squared, t cubed, and so on. And here we have this additional variable t, and we take t, t squared, t cubed, and

so on. And so as a result, if you take the sum of these two, for instance, you're going to get number 10, which will actually... You can do this calculation the way we normally add decimal representations of numbers. You will have a carry, right? You will have a carry because 2 squared plus 2 squared is 2 to the, 2 cubed, right? Can we do that with any number if you just change the base? Sure, and we can do it, that's right, we can absolutely do it with every prime number, but I want to give you this example because it's most obvious in some sense, because here the choice is only 0 and 1. But don't think that these two objects are the same, polynomials in one variable and natural numbers, because the addition and multiplication in the two domains are different. It's just that they look the same.

They both are described, objects are described as sequences of zeros and ones. But with different arithmetic. So this is just to indicate why mathematicians, this is the first indication why these two things are similar. Okay. Remember, I'm explaining an analogy. All right. So this is going somewhere.

You're just giving the first hint. Oh, yes. The first of many, by the way. Okay. Yes, yes. This many, by the way. Okay. Yes, yes. This is the very first observation. But remember, there are two things we are discussing, and it is very important not to confuse them.

There are some correspondences, there are some theorems, there are conjectures where you actually say this set and this set are in one-to-one correspondence. For instance, Shimur-Tanyam way conjecture. What does it say? It says that there is a one-to-one correspondence between cubic equations of certain kind and these modular forms of way two with some properties, right? That is a mathematical fact. It could be a conjecture, it could be a theorem, but it's precise.

What we're talking about now is an analogy. There is no precise statement here. It's an observation that these two fields are very similar. In this case, well, so far it's not fields, it's rings, because we are talking about this ring and actually this ring. But then we say, what is Q? Q consists of all ratios of these guys, right? Where a and b are in z. And what is fq of... So here we two polynomials.

Do you see the analogy? Here you have ratios and here you have ratios. Ratios of what? Here of let's say natural numbers, which can be written expanded in powers of two or a general prime number. And here, f and g are elements of this. Okay. All right. So that's, I'm trying to explain why these two objects are similar. So these two objects are parallel to each other in some sense, give you parallel theories

in a way Rosetta Stone. The field of rational numbers on one side, which is in the first domain, on the first area of the Rosetta Stone, and this function field in the second area of Rosetta Stone. You see? So that's the analogy. But now you can appreciate also why Riemann surfaces appear. Because since we're talking about ratios of things, what other ratios we can write? We can write ratios of two complex polynomials.

You see? Okay, so we don't let it be a finite field any longer. That's right. So here the coefficients were in a finite field. For example, the field with two elements 0 and 1. But then if we do this, do we not lose the translation between the number field and the Riemann surface? No, of course we do.

Of course it's very far away. But you see, you can appreciate on the one side, yes, there is an analogy. On the other side, how far away it is. Because in one case we talk about finite field. In the other case we talk about complex numbers, which is a very infinite field. Yet the general structure is very similar. In both cases, the iterations of polynomials is just that in one case, polynomials are with finite coefficients. In the other case, it's polynomials with complex coefficients.

But it does look like a very tenuous analogy at first glance, you see. Yes. So far it looks like the stereotype of the erratic putting papers on the walls and finding connections and like, okay, sure, there's some resemblances here, but what's the point? Where is this going? What's the upshot? Have you ever heard this expression that the line between madness and genius is very fine? Yes. But see, this is why it is important that at the end of the day we actually come up,

we actually not just being wishy-washy like the way I have been in the last few minutes, but when we actually come up with hard facts, with hard theorems, and then we prove them. That's the beauty of mathematics. It combines both the intuition, which sometimes looks like the ramblings of a madman and then hardcore mathematics. All right. You see? But you cannot get to the hardcore mathematics without being a little mad.

Without saying, you know what, being a little, you know, there's this famous quote which I use sometimes from Alexander Grothendieck, whom we have mentioned, that he says, discovery is a privilege of a child, a child who is not afraid to look like a fool once again, you know, who's just pushing the envelope. So that's what we're doing here. We are like a child. We are being a child and the child has learned the number fields, okay, A over B, where A

and B are integers. And child says, oh, but look, numbers can be written in binary. And in binary form, they look very much like polynomials with coefficients of F2. So then the analog of rational numbers will be this type of ratio, so this function field. And then the child says, you know what, now I'm going to push it even further. And I'm going to replace this by complex numbers. Huh? So then the adult in the room, the adult comes and says, this is madness.

It's not going anywhere. And then yet in 50 years, following this path and being courageous and not being afraid to push this envelope, we come up with some ideas that we can actually prove. That's progress. That's how progress is made in mathematics. All right. So now I'm talking about the objects that we have. So these are the first examples.

And it looks a bit tenuous, I agree, but bear with me. There is going to be more. So what are more general fields? So, so far here, we only can see the rational numbers. But then there is more because for instance square root of 2 is not rational, right? So square root of 2 cannot be written as a fraction of 2 integers. So then we obtain more general fields which are called number fields by adjoining to the field of rational numbers solutions of polynomial

equations with rational coefficients, such as square root of two or the famous I, square root of negative one. The corresponding, the analog of that in the second, again, remember, in the second domain of the Rosetta Stone is a more general function field. So here we talk about just the rational functions, the ratios of polynomials. By the way, here the corresponding Riemann surface, guess what it is? It is actually a sphere. This field is a field of rational functions on the sphere, the simplest Riemann surface.

But now we are considering more general Riemann surfaces. So for instance, it can be surface given by this equation but over complex numbers. And that's an elliptic curve. So it looks like a torus. Now here is one thing that I want to explain, which is that we have already encountered curves over finite fields before. But what we did, we actually looked at this equation, but over fp for all p. And we only were interested in one aspect of this equation, namely the number of solutions.

And now we are, I consider something similar, but very very different because now we fix the ground field. We fix, for example, f5 or f7. And we are considering this equation only over f5 or only over f7. But we are considering not the number of solutions of this equation. We're actually considering the analog of the field of functions on the, what's called, projective line in this case. This is going to be a field very similar to the function field that we considered before. So instead of rational functions like so, we will have something more complicated, which

will be related to this elliptic curve, you see. But this elliptic curve is now defined over finite field, whereas we can replace now our ground field, let's say f5 or f7, with the field of complex numbers. And then we get a Riemann surface because the set of solutions of this equation over complex numbers is going to be exactly the set of points of Homer Simpson's favorite Riemann surface, surface of a donut. More precisely, without the point infinity, we should also add the point infinity, then we get the entire donut, more precisely the surface of the donut.

So that's the fields in general. These are the parallel things. So the theory should develop on parallel, on parallel tracks, where the role of a finite extension of the field of rational numbers will be played by the function field of a curve over a fixed finite field or the function field over a Riemann surface. Okay. All right. Next, we have Galois groups. In number theory, we consider the group of symmetries of what's called algebraic closure of a number field, such as the field of rational numbers, or the field q of square root of 2, when we adjoin square root of 2 and so on.

In the case of curves over finite field, we consider group of symmetries of the algebraic closure of the function field of this curve that I discussed on the previous slide. And likewise for a Riemann surface. But for a Riemann surface now there is a new way to interpret this Galois group. Namely we interpret it as what's called the fundamental group of this Riemann surface, which is denoted by pi one of X. So let's talk about the fundamental group and how it is connected to Galois groups. So again, I'm going to make a... So by the way, what I'm explaining now is basically,

I would say every first year graduate student in a good math department should know this. Great. So, it's not something advanced at all. It's kind of like bread and butter. So this is an analogy which is well established by now. Granted, in 1940, that's what Andre Wey explained. Well, I explain much more because this is just the first steps. The first baby steps of the analogy.

But to get further, you have to kind of grasp more clearly what the objects are. So in here, for instance, you can have a field of rational numbers. So again, on the left, I'm going to have number theory, proper. On the right, I'm going to have curves over finite fields or Riemann surfaces. So here you have the field of rational numbers, which sits inside the field of rational numbers with square root of two adjoined. Right? So here, what are the elements here? They have the form alpha plus beta times square root of two, where alpha and beta are rational numbers, right? So then you notice that there is a symmetry of this field.

Let's call it sigma, which sends square root of two to minus square root of two and vice versa. Under the symmetry, this element goes to alpha minus beta times square root of 2. You see? So the reason why this is a symmetry is that the square root of 2 is a solution of the equation x squared minus two equals zero. We can all agree on this. In real numbers. Yeah.

But so is minus square root of two. So in fact, this equation is a polynomial equation of degree two. It has two different solutions. One of them is the square root of two, and the other one is minus square root of two, which shows you that square root of two and minus square root of two, and that was minus square root of two, which shows you that square root of two and minus square root of two are like two wings of a butterfly. They kind of exist on equal footing,

and therefore exchanging them gives you a symmetry of the field. I talk a lot more about this, by the way, in early pages of Love and Math, so when I introduce groups and so on. So again, we only have so much time, but if people wish to learn more and are looking for a source, so that's one possibility because like I said, most of the things that we're

discussing actually are, I tried to explain it in a written form in love and math. So that's the situation on the number field side of things. How does it generalize to the case of function fields? Here instead of Q now we will have something like Fp of t, right? So that's what we discussed. And so what's the analog of this type of symmetry? And this symmetry by the way is an example of an element of a Galois group. This Galois group appears as a group of symmetries of this field, which preserve the structure

of the field. So now here, the analog is, suppose you take square root of t. You see? It's actually very similar. Here you take square root of something which does not exist in the original field because square root of 2 is not a rational number. But now you take the square root of t. t is your variable here.

It's square, it's not present. But you can enlarge your field by adjoining square root of t. And then you have a very similar situation because an element of this field is going to be very much like an element of this field with square root of 2 replaced by square root of t. And there is again this automorphism, this symmetry, exchanging square root of t and minus square root of t. So again you have a switch, square root of t and minus square root of t are like the wings of a butterfly. And exchanging them gives you an element of the Galois group, the group are like the wings of a butterfly, and exchanging them gives you an element of the Galois group, the group of symmetries of this larger field.

So now do the same for complex, in the complex case. Again you can take square root of t, and again you have a symmetry exchanging. And now why there are two square root of t and minus square root of 2? Same reason why square root of 2 and minus square root of 2, they are solutions of the same equation. And here also they are solutions of the same equation, but the equation instead of x squared minus 2 equals 0, it's going to be x squared minus t equals 0. It's an equation that you can write using the available means.

Available means are elements of this field, likewise here, the field of rational numbers. You see what I mean? And you adjoin new elements by adjoining solutions of such an equation. Okay? So that's how the Galar groups work. But now there is this idea that, in fact, in the case of function fields, we can think of the valor group as what's called a fundamental group. And here's how to explain it.

You see, you can think of geometrically that one, it's kind of a cover of two curves. So you have a curve, which is actually, let's talk about the complex case. In this case, the curve that we have is actually a Riemann surface, which is just a sphere, which is called CP1. But I will approximately draw it as this sort of real curve. And there is a coordinate here, which is t. And if I take now, the functions on it is going to be this field. That's the field of functions here.

What about this one? This one will correspond to what we call a cover, where you have two branches now with coordinates square root of t and minus square root of t. And so now you see geometrically what you're doing is you're just exchanging the two branches. So that's a very nice example of how things become geometric as you move across this Rosetta Stone of Andre Wey. Things that were algebraic like here, you know, here we're talking about some algebraic equation and we have two solutions, square root of two minus square root of two and so

on. It's not clear what kind of geometry we can talk about here. But now we are talking about functions on a Riemann surface. I have now moved two steps from number theory to Riemann surfaces, jumped over the curves over finite fields, and I am now squarely in the Riemann surface world. So this field is responsible for the simplest Riemann surface, which is the sphere, the projected, what's called CP1 projective line complex projective line. And this extension of the field actually corresponds to a cover, a covering of this project line

by which is a double cover. That's right. Because for every point, for every point here, except zero. Okay. Now the reason is that in the middle, there's a dot a dot so the origin it becomes a singularity or something different So is that a marked point? Is that called a ramified double cover or something else? It is a double cover. It's called double cover ramified at this point This point is special

You see? I was gonna say marked It's a special point because over it there is only one point. But for all non-zero points, well actually to be honest, if we talk about CP infinity, there is also a point at infinity where you also have ramification. But I'm only showing you the part outside of infinity so to speak so that then we can see only the zero point as a point ramification. For all non-zero points, there will be two points above in the cover.

And that corresponds to square root of t and minus square root of t. If number is non-zero, there will be two square roots, plus and minus, you see, but square root of zero plus minus is the same thing, it's still zero. That's what I'm trying to explain. So now the point is that you can now realize this Galois group is a group of literally of symmetries of covers, of different covers. And there is one more step that we can make, but now I'm starting to worry a little bit about time, which will show us that if we go to Galois groups, not like this.

These are kind of finite Galois groups here. These are the symmetries of a finite extension of Q. Likewise, here, it's a finite extension. But if we go to the biggest possible extension, the so-called algebraic closure, we get a humongous group of symmetries and likewise here. And the point is that if we insist that we have no ramification anywhere, then this group is what's called a fundamental group. So this is chapter, I think it's chapter nine of Love and Math.

So let me just leave it at that. So what have we done so far? We have fields, we have Galois groups, and we have learned also that there is this object called fundamental group which takes place of the Galois group in the unramified situation for human surfaces. This is something that we'll perhaps need to talk about in more detail next time. But now let me just make a few more steps so that we get to kind of a good place. What other objects are involved?

Remember how we talked about the group and its Lenglitz dual? So first of all, we are now positioned at the beginning of this correspondence, of explaining this correspondence. The correspondence always has two sides, right? So it's one-to-one correspondence between this and that. So this I will call left-hand side, and that I will call the right-hand side. So the left-hand side are easier to explain. If you think this is complicated, wait till you see the right-hand side.

It's even more complicated. So on the left hand side, we have these homomorphisms from the Galois group to the Langlands dual. So remember we had this discussion about the Langlands dual group. This, it appears on the left hand side of the correspondence, which we are discussing now. And we have to consider this homomorphism from the Galois group to the Langlands dual group, both for number fields and for function fields. But for even surface, we now have a replacement for the Galois group called the fundamental group denoted like so.

So we're going to consider homomorphisms from this fundamental group to the Langlands dual group. This is the object on the left-hand side of the Langlands correspondence, because remember you have to have two sides to talk about correspondence. Okay? And this is, so we made a lot of progress. We now know what are the objects on the left-hand side. And now we want to relate them to something else.

By the way, what would these objects be in the case that we considered at the beginning? Remember at the beginning we talked about this counting problem. So the objects on the left-hand side of the counting problem were those numbers of solutions of the cubic equation modulo primes. But behind those numbers stands some homomorphism from the Galois group to the group GL2, which is in this case a language dual group. In other words, the objects on the left hand side, which I have presented as numbers of solutions of the cubic equation, can be realized in a different language.

They can be realized as homomorphisms from the Galois group of the field of rational numbers to the group GL2 of a certain kind. So every elliptic curve gives you such a homomorphism. You see, so now we are using a different language, which is not specific to counting problem, but which is, which can be generalized to others, to much more general case of the language program. Okay. Can you please explain this again? Because let's say you're given an elliptic curve over Q and then you map that to what? To a morphism from the Galois group over Q to some reductive group. So that to me is like a map to a map.

In this case you have the Galois group of Q, and it maps to GL2. And such a map, such an object, such a homomorphism, such a homomorphism can be obtained from an elliptic curve over Q. That in particular, for example, the y squared plus y equals x cubed minus x. This curve gives rise to such a homomorphism. And the numbers of solutions can be interpreted in terms of this homomorphism as the so-called traces of Frobenius for prime numbers. You see, so in other words, in the traces of Frobenius. Yes, okay.

So there are certain elements, more precise conjugacy classes here, which are called the Frobenius conjugacy classes. We are considering their images under these homomorphisms and taking the traces. And the result is precisely this number AP that we talked about. So in the- Interesting. I could get by with some, with more classical notions in this case, namely I could talk without ever mentioning function field or, and actually in this case there is no function

field but number field, field of rational numbers or the Galois group or the Langlis-Doux group. I could completely put this aside and speak about cubic equations and the numbers of solutions of these equations, modulo primes, which is what I did at the beginning because I wanted to explain it in more down to earth terms. But now we are embarking in a much more general correspondence. And in this general correspondence, you can no longer get by with some equations and counting numbers of solutions of those equations. Instead, you're considering a number field in this scenario.

You're considering a number field F. You're considering its Galois group, more precisely the Galois group of its algebraic closure. And you're considering homomorphisms from this Galois group to the Langlands dual group. So then somebody can say, well, what's the connection between these objects and the objects we discussed at the beginning? Here I explained briefly what the connection is. Given an elliptic curve, such as the one defined by this equation, an elliptic curve over the rational numbers, we can assign to it a two-dimensional representation of the Galois group of Q or equivalently a

homomorphism from the Galois group of Q to the group GL2, which is the Langlands-Duhl group in this case. And the numbers that we talked about where P is a prime can be obtained from this homomorphism as the so-called traces of the so-called Frobenius-Kantzschi classes. And briefly, what dictates the group on the reductive group on the right-hand side, so the GL2 in this case? Is it the elliptic curve or the field that it's over? These are two separate parameters.

Oh, sorry. So elliptic curve gives GL2. Why elliptic curve or the field that it's over? No, so these are two separate parameters. Oh, sorry, so elliptic curve gives GL2. Why elliptic curve gives GL2? Oh, yes, that's a very good question. Okay, let's talk about this briefly. So here is an explanation is the following, that there is certain structure in this elliptic curve which is two dimensional. And this structure is easy to, is not easy, but is useful to explain in terms of the analogy we talked about.

The idea is that let's, instead of looking at an elliptic curve over the rational numbers, let's look at the corresponding elliptic curve over the complex numbers. In this case, it is just the surface of a donut. It's a torus. And the torus has two cycles which cannot be contracted. One of them is a cross and one of them is a long. These cycles generate what is called the first homology of this torus.

Now it turns out that you can define the notion of homology or cohomology in the context of curves over a number field. They are called etal cohomology. And they have surprisingly similar structure to what you expect from the biology with complex Riemann surfaces. In particular, in this case, it's going to be two-dimensional. And because it's two-dimensional, you get a representation of the Galois group of the field of rational numbers on a two-dimensional vector space.

A two-dimensional representation is the same as a homomorphism to GL2 because you assign to every element of your group a two-by-two matrix which acts on this two-dimensional vector space. You see? So you ask me why the Galois representations associated to elliptic curve are two dimensional, which is to say why do we get homomorphism to GL2 and not to GL3? And the answer is, is because the elliptic curve sports a two dimensional first homology. So if we were in a different genus on a Riemann surface, would we get a different group? The homology groups. So then what does it look like on a sphere?

But on a sphere there are no non-contractable cycles. So that's why the story starts with elliptic curves, you see, on the number field side. So actually, you know, it's interesting because I want to mention something because someone asked me whether there is a link between the LENK Linux program and the Moonstyne, what's it called? Moonshine. Moonshine. Moonshine is my DJ name, sorry. That's funny.

Okay, we'll put a link to that in the description as well. We didn't promote that last time. I think after this long conversation, I think viewers will need to relax. And what better way to relax than to listen to some of my mixes on SoundCloud. So you can find a link under the video, if you want. But there is such a thing as a moonshine conjecture, which by the way, I think you interviewed Richard Borchardt. Right.

Yeah. Who is another, who is a colleague of mine here at UC Berkeley. And he actually gave a proof of this conjecture and received Fields Medal for it, which is really a very important achievement. So modular forms actually make appearance in his work as well. And someone asked me whether those modular forms have something to do with the language correspondence. And the answer is no. At least I don't know how, but now I can explain why.

Because they essentially correspond to the Riemann surface, which is a sphere. And the sphere does not have any useful Galois representations. They only have basically trivial representation, Galois representation associated to it. That's why the so-called Hauptmodule, which are the modular forms arising in the moonshine conjecture, a priori have no bearing on the Langlands correspondence. Elliptic curves do because they give rise to non-trivial two-dimensional representations of the Galois group. And then for higher genus curves, you will also get higher dimensional representation. But you can also consider algebraic varieties of higher dimension.

You don't have to be stuck with curves. But you can see how the simplest example of the Lagrange's correspondence in the number field context would be the one corresponding to an elliptic curve. Because this is the simplest algebraic variety for which there is a non-trivial representation of the Galois group. because there are these two cycles. You see? For the CP1, for the projective line, for the sphere, there are no non-contractable cycles.

They can all be contracted to a point. Therefore, there is no homology. There is no first homology or co-homology. But for a service of a donut, you have two cycles, two independent cycles, which are non-contractible. The one which goes across and one which goes along the torus. And they give rise to two-dimensional representation of the Galois group. That's why we get a non-trivial example of the Langlands correspondence. All right? Okay, well, for people who have watched this far, congratulations.

And part three will be out in maybe a month or two months, maybe shorter. Yeah, for the die-hards, where we will explain actually... If you've watched this far, it's for you. That's right. So we will explain actually what is on the right-hand side, you see. And so, by the way, maybe I just give you kind of a teaser. So in the number field, number theory context, these are the automorphic functions, which are the generalizations of modular forms in our example, in our basic example, what corresponds to the elliptic curve on the left-hand side is

a modular form on the right-hand side, right? In our example. But in general, for a general Galois representation, you will have automorphic functions for the group G. Here you have the Langlands dual group. Here you will have the group G itself. And then for curves over a finite field, there will be something similar. But here already they will have a nice geometric interpretation as functions on what's called band G. And this I will explain next time. But now the punchline is that for a Riemann surface, now there are two versions. In the analytical language, which is this recent work by a piloting of David Kasdan and myself, we have functions on

Bungie. So we actually have a Hilbert space of actually half, what they call half densities on Bungie. And we have some commuting operators, which are the annuls of the Hacke operators of the classical theory, as well as some differential operators, which are actually very similar to the kind of Schrodinger operators, and higher order differential operators.

And so there is a well-defined spectral problem where you can talk about the eigenfunctions and eigenvalues of those operators. They commute with each other so we can see the joint eigenfunctions and the corresponding eigenvalues, which are given in terms of the Langlands dual group. So that's one formulation. But in the geometric Langlands corresponds, instead of functions, you can see the sheaves. So instead of the Hilbert space of functions or more properly half densities on band G,

you have a category of sheaves of a certain kind on band G. And it is what the object on the right-hand side. So we will, next time I will explain what this moduli space of G bundles is, what those are functions versus sheaves, vector spaces versus categories, and Langlands correspondence as a Fourier transform. Okay? All right. So, then we'll have a rough formulation of both stories.

Maybe this question will take longer than 30 seconds to answer, but if you go back, keep going back to where you introduced Bungee. So over here, the continuous functions are a sheaf. So on a space, continuous functions are an example of a sheaf. So why is it so difficult for Robert Langlands to accept sheaves when he's putting forward functions, but functions, continuous functions, for instance, are the prototypical example of sheaves. But let's be careful.

A sheave is not one vector space. It's a vector space attached to every open subset. It's a much more sophisticated object. So a function is actually a rule which assigns a number to every point. But the sheave assigns a vector space to every point. You see, this is a much more physical. So what you're saying is,

there is the space in your example, there is a space of all functions on a given space. That's a vector space. Each object is a function on the entire space, which is a rule which assigns to every point some number, be it real number or complex number and so on. But there is also a notion of a shift of functions, which is an object which assigns to every point a vector space of what's called the germs of functions in the formal neighborhood of this point.

It's a much more sophisticated object, Even though the word functions is used, but the object itself is much more sophisticated. Where instead of a number assigned to a point, you have a vector space assigned to a point. That sounds like it's not much more complicated. Sure, you're dealing with higher dimensional spaces, but still. How about this?

Number five versus five dimensional vector space. I think it's way more sophisticated. In a five dimensional vector space, you have how many vectors? You have continuum of vectors. And now on the other side, you have number 5. A given function will have a value 5 at a given point. A given shift will have a value which is a 5 dimensional vector space. Yes, way more complicated.

Way more complicated. All right. Okay. Thank you, professor. You're welcome. a value which is a five dimensional vector space. Yes, way more complicated, way more complicated. Okay, thank you, professor. You're welcome. And I hope that was more, how to say, I added to people's understanding and did not subtract. You always hope that you don't do harm. You know what I mean? So I hope that if anything,

I kind of spurred more curiosity and did not, and not the other way around. Of course man, the audience appreciates it and I appreciate it as well Edward Also, thank you to our partner the Economist Firstly thank you for watching. Thank you for listening There's now a website KurtJaimungal.org and that has a mailing list The reason being that large platforms like YouTube, like Patreon, they can disable you for whatever reason, whenever they like. That's just part of the terms of service. Now a direct mailing list ensures

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This is Uncharted territory. We are really, today we really talked, you know, last time we talked about some stuff which is, you know, technical but not too technical. So you talk about some equations, some solutions of equations, some serious, you know, that is pretty much high levels, high school level. But today we already talked about some more serious stuff. And so we'll see how it goes. But I think it's worthwhile to do this experiment because to see how far we can actually go in how deep we can go I think it's important to to see that because I think there will be people who

actually will dig this oh yeah and this is unprecedented in podcast form yes so it'll be interesting to see the results absolutely