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Edward Frenkel: Monumental Breakthrough in Mathematics (Part 2)
October 2, 2024
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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, your dedicated audience of your podcast theories of everything. I was very pleased by the response to our first part and the language program. There was a lot 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 and you 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 unfamiliar terms, analytic Langlands, number fields, Galois groups. Explain why this recent proof is monumental. Uh, the way mathematics develops is that is, is like you're searching for something in the dark room or, you know, in the dark space. Uh, we may be searching for a key, you know, uh, and, uh,
you the way you would proceed usually is that we trying to imagine how things could work out and that 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 analogies and we will also have theorems a theorem is something which is proved
which is valid universally that there is no argument in mathematics that something's 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
So there is a very, um,
um there is a scrutiny for all these papers so they are all subjected to scrutiny by peers not necessarily through 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 uh 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 a hundred 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 a hundred years and so on. That's the that's mathematical rigor. So you see, you have this, we have this process of progress in mathematics. You don't immediately come with theorems. You usually come with certain visions, with some intuitions, with
Uh, 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 kind of 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 this visions, um, this 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 Geisgurje, 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,
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 into the into the topic. Okay, so why don't we go straight into the slides?
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 language 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? 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. 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, 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 in at least in North America, don't say 14 o'clock, we say two o'clock. So in other words, we take numbers, modular 12, 13 means one, 14 means two, 15 means three, 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
um, arithmetic with a clock that has seven hours, right? So, 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. And, uh, right. 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 number of p hours where p is one of those primes 2 3 5 7 11 13 and then we get a new numerical 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 two and a three. And some people were thinking, wait a moment, I thought Fermat's last theorem had to do with any n greater than three. 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 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
this the cubic equations that give us what's called elliptic curves. But the idea is that we are considering the same equation, but we're looking for solutions, modular, every prime number you see, because we can 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 the here's a little table which shows you how many solutions this equation has.
So that the difference is divisible by five, right? And likewise for the prime number seven we have nine 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 two is two minus four 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 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 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 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 the computer and you can find
this 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 this 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, you see, you have, 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 one minus Q squared, there's one minus Q squared, one minus Q cubed squared, Q to the fourth squared. Then there will also be one 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 minus Q to the 11 squared.
one minus q to the 22 squared one minus q to the 33 squared and so on. So it's very easy to write down these factors right up to any 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 two or q to the fifth with coefficient one or q to the sixth 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. So one was that we made plenty of references to signs and cosines when talking about harmonic analysis. And then we go to this Q polynomial, and there are no signs or cosines. So someone wants to know what is the connection between this and harmonic analysis precisely? Right. So I explained in the in the in part one,
that a prototype that 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, what 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.
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 and there are more namely you can say cos 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 shift by by 2 pi
And so the idea of harmonic analysis is to use this trigonometric functions as sort of a, as a basis for the space of all functions and try to decompose general functions as linear combinations of this basic harmonics they're called. Right. And so, um, it's, it is like decomposing the sound of a symphony into the nodes of, of specific instruments, which each node being a kind of a sine or cosine function. Right.
with a particular grade responding to 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? So, and basically the statement is that this coefficient are precisely the sort out numbers AP, which is, which are essentially the numbers of solutions of the cubic equation, modular, the corresponding prime, right?
and so you see just just just to kind of the to 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 one formula to rule them all you see and kind of a colossal compression of information or you 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's just an infinite series algebraic like an arbitrary algebraic equation it's a kind of algebraic equation where q is actually just some formal variable right but actually turns out that you can actually substitute for q any number which is less than say positive real number which is less than less than one
And moreover, you can substitute any real number, which is between negative one and one. So it's 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 that is whose absolute value is less than one. So here I want to remind you that complex numbers
can be can be represented as points on the plane right where 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 one geometrically, it just means taking the taking a disk of radius one, right? So these are exactly all the points whose 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 open disk, in other words, without the boundary, any point, which is a point, which is a complex number corresponding to this, 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 one plus two plus three plus four and so on where it can give you minus one over twelve is 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 one, you see. If you take Q with absolute value greater than one, it will diverge same way as one plus two plus three plus four 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 of this group PSL to Z or perhaps it's subgroup. Sure, these groups have this group has a family of subgroups which are appear naturally in this key in this setting. Okay, so that is this notion of a modular form. And what I'm trying to say is this infinite series, which, you know, contains the answer to all of the counting problems for this specific cubic equation.
This infinite 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 module 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 second recap of that. Okay. This is called a Q series. This is a
A generalization of sine, cosine, which is a Fourier series. So the 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 know how people are familiar with Taylor series expansion, right? 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 you and you 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 coordinate naming appears in mathematics.
Yes.
in in single variable calculus. For example, we know that one divide by one minus x, maybe I should give this example, you can have this one divide by one minus x, which you can write as a sum one plus x plus x squared plus x cubed and so on. But 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. You see,
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, right? So we're doing something very similar, except we are now working with a complex
playing 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 one. Actually, it also converges when X is a complex number and whose absolute value is less than one. 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 whose which satisfy this inequality. So is this series, which is another way is just expanding this product is converging when Q has the same property satisfies the same inequality doesn't 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? Well, so you're jumping ahead, right? Because so far, so far, so far, there is only one cubic equation, there is only one 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 the subject of what is called the Shimura Taniyama Vey 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. That's 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 on how this ideas play out in geometry. Right. So in principle, if you're most interested in that subject, you can kind of just watch this, 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, 12, 13 years, 12, 11, 12 years ago. It was published 2013 precisely to explain the ideas of the language 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 are, if you would like to really go deeper and to really understand this concepts and ideas,
more precisely than, than this is one source for you. Uh, 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, several, several resources for that. Um, and in love, love and math, I try to explain it for the general audience. In other words, not for non-mathematicians, which is the idea today as well. People interested in mathematics, but not, not necessarily specialists, not necessarily experts.
And, uh, but we will include as also some sources for more advanced, uh, 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's
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 modular form that is associated to this cubic equation or this cubic or this elliptic curve over the field of rational numbers you see and this way one updates 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 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 they could be maybe it covers all primes except number p equal 11 or something like that you see so almost all yeah but that's not that's not such a big deal given that it covers it for all
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 example, 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 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 let's, I want to mention
the link to the between the shimur tanayama way congestion. I should say this conjecture was formulated uh initially uh in 1955 by uh by this japanese mathematician yutaka tanayama interesting interesting interesting human story so the original formulation was actually incorrect and then um
These two other mathematicians, Goro Shimura, who was actually his friend, also a 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 contributions. They kind of corrected the initial formulation by Taniyama. But what Taniyama 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 in the commit suicide at the age of 31. This is brilliant mathematician who died very young. And there is a beautiful tribute to him written by his friend and colleague, Gor Shimura. And I remember one quote from there that Shimura says that Tanyama has this unique ability to make good mistakes. Right. And he said, it's not, 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 with 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 the story because it shows you the drama, the, you know, the of mathematical ideas, the drama of mathematicians trying to come up with this idea is coming to come up with this conjecture is trying to prove them. Oftentimes this conjecture is maybe wrong. So then other mathematicians come along who disprove them.
But at the end of the day, it's this co it is collective enterprise of sharing this 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 Tony Alma was doing the stream of consciousness. No, no, not at all. It was, uh, no, he was 90% correct. Let's just say,
Imagine that, imagine a 90% correct green play on the first day. Maybe 80%. Okay, it's hard to measure. 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 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.
also known as modularity conjecture, or today 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 Shmur-Tanyama-Wei Conjecture in the same 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 Shmur-Tanyamvay Conjecture. And then Andre Weiss and Richard Taylor proved Shmur-Tanyamvay 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 Shmur-Tanyamvay.
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, you know. It's much more subtle. All right. And so now, what does it tell us about the Langlands program? Well, guess what? 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 more than Yam of a by itself is a vast realization of this one example that we started with for a specific cubic equation and specific modular form. But the Langlands correspondence or the Langlands program rankers conjectures are vast realization of this. She more than Yam of a conjecture. You see. All right. So now what do I mean by this? Um, you see,
Let's recap. The original Langlands program was about difficult questions 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 questions in harmonic analysis, like finding coefficients of modular forms. So, Schmortani M of A 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 numbers AP, which is remind you is not exactly the number of solutions, but it's P minus the number of solutions. Right. The error, the error, the difference between the prime number itself and the number of solutions. Uh, and on the right hand side, you have objects called modular forms, more specifically, uh, 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 numbers of solutions or approximate errors of a number of solutions AP match with coefficients of the modular form in front of so you have here 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, more general representations of what's called a Galois group on the left in this, in the special case that gives us this, this representations correspond to cubic equations and the certain, certain numbers associated with them correspond to this 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 Langlands correspondence or Langlands 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 that has two arrows. Right. By the way, there is a lot more. This is only the beginning. Okay. So it's going to be even more. It's 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 in the is a kind of an upper index on the left.
Yeah, why is it on the left? Langlands introduced this notation.
And in fairness, it 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 it 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 cold Joyce manifold.
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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 the subject, myself included, is to understand why the Langlands dual group appears.
There are some explanations. I came up with an explanation with my co-author and former advisor, Boris Fagan, 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 Schmurtan-Yamaway 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, uh, it's one of the simplest groups that, that mathematicians study. And in this case, it just so happens that the dual group is the same. So at the level of the Schmur-Tanyama-Wei 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.
uh, which appears in the study in some physical theories and so on. Uh, then you will discover that there is this duality appearing different, two different groups appearing on the two sides of the language correspondence. So here, here's one rabbit hole that we could
go into, which is classifying all possible groups that appear here. So here there's this what we call they are what's called reductively groups or reductive algebraic groups. And those are essentially products of a billion groups and what's called semi simple groups. And so the semi simple groups in turn are kind of like products
of simply groups or simple algebraic groups. And finally, those are classified by what's called dynkin diagrams, more precisely, the so-called simply connected, simply connected to 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 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 what this results to, for instance, 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
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 Dynkin diagrams and the centers, the finite centers of the corresponding algebraic groups.
It's not always the case that a reductive groups dual or Langlands dual, if that's what it's called, is also a reductive group. It's always a reductive group, but in particular always reductive. So Langlands duality is an involution on the set of all the 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 and they, 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. And 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 two, 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, but about the exact, exact, how the dual 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 if it
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 odd-dimensional vector space, and that 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 Lengar's group to the Lengar's dual group as we go from one side to the other side of Lengar's correspondence suggests that this correspondence is extremely sophisticated. Close your eyes, exhale, feel your body relax,
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. So let's find out where else in what other areas of mathematics
Can this 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 language program. So that explains why we're interested in the reformulations of this original ideas, but in other domains, the other areas of mathematics. Okay. And so what helps us here is, is this Rosetta stone of mathematics, which I also talked about,
Last time, uh, which was proposed by Andre way, whose name has already been mentioned several times in a letter to his sister, Simone way from prison, actually in France in 1940, he wrote about analogies between these three areas of mathematics. One of them is number theory. And that's what has, what appears in the original formulation of the language 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. Andre Wey 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 columns, 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 this automorph functions, right? That's the original. That's the original version. But because we now know that this, these two other areas 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 sort of 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 from totally from the left field. And the only reason why only reason why we even even
there to believe that there is some kind of Langlands 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 Andreweiss 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 the kind of a sweet spot where, which kind of is a turntable, as Andre we called it is a kind of a bridge between these two. In the case of in the case of curves over a finite field on the one 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, the one of the biggest developments in the language program across all three domains are all three areas of mathematics has been the proof of the
Langlands correspondence in the in the basic case of the group GLN, we have talked about earlier about the group GL2, which appears in the Shimur-Tanyama-Wei conjecture from the perspective of the Langlands correspondence. GL2 is the group of two by two 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
you know, positive integer, which have non-zero determinant. It's important to have non-zero determinant because then you have the inverse, the multiplicative inverse. So if you consider all n by n matrices with non-zero 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 is actually very hard, from n equal two to arbitrary n.
in the early 2000s, both of them received fiddle fields medals for their works. So this is a highly kind of highly prized achievements 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 the, 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 language correspondence is. That's where, that's what this recent series of works by Gates, Gury, Raskin and others is about. So what are the Riemann surfaces? First of all, these are the Riemann surfaces.
These are examples of reman surfaces, the sphere, the surface of a donut, surface of you could say Danish pastry or something. Yes. So I guess I guess this one is Homer Simpson's favorite. Sure. Now these are compact reman 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
response to having finitely many marked points and kind of removing those points so that you allow some sort of singularities at those marked points mark means that you mark 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 is the second one the third one this is just the way people say people say it you can say chosen or specified it's the same
All right. So these are the human surfaces. And the question we're going to discuss now is to how to formulate the language program for this object for this human surfaces, as if the language program was not complicated enough already. We're now going to move to a different domain. We're going to move under the streetlight, if you will. All right. And try to find the key there. You see,
So, for Riemann surfaces, here is one more twist. In fact, for Riemann surfaces, there are two versions. There are two versions. The first one was initiated by great mathematicians, De Ligne, Pierre De Ligne, Vladimir Dreinfeldt, whom I have already mentioned, and Gerard Lamond in the 80s. And it is called the Geometric Languages 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 geometric Langlands correspondence for human surfaces balance from the dream felt 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 goodie 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 sheaves. But interesting enough, in the last five years, a new version of the language correspondence for human surfaces was proposed. And I've been involved in this.
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.
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 theoretic version. And in some sense, I feel guilty because I spent a lot of time trying to explain it to him.
And I guess I didn't do a good job. Why didn't he like the sheaf version? He's more traditional mathematician for for mathematicians of his generation. Sheaves are kind of not exactly anathema, but it's like additional bells and whistles that you don't need. But he's of the generation of growth in deep, no? Yeah, but he comes, he's sort of the other side of the of the spectrum.
Oh, the Sarah side? Well, Sarah is closer to Grotendieck. The division between Sarah and Grotendieck is not that big. Grotendieck, Alexander Grotendieck, one of the most brilliant mathematicians 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. Grotendieck 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 shifts. 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
of functions of spaces, operators, and so on, as opposed to shifts, 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? Yes, we did. We started it. So six years ago, there was no analytic Langlands program, if I must search that. The first time I talked about it was actually in 2018, soon after Langlands received the
The main point of his talk was there has to be a function theoretic version.
and he and i had a little tension about this but he had absolutely correct idea that this version has to exist and in many ways that inspired me to look closer into this okay 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 in maybe a year earlier about some special cases of this how this could work out and
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 when we were kids, basically, when we were in college, and David Kasdan, who is one of the luminaries in representation theory, he was
The favorite student of Israel Gelfand. So in some sense, he's a kind of 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. So where he is now. And so with piloting golf and David Kajdan, we have worked on this for the last five years. We've published
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,
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So let me actually show you this is the web page of dennis gays gary who by the way has been my co-author my collaborate 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 web page 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 the 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 refereed yet, but you know, there's still enough preprint form, but you know, 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's so many, there was so many technical issues that one needed to deal with. And remarkably this, they are addressed in this new series of papers, but those papers, I think were written over the years, not like they didn't just dump them all at once. Okay. So here's one of, here's what the first page of one of them looks like. It's a second one, second paper in the series. Okay. And so
I want to show you something which is kind of to give you an indication of what a great, what a grand project that was and how many people have been involved. Just to give you an idea of how much goes into this. It really takes a village, you know, if you will. So this as the proverb says, so the first thing they mentioned here, this is the acknowledgement 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 are using, which by the way, you know, I'm impressed by the way they are kind of acknowledging the contributions of other mathematicians is really, they're showing a great example for young mathematicians. This is how it's done. And so the first thing, the first two people they mentioned are Alexander Bellinson 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, 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 newer 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. I see. But we, we, we use BRST in a more generalized sense these days.
And then there is this Fagin-Frankel Center, which is kind of plays the central role, as they say, you know, interesting. And so then, of course, then there is another deep idea of factorization. And so they mentioned Jacob Lurie down there and Jacob Lurie. Yeah, Jacob Lurie tremendous works of tremendous importance, which kind of like gives you proper language for what's called higher algebra.
So I 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, 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, Denis and I, Denis Gait-Gurion, I actually wrote eight papers together on this subject closely related to this Katz-Mudi Algebra at the critical level, which give the
the tool for connecting for the establishing of geometric language correspondence. So in fact, some of those are important in this in this series of papers, I have been involved in this project. And up until, you know, a few years ago, when I switched to this new version, which is the function theoretic version called the analytical language 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, you cannot say that it's, 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, consequences of this series of works by Dennis Gaizko, 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 Letingov and Dave Cash done about the analytic language correspondence, you see, which was recently published in this journal and was actually a special issue in honor of a wonderful Italian mathematician, Corrado de Conchini. You mentioned writing screenplays earlier.
Fun fact, Corrado de Cancini's father, Ennio de Cancini, was a famous Italian filmmaker. He actually got an Oscar. He co-wrote a screenplay of the film Divorce Italian Style 1961 with Marcello Mastroianni. And he received an Academy Award with his co-authors for the screenplay. Anyway, it's just kind of a
Interesting connection to what was his son's contribution to this. So there was a there was this paper appeared in a volume in a in a issue of a journal, pure and applied mathematics in honor of his 70th birthday. So we dedicate dedicate the paper to him with admiration. He's a great mathematician, great human being as well. So
So that so there is this now this other this other thread. In addition to geometric Langlands correspondence, there is now this analytical Langlands correspondence. It doesn't mean that it supersedes. Just to be clear, it doesn't mean that it's better than the other one is different. And the two are connected. So they complement each other. And I will comment more on that as we as we proceed. All right. Sure. The Langlands would probably say this one's better. So the Langlands would say is better. Yes. So you know, this is what happened is like,
You know, I mentioned in part one that I collaborated with Langlands. We wrote a paper together and this was about from 2008 to 2010. And I spent a lot of time talking to him. And then he, our work was not in the geometrical language because it was more like a traditional one, but for 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 the 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 is about? What is this about? What is this? What are these shifts? How do they fit? And I tried to explain
And the result of it was that he hated it. Okay, so sorry to say 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 are not have no place. There has to be a function theoretic version. He even went
As far as saying during his talk at the Abel prize symposium, you know, not symposium, but it was kind of acceptance speech after the, he said that they might as well remove my name from geometric correspondence. I don't want it to be, I think it's a little too much because it is really, to me, geometric is right. It has a place in this general overarching theory.
but different matters have different tastes and they have different styles. This is the reason why I'm sharing with you this stories is to show you what a robust dialogue we have with each other that even though we respect each other tremendously, you know, 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 than other and so on. So in this case, here is Langlands.
He's in his eighties, right? And so, but he's still very full of energy and he's like, no, I don't like, uh, I don't like this shift formulation. There has to be a function theoretic formulation, which by the way, at that time in the subject, nobody believed 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 coauthors to, to try to do it.
And it turns out indeed that it was possible to do. I should mention, as I mentioned, there was, there was a York Tashner was another mathematician in Hamburg, but Michael physicist who tried to do something similar. There was another Russian born mathematician who works in France, uh, 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 and as a kind of a,
really a branch of the Langlands program, which is what we called the Analytical Langlands program. Now is that the same Koncevich who generalized mirror symmetry to homological mirror symmetry? So Koncevich is a very brilliant mathematician who has worked in many different areas and made very important contributions, but at some point he got interested in the Langlands program and so he tried to do it 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 just we are considering as the analytical language correspondence. Got it. Yeah. So 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 that 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 Gelfand's, but they were closely connected and they were extremely influential, not only in the Soviet Union, but all over the world. And I want to mention them because
In many ways, all of what all these ideas that I'm going to talk about that have to do with the Hellenic correspondence for human surfaces, both the geometric formulation and analytic formulation, in some ways are can be traced to these two giants, Kelfand and and Manin. So, for example, Bellinson and Drenfeld, these two mathematicians who are perhaps
most responsible for these developments were his students, you see, of Eurimania. They were also very closely connected to Gelfand. So Gelfand had this famous seminar. By the way, in my, 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. And, you know, it's really hard to, to, to capture it in a few words, but I really wanted to, 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, I have been greatly influenced by his, not only his ideas, but his general approach to mathematics.
one of the things
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 Neumann, you know, so there is this famous Guelph-Neimark theorem about sister algebras and so on. So he really, but he worked in so many different fields. He also worked with computer scientists. He worked with biologists. He was really in the 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 Gelfand. For instance, David Cashdown, my co-author and collaborator on the analytic language correspondence was, one could say, his favorite student, favorite student of Gelfand. And in some ways, he's perhaps the most connected to this whole circle of ideas. My other co-author Pavel Letingov was a student of Gelfand as well, to some extent.
but also Yuriy 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 Yuriy 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.
Interesting.
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Alright, 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, 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 was p was 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. Uh, the highlighting the objects, which are analogous to each other. Uh, the first thing to start with on the left hand side left has left hand side has to do with number theory. We have the set of integers, right? So you have like, you know, minus one, zero, one, two, and so on. Right.
So the analog of this is what's called FP of T, which is where the T is with square brackets, which is all polynomials, polynomials over FP. So what does it look like? So for example, you can have polynomial. So you have a zero plus a one T plus a two T squared and so on plus some a N
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 school polynomials with real coefficients right so a typical such polynomial would be would have this form where a zero a one 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 what is it? 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 zero. Sorry, p is equal to two. Let's consider a special case when p is equal to two. In this case, ai
is either zero or one because the field F2 actually has only two elements zero and one because two is already back to takes us back to zero right so then a polynomial would look like this for example it would be like one plus zero times t plus one times t squared and so on for example here's an example one plus t squared is an example of a polynomial over F2 right
so what you see is that you have a sequence of zeros and ones really what 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 the 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 at least 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
you can kind of say all right well here you have one and here you have one here you have zero okay and here you have one the difference is that here we we take two and then two squared and two cube 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 two squared plus two squared is two 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 zero and one. 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're both are described
Objects are described as sequences of zeros and ones. Okay. But with different arithmetic. So this is just to indicate why we believe why mathematicians 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 is the very first observation. And 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, the conjectures, where you actually say this set and this set are in one to one correspondence, for instance, she more than I am 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 weight 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 have polynomials, but now we also have rational functions or
It consists of ratios of 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 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 a 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 you once since you know, 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 of two with two elements zero and one.
Right. Then if we do this, do we not lose the translation between the number field and the remote surface? Of course we do. Of course it's very far away, but you see, you see that you can appreciate on the one side that yes, there is an analogy on the other side, how far away it is, because in one case we talk about finite field and in that 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 always finite coefficients.
In the other case, it's polynomials with complex coefficients, but it does look like a very tenuous analogy at first glance.
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 the 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 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're being a child and the child has learned the number fields okay a over b where a and b are integers and just oh but look numbers can be written binary and in binary form they look very much like polynomials with coefficients and 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, 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 then 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 two is not rational, right? So square root of two cannot be written as a fraction of two 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 agreement surface, guess what it is? It is actually sphere. The
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, if it can be a surface, Riemann 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 so I can see there's something similar, but very different because now we fix the fix the ground field, we fix for example, f five or f seven. And we're considering this equation only over five or only over f seven.
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 on the 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.
Okay. So that's the fields in general. These are the parallel things. So the theory should develop in 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 over 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 are joined square root of 2 and so on. In the case of curves over a 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 the streaming surface, which is denoted by pi one of X. So let's talk about the fundamental group and how it is connected to 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 like, this is this is an analogy, which is well established by now. Granted, in 1940, that's what Andre we explained, but 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, more, more clearly what 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, 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 There is a symmetry of this of this field Let's call a 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 two. You see, so the reason why this is a symmetry is that the square root of two is a solution of the equation. X squared minus two equals zero. We can all agree on this in real numbers. 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 that was minus square root of two, which shows you that square root of two 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 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
to the case of function fields here instead of Q. Now we will have something like, um, FP of T, right? So that's what we discussed. Uh, and so what's the analog of this type of symmetry and the symmetry, by the way, is an example of a, of a, an element of a Galois group is Galois group is appears as a group, as a group of symmetries of this field, which preserve, preserve, uh, the, 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 two 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 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.
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 of symmetries of this larger field. So now do the same for complex in the complex case again you can take a 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
The group is what's called a fundamental group. And here's how to explain it. You see, you can think of geometrically that one curve here is below is kind of a cover of two curves. So you have you have a curve, which I, which is actually the 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 CP one.
but I will approximately
And so now you see geometrically what you're doing is you're just exchanging the two branches. Right. So that's a, that's a, it's a very nice example of how things become geometric. As you move across this Rosetta stone of Andre V things that were algebraic, like here, you know, um, here this, we're talking about some algebraic equation and we have two solutions, square root of two minus square root of two. 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 was called CP1 projective line, complex projective line.
And this extension of the field actually corresponds to a cover, a covering of this project flying 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 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 so this point is special you see it's a 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, which is also where you also have ramification. Right. So, but I'm only showing you the part outside of infinity, so to speak, so that then we can see one of the zero point as a point ramification for all non zero points, there will be two points above in the cover. And that's that that corresponds to square root of T and minus square root of T.
If t if number is non zero, that 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. Okay. 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.
So which will show us that if we go to Galois groups, not of like this, the second 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 unranified 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. Okay. What other objects are involved? Remember how we talked about the group and its language dual? So first of all, we are now positioned at the beginning of this correspondence of explaining this correspondence 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 the left on the left hand side, we have this 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 language dual group. These are the left hand side of the language correspondence, because remember you have to have two sides to talk about correspondence. Okay. And this is, 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 the Langlands dual group. In other words, the objects on the left hand side, which I have presented as numbers of solutions of a 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. So the special case, that's a special case. In this case,
F is Q, so you have the Galois group of Q, and it maps to GL2. And such a map, such an object, 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 interesting, I could get by with some, with more classical notions in this case, namely, I could, I could talk without ever mentioning function field or actually in this case, there's no function field, but number field, a field of rational numbers or the Galois group or the language do 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 conjugate classes.
And briefly, what dictates the group on the reductive group on the right hand side? So the GL two in this case, is it the elliptic curve or the field? So these are two separate parameters. Oh, sorry. So elliptic curve gives GL two, why elliptic curve gives GL two? Oh, yes, that's very good question. Okay, let's talk about this briefly. So here is the here is the explanation is the following that there are certain structure in this elliptic curve, which is two dimensional.
And this structure is not easy, but is useful to explain in terms of the analogy we talked about. The idea is that 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.
This 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 by analogy 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 homomorphous to GL two and not to GL three? Yes. And the answer is, is because the elliptic curve, 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? Yes, absolutely. We would get so for a, for a,
Higher genus curves, you will have a higher dimensional 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 next program and the Moonstyne. What's it called? 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. 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, 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 human surface, which is a sphere. And the sphere does not have
Any useful color representations. They only have trivial, basically trivial representation. Gallo representation associated to it. That's why the so-called helped module, 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 Gallo group. And then for higher genus,
You see,
For the CP one for the project flying for the sphere, there are there are no noncontractable cycles. They can all be contracted to a point. Therefore, there is no homology. There is no first homology or a coma. Right. But for a for a service of a donut, you have two cycles, two independent cycles, which are noncontractable, the one which goes across and one which goes along the the Taurus, and they give rise to two dimensional representation of the Galois group.
That's why we get, we get a non-trivial example of the language 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. For the diehards, 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 a kind of a teaser. So in the number field number theory context, these are the water morphic functions, which are the generalizations of modular forms. In our example, in our basic example, right? In our basic example, what corresponds to
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.
Here already, they will have a nice geometric interpretation as functions on what's called Bungie. And this I will explain next time. But now the punchline is that for a Riemann surface, now there are two versions in the analytic Langlands, 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 bungee and we have some computing operators which are the annals of the HECA operators of the classical theory as well as some differential operators which are actually very similar to the kind of kind of Schrodinger operators and higher and higher order differential operators and so there is a well-defined spectral problem where you can talk about the
Eigen functions and Eigen values of those operators, they commute with each other. So we can see the joint Eigen functions and the corresponding Eigen values, which are given in terms of the Langlands dual group. So that's one formulation, but the in the geometric Langlands corresponds instead of functions, you can see the sheaves. So instead of the Hilbert space of functions or more properly have densities on bungee, you have a category of sheaves of a certain kind on bungee.
And it is what the object on the right hand side. So we will, next time I will explain what this modular space of G bundles is, what those, then what those are functions versus sheaves, vector spaces versus categories and Langlands correspondence as a Fourier transform. Okay. All right. So, so then we'll have a rough formulation of both, 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 shift 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 shift assigns a vector space to every point. You see, this is a much more sophisticated object. 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 neighborhood, 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 five dimensional vector space, you have how many vectors? You have continuum of vectors. And now on the other side, you have number five.
a given function will have a value five at a given point. A given shift will have 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, I hope, I hope there 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
Also, thank you to our partner, The Economist. Firstly, thank you for watching, thank you for listening. There's now a website, kurtjymungle.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 that I have an untrammeled communication with you. Plus, soon I'll be releasing a one-page PDF of my top 10 toes. It's not as Quentin Tarantino as it sounds like. Secondly, if you haven't subscribed or clicked that like button, now is the time to do so. Why? Because each subscribe, each like helps YouTube push this content to more people
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Every dollar helps far more than you think. Either way, your viewership is generosity enough. Thank you so much. 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 level, 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 in how deep we can go i think it's important to um 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 i think so it'll be interesting to see the results absolutely
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▶ View Full JSON Data (Word-Level Timestamps)
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"text": " 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"
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"text": " with you, Kurt and your, your dedicated audience of your podcast theories of everything. I was very pleased by the response to our first part and the language program. There was a lot of"
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"text": " I'm really excited about continuing along these lines and revealing more secrets of the Langlands program."
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"text": " Oh, wonderful. So one of the questions, and by the way, I was happily surprised about the reception and you 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."
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"text": " Okay. So professor, 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. Uh, the way mathematics develops is that is, is like you're searching for something in the dark room or, you know, in the dark space. Uh, we may be searching for a key, you know, uh, and, uh,"
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"text": " you the way you would proceed usually is that we trying to imagine how things could work out and that 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 analogies and we will also have theorems a theorem is something which is proved"
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"text": " which is valid universally that there is no argument in mathematics that something's 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"
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"text": " So there is a very, um,"
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"text": " um there is a scrutiny for all these papers so they are all subjected to scrutiny by peers not necessarily through 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 uh once this is done we say okay it's a theorem so then"
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"text": " The expectation is it's something unassailable. If it's true today, it will be true in a hundred 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 a hundred years and so on. That's the that's mathematical rigor. So you see, you have this, we have this process of progress in mathematics. You don't immediately come with theorems. You usually come with certain visions, with some intuitions, with"
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"text": " Uh, 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 kind of 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 this visions, um, this intuitions become concrete. You see,"
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"text": " 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 Geisgurje, 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,"
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"text": " 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 into the into the topic. Okay, so why don't we go straight into the slides?"
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"text": " 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 language program, as I mentioned, is a kind of a giant project aimed at finding common patterns in different fields of mathematics."
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"text": " 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,"
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"text": " a great mathematician on his own, right? 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,"
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"text": " 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. 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."
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"text": " 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, kind of what is often called clock arithmetic, just like our clock has 12 hours. And when we talk about time,"
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"text": " We usually don't say in at least in North America, don't say 14 o'clock, we say two o'clock. So in other words, we take numbers, modular 12, 13 means one, 14 means two, 15 means three, 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"
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"text": " um, arithmetic with a clock that has seven hours, right? So, 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. And, uh, right. So seven is a prime number. It's not divisible by any other whole number other than itself. And number one,"
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"text": " And we can likewise consider clocks with p number of p hours where p is one of those primes 2 3 5 7 11 13 and then we get a new numerical 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"
},
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"text": " 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 two and a three. And some people were thinking, wait a moment, I thought Fermat's last theorem had to do with any n greater than three. 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."
},
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"text": " 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."
},
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"text": " 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 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"
},
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"text": " this the cubic equations that give us what's called elliptic curves. But the idea is that we are considering the same equation, but we're looking for solutions, modular, every prime number you see, because we can 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."
},
{
"end_time": 739.599,
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"start_time": 730.418,
"text": " And what we want to do is count the number of solutions, right? And so the here's a little table which shows you how many solutions this equation has."
},
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"text": " So that the difference is divisible by five, right? And likewise for the prime number seven we have nine solutions and so on."
},
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"text": " 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 two is two minus four and so on. It's p minus the number of solutions. A general expectation is that the number of solutions is going to grow"
},
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"text": " the same way as 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 you have to simply take this number"
},
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"text": " You know, you take P minus this number and then you'll get number of solutions. So in other words, we 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 the computer and you can find"
},
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"text": " this 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"
},
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"start_time": 859.189,
"text": " translating complicated questions of number theory, such as finding this 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, you see, you have, you have this variable Q and you consider Q multiplied by"
},
{
"end_time": 912.056,
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"start_time": 889.189,
"text": " a product of factors. So there are two types of factors. There is like one minus Q squared, there's one minus Q squared, one minus Q cubed squared, Q to the fourth squared. Then there will also be one 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 minus Q to the 11 squared."
},
{
"end_time": 942.739,
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"start_time": 913.404,
"text": " one minus q to the 22 squared one minus q to the 33 squared and so on. So it's very easy to write down these factors right up to any 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 two or q to the fifth with coefficient one or q to the sixth coefficient two and so on."
},
{
"end_time": 970.043,
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"start_time": 943.097,
"text": " 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"
},
{
"end_time": 979.002,
"index": 38,
"start_time": 970.52,
"text": " 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?"
},
{
"end_time": 1003.473,
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"start_time": 979.462,
"text": " Yes, yes. There were two quick questions that the audience had here, if you don't mind. So one was that we made plenty of references to signs and cosines when talking about harmonic analysis. And then we go to this Q polynomial, and there are no signs or cosines. So someone wants to know what is the connection between this and harmonic analysis precisely? Right. So I explained in the in the in part one,"
},
{
"end_time": 1029.172,
"index": 40,
"start_time": 1003.968,
"text": " that a prototype that 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, what 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."
},
{
"end_time": 1059.872,
"index": 41,
"start_time": 1030.009,
"text": " 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 and there are more namely you can say cos 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 shift by by 2 pi"
},
{
"end_time": 1086.852,
"index": 42,
"start_time": 1060.162,
"text": " And so the idea of harmonic analysis is to use this trigonometric functions as sort of a, as a basis for the space of all functions and try to decompose general functions as linear combinations of this basic harmonics they're called. Right. And so, um, it's, it is like decomposing the sound of a symphony into the nodes of, of specific instruments, which each node being a kind of a sine or cosine function. Right."
},
{
"end_time": 1114.497,
"index": 43,
"start_time": 1087.585,
"text": " with a particular grade responding to 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"
},
{
"end_time": 1142.449,
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"start_time": 1114.906,
"text": " 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? So, and basically the statement is that this coefficient are precisely the sort out numbers AP, which is, which are essentially the numbers of solutions of the cubic equation, modular, the corresponding prime, right?"
},
{
"end_time": 1172.773,
"index": 45,
"start_time": 1143.217,
"text": " and so you see just just just to kind of the to 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 one formula to rule them all you see and kind of a colossal compression of information or you could say finding order and seeming chaos that's one of the examples of what the langlands program is about"
},
{
"end_time": 1202.125,
"index": 46,
"start_time": 1173.899,
"text": " 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's just an infinite series algebraic like an arbitrary algebraic equation it's a kind of algebraic equation where q is actually just some formal variable right but actually turns out that you can actually substitute for q any number which is less than say positive real number which is less than less than one"
},
{
"end_time": 1232.585,
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"start_time": 1202.961,
"text": " And moreover, you can substitute any real number, which is between negative one and one. So it's 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 that is whose absolute value is less than one. So here I want to remind you that complex numbers"
},
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"index": 48,
"start_time": 1233.08,
"text": " can be can be represented as points on the plane right where 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"
},
{
"end_time": 1288.848,
"index": 49,
"start_time": 1260.691,
"text": " 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 one geometrically, it just means taking the taking a disk of radius one, right? So these are exactly all the points whose such that the distance from the point to the origin is less than one."
},
{
"end_time": 1310.452,
"index": 50,
"start_time": 1289.275,
"text": " 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 open disk, in other words, without the boundary, any point, which is a point, which is a complex number corresponding to this, this inequality or satisfying this inequality,"
},
{
"end_time": 1339.531,
"index": 51,
"start_time": 1311.084,
"text": " 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 one plus two plus three plus four and so on where it can give you minus one over twelve is 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"
},
{
"end_time": 1368.217,
"index": 52,
"start_time": 1340.145,
"text": " is absolute value is less than one, you see. If you take Q with absolute value greater than one, it will diverge same way as one plus two plus three plus four 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,"
},
{
"end_time": 1372.381,
"index": 53,
"start_time": 1368.609,
"text": " As the real line is relevant to the harmonic analysis,"
},
{
"end_time": 1402.005,
"index": 54,
"start_time": 1372.875,
"text": " 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."
},
{
"end_time": 1432.363,
"index": 55,
"start_time": 1402.671,
"text": " 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."
},
{
"end_time": 1461.015,
"index": 56,
"start_time": 1433.302,
"text": " With respect to the action of of this group PSL to Z or perhaps it's subgroup. Sure, these groups have this group has a family of subgroups which are appear naturally in this key in this setting. Okay, so that is this notion of a modular form. And what I'm trying to say is this infinite series, which, you know, contains the answer to all of the counting problems for this specific cubic equation."
},
{
"end_time": 1480.947,
"index": 57,
"start_time": 1461.596,
"text": " This infinite 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,"
},
{
"end_time": 1508.968,
"index": 58,
"start_time": 1481.323,
"text": " Counting the numbers of solutions of this specific cubic equation module 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 second recap of that. Okay. This is called a Q series. This is a"
},
{
"end_time": 1530.213,
"index": 59,
"start_time": 1509.309,
"text": " A generalization of sine, cosine, which is a Fourier series. So the 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."
},
{
"end_time": 1545.026,
"index": 60,
"start_time": 1530.725,
"text": " You know how people are familiar with Taylor series expansion, right? 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."
},
{
"end_time": 1561.988,
"index": 61,
"start_time": 1545.503,
"text": " and you have a point and you and you 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."
},
{
"end_time": 1584.582,
"index": 62,
"start_time": 1562.5,
"text": " 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 coordinate naming appears in mathematics."
},
{
"end_time": 1602.568,
"index": 63,
"start_time": 1585.06,
"text": " Yes."
},
{
"end_time": 1631.527,
"index": 64,
"start_time": 1603.2,
"text": " in in single variable calculus. For example, we know that one divide by one minus x, maybe I should give this example, you can have this one divide by one minus x, which you can write as a sum one plus x plus x squared plus x cubed and so on. But 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. You see,"
},
{
"end_time": 1652.466,
"index": 65,
"start_time": 1631.749,
"text": " 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, right? So we're doing something very similar, except we are now working with a complex"
},
{
"end_time": 1674.991,
"index": 66,
"start_time": 1652.756,
"text": " playing 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 one. Actually, it also converges when X is a complex number and whose absolute value is less than one. So it's actually very similar. You can actually allow to expand from"
},
{
"end_time": 1695.742,
"index": 67,
"start_time": 1675.265,
"text": " 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 whose which satisfy this inequality. So is this series, which is another way is just expanding this product is converging when Q has the same property satisfies the same inequality doesn't make sense."
},
{
"end_time": 1714.224,
"index": 68,
"start_time": 1696.357,
"text": " 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"
},
{
"end_time": 1736.63,
"index": 69,
"start_time": 1714.462,
"text": " a corresponding elliptic curve and also backward. How do you find the Q series? Well, so you're jumping ahead, right? Because so far, so far, so far, there is only one cubic equation, there is only one 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,"
},
{
"end_time": 1762.073,
"index": 70,
"start_time": 1737.056,
"text": " 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 the subject of what is called the Shimura Taniyama Vey Conjecture. In our first conversation,"
},
{
"end_time": 1781.186,
"index": 71,
"start_time": 1762.585,
"text": " 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. That's the current working title. It's a fantastic title, by the way."
},
{
"end_time": 1807.944,
"index": 72,
"start_time": 1781.698,
"text": " 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 on how this ideas play out in geometry. Right. So in principle, if you're most interested in that subject, you can kind of just watch this, this short recap and then"
},
{
"end_time": 1834.428,
"index": 73,
"start_time": 1808.422,
"text": " 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."
},
{
"end_time": 1862.995,
"index": 74,
"start_time": 1834.616,
"text": " Thank you, Kurt. I recommend you check it out. This book, I wrote this book about 12 years ago, 12, 13 years, 12, 11, 12 years ago. It was published 2013 precisely to explain the ideas of the language 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 are, if you would like to really go deeper and to really understand this concepts and ideas,"
},
{
"end_time": 1890.879,
"index": 75,
"start_time": 1863.285,
"text": " more precisely than, than this is one source for you. Uh, 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, several, several resources for that. Um, and in love, love and math, I try to explain it for the general audience. In other words, not for non-mathematicians, which is the idea today as well. People interested in mathematics, but not, not necessarily specialists, not necessarily experts."
},
{
"end_time": 1914.65,
"index": 76,
"start_time": 1891.493,
"text": " And, uh, but we will include as also some sources for more advanced, uh, 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,"
},
{
"end_time": 1926.425,
"index": 77,
"start_time": 1914.872,
"text": " 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?"
},
{
"end_time": 1954.343,
"index": 78,
"start_time": 1926.766,
"text": " it's"
},
{
"end_time": 1979.462,
"index": 79,
"start_time": 1954.804,
"text": " 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"
},
{
"end_time": 2001.459,
"index": 80,
"start_time": 1979.906,
"text": " Solved by a particular modular modular form that is associated to this cubic equation or this cubic or this elliptic curve over the field of rational numbers you see and this way one updates one to one correspondence between those elliptic curves over Q and what's called modular forms of weight two with integer coefficients"
},
{
"end_time": 2027.807,
"index": 81,
"start_time": 2002.193,
"text": " 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"
},
{
"end_time": 2057.056,
"index": 82,
"start_time": 2028.319,
"text": " 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 they could be maybe it covers all primes except number p equal 11 or something like that you see so almost all yeah but that's not that's not such a big deal given that it covers it for all"
},
{
"end_time": 2080.606,
"index": 83,
"start_time": 2057.312,
"text": " 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 example, 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."
},
{
"end_time": 2103.404,
"index": 84,
"start_time": 2081.34,
"text": " 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 let's, I want to mention"
},
{
"end_time": 2126.34,
"index": 85,
"start_time": 2103.712,
"text": " the link to the between the shimur tanayama way congestion. I should say this conjecture was formulated uh initially uh in 1955 by uh by this japanese mathematician yutaka tanayama interesting interesting interesting human story so the original formulation was actually incorrect and then um"
},
{
"end_time": 2155.93,
"index": 86,
"start_time": 2126.971,
"text": " These two other mathematicians, Goro Shimura, who was actually his friend, also a 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 contributions. They kind of corrected the initial formulation by Taniyama. But what Taniyama did was kind of like a quantum leap, if you will. So he really came up with something which nobody expected."
},
{
"end_time": 2185.589,
"index": 87,
"start_time": 2155.93,
"text": " And Shimura, so Tanyama unfortunately took his life in the commit suicide at the age of 31. This is brilliant mathematician who died very young. And there is a beautiful tribute to him written by his friend and colleague, Gor Shimura. And I remember one quote from there that Shimura says that Tanyama has this unique ability to make good mistakes. Right. And he said, it's not, it's not easy. He said, I tried to make good mistakes, but I couldn't, I failed."
},
{
"end_time": 2214.48,
"index": 88,
"start_time": 2185.981,
"text": " It is a very unique talent. So what does it mean with 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"
},
{
"end_time": 2245.333,
"index": 89,
"start_time": 2215.606,
"text": " 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 the story because it shows you the drama, the, you know, the of mathematical ideas, the drama of mathematicians trying to come up with this idea is coming to come up with this conjecture is trying to prove them. Oftentimes this conjecture is maybe wrong. So then other mathematicians come along who disprove them."
},
{
"end_time": 2263.148,
"index": 90,
"start_time": 2245.964,
"text": " But at the end of the day, it's this co it is collective enterprise of sharing this 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."
},
{
"end_time": 2285.418,
"index": 91,
"start_time": 2263.473,
"text": " 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 Tony Alma was doing the stream of consciousness. No, no, not at all. It was, uh, no, he was 90% correct. Let's just say,"
},
{
"end_time": 2314.155,
"index": 92,
"start_time": 2286.084,
"text": " Imagine that, imagine a 90% correct green play on the first day. Maybe 80%. Okay, it's hard to measure. 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 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."
},
{
"end_time": 2339.787,
"index": 93,
"start_time": 2314.65,
"text": " also known as modularity conjecture, or today 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"
},
{
"end_time": 2366.988,
"index": 94,
"start_time": 2340.657,
"text": " 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 Shmur-Tanyama-Wei Conjecture in the same stable case. But because of the work by Ken Ribbett,"
},
{
"end_time": 2396.971,
"index": 95,
"start_time": 2367.944,
"text": " 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"
},
{
"end_time": 2426.988,
"index": 96,
"start_time": 2397.312,
"text": " without success. And so finally Ken Ribbett was able to connect it to the Shmur-Tanyamvay Conjecture. And then Andre Weiss and Richard Taylor proved Shmur-Tanyamvay 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 Shmur-Tanyamvay."
},
{
"end_time": 2457.142,
"index": 97,
"start_time": 2427.551,
"text": " 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, you know. It's much more subtle. All right. And so now, what does it tell us about the Langlands program? Well, guess what? The link between the counting problem for cubic equations and modular forms of way two is a very special case of the Langlands program."
},
{
"end_time": 2483.49,
"index": 98,
"start_time": 2458.08,
"text": " So more than Yam of a by itself is a vast realization of this one example that we started with for a specific cubic equation and specific modular form. But the Langlands correspondence or the Langlands program rankers conjectures are vast realization of this. She more than Yam of a conjecture. You see. All right. So now what do I mean by this? Um, you see,"
},
{
"end_time": 2513.456,
"index": 99,
"start_time": 2484.138,
"text": " Let's recap. The original Langlands program was about difficult questions 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 questions in harmonic analysis, like finding coefficients of modular forms. So, Schmortani M of A 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."
},
{
"end_time": 2542.858,
"index": 100,
"start_time": 2513.746,
"text": " And we have numbers of solutions or more precisely this numbers AP, which is remind you is not exactly the number of solutions, but it's P minus the number of solutions. Right. The error, the error, the difference between the prime number itself and the number of solutions. Uh, and on the right hand side, you have objects called modular forms, more specifically, uh, normalized new forms of weight two with integer coefficients. And it turns out that there is a bijection or"
},
{
"end_time": 2572.363,
"index": 101,
"start_time": 2543.217,
"text": " As we say one-to-one correspondence between objects on the left and objects on the right under which this numbers of solutions or approximate errors of a number of solutions AP match with coefficients of the modular form in front of so you have here BPQ to the P. That's the term in the modular form corresponding to the peace power where P is a prime."
},
{
"end_time": 2600.196,
"index": 102,
"start_time": 2573.78,
"text": " and Langlands correspondence, which is a giant generalization of this one to one correspondence is about more general, more general representations of what's called a Galois group on the left in this, in the special case that gives us this, this representations correspond to cubic equations and the certain, certain numbers associated with them correspond to this numbers of solutions that we discussed."
},
{
"end_time": 2622.892,
"index": 103,
"start_time": 2600.776,
"text": " 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 Langlands correspondence or Langlands conjecture in the original formulation. It's just so remarkable that firstly, the top even has an arrow in any direction."
},
{
"end_time": 2650.213,
"index": 104,
"start_time": 2623.336,
"text": " and then it's remarkable that it's in both directions. It's remarkable that you can generalize that and that that has two arrows. Right. By the way, there is a lot more. This is only the beginning. Okay. So it's going to be even more. It's 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."
},
{
"end_time": 2675.401,
"index": 105,
"start_time": 2650.589,
"text": " 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 in the is a kind of an upper index on the left."
},
{
"end_time": 2681.51,
"index": 106,
"start_time": 2675.964,
"text": " Yeah, why is it on the left? Langlands introduced this notation."
},
{
"end_time": 2706.203,
"index": 107,
"start_time": 2682.176,
"text": " And in fairness, it 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 it 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."
},
{
"end_time": 2712.415,
"index": 108,
"start_time": 2706.442,
"text": " And then someone's like, Oh, no, why are you naming it after yourself? No, no, no, that's a cold Joyce manifold."
},
{
"end_time": 2736.988,
"index": 109,
"start_time": 2713.336,
"text": " As you know, on Theories of Everything, we delve into some of the most reality-spiraling concepts from theoretical physics and consciousness to AI and emerging technologies. To stay informed, in an ever-evolving landscape, I see The Economist as a wellspring of insightful analysis and in-depth reporting on the exact topics explored here and even more."
},
{
"end_time": 2761.852,
"index": 110,
"start_time": 2736.988,
"text": " The economist's commitment to rigorous journalism means you get a clear picture of the world's most significant developments, whether it's the latest in scientific innovation or the shifting tectonic plates of global politics. The economist provides comprehensive coverage that goes beyond the headlines. What sets the economist apart is their ability to make complex issues accessible and engaging, much like we strive to do in this podcast."
},
{
"end_time": 2783.558,
"index": 111,
"start_time": 2761.852,
"text": " If you're passionate about expanding your knowledge and gaining a deeper understanding of the forces that shape our world, then I highly recommend subscribing to The Economist. It's an investment into intellectual growth, one that you won't regret. As a listener of Toe, you get a special 20% off discount. Now you can enjoy The Economist and all it has to offer for less."
},
{
"end_time": 2801.408,
"index": 112,
"start_time": 2783.558,
"text": " Head over to their website www.economist.com slash totoe to get started. Thanks for tuning in. And now back to our explorations of the mysteries of the universe. And the notation kind of caught on. So people like this in the we've been using this for more than 50 years."
},
{
"end_time": 2825.845,
"index": 113,
"start_time": 2801.732,
"text": " 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 the subject, myself included, is to understand why the Langlands dual group appears."
},
{
"end_time": 2850.93,
"index": 114,
"start_time": 2826.34,
"text": " There are some explanations. I came up with an explanation with my co-author and former advisor, Boris Fagan, 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."
},
{
"end_time": 2881.425,
"index": 115,
"start_time": 2851.527,
"text": " In any case, there is this phenomenon. So you can ask, what about the Schmurtan-Yamaway 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."
},
{
"end_time": 2908.234,
"index": 116,
"start_time": 2881.903,
"text": " And, uh, it's one of the simplest groups that, that mathematicians study. And in this case, it just so happens that the dual group is the same. So at the level of the Schmur-Tanyama-Wei 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."
},
{
"end_time": 2928.148,
"index": 117,
"start_time": 2908.882,
"text": " uh, which appears in the study in some physical theories and so on. Uh, then you will discover that there is this duality appearing different, two different groups appearing on the two sides of the language correspondence. So here, here's one rabbit hole that we could"
},
{
"end_time": 2954.275,
"index": 118,
"start_time": 2928.49,
"text": " go into, which is classifying all possible groups that appear here. So here there's this what we call they are what's called reductively groups or reductive algebraic groups. And those are essentially products of a billion groups and what's called semi simple groups. And so the semi simple groups in turn are kind of like products"
},
{
"end_time": 2980.64,
"index": 119,
"start_time": 2954.667,
"text": " of simply groups or simple algebraic groups. And finally, those are classified by what's called dynkin diagrams, more precisely, the so-called simply connected, simply connected to 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."
},
{
"end_time": 3001.698,
"index": 120,
"start_time": 2981.032,
"text": " You see, and so it's 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 what this results to, for instance, B and C and get replaced, the arrow has to be reversed."
},
{
"end_time": 3012.5,
"index": 121,
"start_time": 3001.937,
"text": " 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"
},
{
"end_time": 3036.391,
"index": 122,
"start_time": 3012.841,
"text": " 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 Dynkin diagrams and the centers, the finite centers of the corresponding algebraic groups."
},
{
"end_time": 3064.821,
"index": 123,
"start_time": 3036.783,
"text": " It's not always the case that a reductive groups dual or Langlands dual, if that's what it's called, is also a reductive group. It's always a reductive group, but in particular always reductive. So Langlands duality is an involution on the set of all the 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"
},
{
"end_time": 3087.312,
"index": 124,
"start_time": 3065.589,
"text": " 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,"
},
{
"end_time": 3116.732,
"index": 125,
"start_time": 3087.654,
"text": " If this Lie group is simple and they, 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. And you can see here, there is also an arrow, but if you reverse it, you get the same diagram."
},
{
"end_time": 3140.52,
"index": 126,
"start_time": 3118.439,
"text": " Likewise here. But here, if n is greater than two, 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, but about the exact, exact, how the dual is."
},
{
"end_time": 3164.838,
"index": 127,
"start_time": 3141.288,
"text": " 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 if it"
},
{
"end_time": 3193.558,
"index": 128,
"start_time": 3165.316,
"text": " 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 odd-dimensional vector space, and that 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 Lengar's group to the Lengar's dual group as we go from one side to the other side of Lengar's correspondence suggests that this correspondence is extremely sophisticated. Close your eyes, exhale, feel your body relax,"
},
{
"end_time": 3222.654,
"index": 129,
"start_time": 3193.951,
"text": " All right. So now,"
},
{
"end_time": 3247.517,
"index": 130,
"start_time": 3223.2,
"text": " 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."
},
{
"end_time": 3277.022,
"index": 131,
"start_time": 3248.575,
"text": " 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. So let's find out where else in what other areas of mathematics"
},
{
"end_time": 3306.852,
"index": 132,
"start_time": 3277.91,
"text": " Can this 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 language program. So that explains why we're interested in the reformulations of this original ideas, but in other domains, the other areas of mathematics. Okay. And so what helps us here is, is this Rosetta stone of mathematics, which I also talked about,"
},
{
"end_time": 3335.657,
"index": 133,
"start_time": 3307.227,
"text": " Last time, uh, which was proposed by Andre way, whose name has already been mentioned several times in a letter to his sister, Simone way from prison, actually in France in 1940, he wrote about analogies between these three areas of mathematics. One of them is number theory. And that's what has, what appears in the original formulation of the language program. Right. But then there are two other areas."
},
{
"end_time": 3363.285,
"index": 134,
"start_time": 3336.561,
"text": " 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. Andre Wey wrote in this letter, my work consists in deciphering a trilingual text. So it is indeed just like Rosetta Stone in some sense."
},
{
"end_time": 3388.899,
"index": 135,
"start_time": 3364.514,
"text": " Of each of the three columns, 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"
},
{
"end_time": 3417.159,
"index": 136,
"start_time": 3390.111,
"text": " 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"
},
{
"end_time": 3447.09,
"index": 137,
"start_time": 3418.712,
"text": " And this automorph functions, right? That's the original. That's the original version. But because we now know that this, these two other areas 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,"
},
{
"end_time": 3472.278,
"index": 138,
"start_time": 3447.654,
"text": " 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 sort of the other sides."
},
{
"end_time": 3494.991,
"index": 139,
"start_time": 3472.722,
"text": " 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 from totally from the left field. And the only reason why only reason why we even even"
},
{
"end_time": 3522.585,
"index": 140,
"start_time": 3495.538,
"text": " there to believe that there is some kind of Langlands 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."
},
{
"end_time": 3551.664,
"index": 141,
"start_time": 3523.677,
"text": " 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 Andreweiss 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."
},
{
"end_time": 3579.548,
"index": 142,
"start_time": 3552.329,
"text": " The best understood, in fact, is the middle area, middle area, meaning curves over finite fields. That's the kind of a sweet spot where, which kind of is a turntable, as Andre we called it is a kind of a bridge between these two. In the case of in the case of curves over a finite field on the one on the one hand, you don't have some of the difficult aspects of number theory."
},
{
"end_time": 3604.838,
"index": 143,
"start_time": 3581.237,
"text": " 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, the one of the biggest developments in the language program across all three domains are all three areas of mathematics has been the proof of the"
},
{
"end_time": 3634.48,
"index": 144,
"start_time": 3605.555,
"text": " Langlands correspondence in the in the basic case of the group GLN, we have talked about earlier about the group GL2, which appears in the Shimur-Tanyama-Wei conjecture from the perspective of the Langlands correspondence. GL2 is the group of two by two 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"
},
{
"end_time": 3662.824,
"index": 145,
"start_time": 3634.821,
"text": " you know, positive integer, which have non-zero determinant. It's important to have non-zero determinant because then you have the inverse, the multiplicative inverse. So if you consider all n by n matrices with non-zero 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."
},
{
"end_time": 3687.637,
"index": 146,
"start_time": 3663.166,
"text": " 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 is actually very hard, from n equal two to arbitrary n."
},
{
"end_time": 3714.548,
"index": 147,
"start_time": 3688.046,
"text": " in the early 2000s, both of them received fiddle fields medals for their works. So this is a highly kind of highly prized achievements 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,"
},
{
"end_time": 3741.561,
"index": 148,
"start_time": 3714.787,
"text": " which arises in the, 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 language correspondence is. That's where, that's what this recent series of works by Gates, Gury, Raskin and others is about. So what are the Riemann surfaces? First of all, these are the Riemann surfaces."
},
{
"end_time": 3769.616,
"index": 149,
"start_time": 3742.79,
"text": " These are examples of reman surfaces, the sphere, the surface of a donut, surface of you could say Danish pastry or something. Yes. So I guess I guess this one is Homer Simpson's favorite. Sure. Now these are compact reman surfaces. Is that important? Yes. Well, there is a more general formulation where we consider non-compact ones where we remove points."
},
{
"end_time": 3791.118,
"index": 150,
"start_time": 3770.606,
"text": " 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"
},
{
"end_time": 3817.619,
"index": 151,
"start_time": 3791.715,
"text": " response to having finitely many marked points and kind of removing those points so that you allow some sort of singularities at those marked points mark means that you mark 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 is the second one the third one this is just the way people say people say it you can say chosen or specified it's the same"
},
{
"end_time": 3843.234,
"index": 152,
"start_time": 3819.155,
"text": " All right. So these are the human surfaces. And the question we're going to discuss now is to how to formulate the language program for this object for this human surfaces, as if the language program was not complicated enough already. We're now going to move to a different domain. We're going to move under the streetlight, if you will. All right. And try to find the key there. You see,"
},
{
"end_time": 3872.944,
"index": 153,
"start_time": 3844.684,
"text": " So, for Riemann surfaces, here is one more twist. In fact, for Riemann surfaces, there are two versions. There are two versions. The first one was initiated by great mathematicians, De Ligne, Pierre De Ligne, Vladimir Dreinfeldt, whom I have already mentioned, and Gerard Lamond in the 80s. And it is called the Geometric Languages Program. I should also mention another mathematician who is very heavily involved in this, Alexander Bellinson. Bellinson and Dreinfeldt"
},
{
"end_time": 3902.824,
"index": 154,
"start_time": 3874.309,
"text": " actually made perhaps the biggest contribution to this geometric Langlands correspondence for human surfaces balance from the dream felt 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 goodie 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"
},
{
"end_time": 3926.8,
"index": 155,
"start_time": 3903.234,
"text": " 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,"
},
{
"end_time": 3957.449,
"index": 156,
"start_time": 3927.534,
"text": " 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 sheaves. But interesting enough, in the last five years, a new version of the language correspondence for human surfaces was proposed. And I've been involved in this."
},
{
"end_time": 3977.927,
"index": 157,
"start_time": 3957.79,
"text": " 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."
},
{
"end_time": 4000.503,
"index": 158,
"start_time": 3978.217,
"text": " 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,"
},
{
"end_time": 4021.63,
"index": 159,
"start_time": 4001.049,
"text": " 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."
},
{
"end_time": 4050.009,
"index": 160,
"start_time": 4022.056,
"text": " 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 theoretic version. And in some sense, I feel guilty because I spent a lot of time trying to explain it to him."
},
{
"end_time": 4074.65,
"index": 161,
"start_time": 4050.845,
"text": " And I guess I didn't do a good job. Why didn't he like the sheaf version? He's more traditional mathematician for for mathematicians of his generation. Sheaves are kind of not exactly anathema, but it's like additional bells and whistles that you don't need. But he's of the generation of growth in deep, no? Yeah, but he comes, he's sort of the other side of the of the spectrum."
},
{
"end_time": 4104.36,
"index": 162,
"start_time": 4075.299,
"text": " Oh, the Sarah side? Well, Sarah is closer to Grotendieck. The division between Sarah and Grotendieck is not that big. Grotendieck, Alexander Grotendieck, one of the most brilliant mathematicians 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. Grotendieck died in 2014."
},
{
"end_time": 4131.271,
"index": 163,
"start_time": 4105.094,
"text": " 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 shifts. 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."
},
{
"end_time": 4161.493,
"index": 164,
"start_time": 4131.732,
"text": " 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"
},
{
"end_time": 4191.578,
"index": 165,
"start_time": 4161.817,
"text": " of functions of spaces, operators, and so on, as opposed to shifts, 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? Yes, we did. We started it. So six years ago, there was no analytic Langlands program, if I must search that. The first time I talked about it was actually in 2018, soon after Langlands received the"
},
{
"end_time": 4208.08,
"index": 166,
"start_time": 4192.295,
"text": " The main point of his talk was there has to be a function theoretic version."
},
{
"end_time": 4236.834,
"index": 167,
"start_time": 4208.746,
"text": " and he and i had a little tension about this but he had absolutely correct idea that this version has to exist and in many ways that inspired me to look closer into this okay 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 in maybe a year earlier about some special cases of this how this could work out and"
},
{
"end_time": 4265.845,
"index": 168,
"start_time": 4237.5,
"text": " 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 when we were kids, basically, when we were in college, and David Kasdan, who is one of the luminaries in representation theory, he was"
},
{
"end_time": 4292.483,
"index": 169,
"start_time": 4266.34,
"text": " The favorite student of Israel Gelfand. So in some sense, he's a kind of 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. So where he is now. And so with piloting golf and David Kajdan, we have worked on this for the last five years. We've published"
},
{
"end_time": 4315.623,
"index": 170,
"start_time": 4292.773,
"text": " 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,"
},
{
"end_time": 4342.654,
"index": 171,
"start_time": 4315.947,
"text": " Ford Blue Cruise hands-free highway driving takes the work out of being behind the wheel, allowing you to relax and reconnect while also staying in control."
},
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"end_time": 4360.981,
"index": 172,
"start_time": 4343.78,
"text": " Enjoy the drive in Blue Cruise enabled vehicles like the F-150, Explorer and Mustang Mach-E. Available feature on equipped vehicles. Terms apply. Does not replace safe driving. See Ford.com slash Blue Cruise for more details. Okay."
},
{
"end_time": 4379.411,
"index": 173,
"start_time": 4361.92,
"text": " So let me actually show you this is the web page of dennis gays gary who by the way has been my co-author my collaborate 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"
},
{
"end_time": 4398.336,
"index": 174,
"start_time": 4379.889,
"text": " So Dennis has this web page 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 the statement for arbitrary groups, which is a really big deal. It's a really wonderful achievement."
},
{
"end_time": 4416.783,
"index": 175,
"start_time": 4398.336,
"text": " So there are five different papers listed on this website, haven't been published yet, haven't been refereed yet, but you know, there's still enough preprint form, but you know, 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"
},
{
"end_time": 4436.254,
"index": 176,
"start_time": 4417.244,
"text": " 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"
},
{
"end_time": 4462.398,
"index": 177,
"start_time": 4437.142,
"text": " the foundation was laid out already, but there's so many, there was so many technical issues that one needed to deal with. And remarkably this, they are addressed in this new series of papers, but those papers, I think were written over the years, not like they didn't just dump them all at once. Okay. So here's one of, here's what the first page of one of them looks like. It's a second one, second paper in the series. Okay. And so"
},
{
"end_time": 4485.93,
"index": 178,
"start_time": 4463.285,
"text": " I want to show you something which is kind of to give you an indication of what a great, what a grand project that was and how many people have been involved. Just to give you an idea of how much goes into this. It really takes a village, you know, if you will. So this as the proverb says, so the first thing they mentioned here, this is the acknowledgement to one of the papers."
},
{
"end_time": 4512.91,
"index": 179,
"start_time": 4486.459,
"text": " one of the papers in which they give tribute to some of the mathematicians who came before them, whose results they are using, which by the way, you know, I'm impressed by the way they are kind of acknowledging the contributions of other mathematicians is really, they're showing a great example for young mathematicians. This is how it's done. And so the first thing, the first two people they mentioned are Alexander Bellinson and Vladimir Dreamfeld, whom I have already mentioned."
},
{
"end_time": 4541.135,
"index": 180,
"start_time": 4513.729,
"text": " 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,"
},
{
"end_time": 4556.578,
"index": 181,
"start_time": 4541.63,
"text": " So indeed, we contributed on the side of representation theory, 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,"
},
{
"end_time": 4580.162,
"index": 182,
"start_time": 4557.21,
"text": " 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 newer 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. I see. But we, we, we use BRST in a more generalized sense these days."
},
{
"end_time": 4603.046,
"index": 183,
"start_time": 4580.776,
"text": " And then there is this Fagin-Frankel Center, which is kind of plays the central role, as they say, you know, interesting. And so then, of course, then there is another deep idea of factorization. And so they mentioned Jacob Lurie down there and Jacob Lurie. Yeah, Jacob Lurie tremendous works of tremendous importance, which kind of like gives you proper language for what's called higher algebra."
},
{
"end_time": 4622.705,
"index": 184,
"start_time": 4604.002,
"text": " So I 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, okay, we have proved this conjecture and this subject has been developing for like 40 years, you know, so like what kind of"
},
{
"end_time": 4643.985,
"index": 185,
"start_time": 4622.875,
"text": " 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, Denis and I, Denis Gait-Gurion, I actually wrote eight papers together on this subject closely related to this Katz-Mudi Algebra at the critical level, which give the"
},
{
"end_time": 4667.022,
"index": 186,
"start_time": 4644.753,
"text": " the tool for connecting for the establishing of geometric language correspondence. So in fact, some of those are important in this in this series of papers, I have been involved in this project. And up until, you know, a few years ago, when I switched to this new version, which is the function theoretic version called the analytical language correspondence, but I'm very pleased and impressed that my"
},
{
"end_time": 4696.698,
"index": 187,
"start_time": 4667.312,
"text": " 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, you cannot say that it's, 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,"
},
{
"end_time": 4723.046,
"index": 188,
"start_time": 4696.903,
"text": " 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, consequences of this series of works by Dennis Gaizko, Sam Raskin and others. All right."
},
{
"end_time": 4747.739,
"index": 189,
"start_time": 4724.633,
"text": " So then I want to, well, I also want to show you the first page of my recent paper with Pavel Letingov and Dave Cash done about the analytic language correspondence, you see, which was recently published in this journal and was actually a special issue in honor of a wonderful Italian mathematician, Corrado de Conchini. You mentioned writing screenplays earlier."
},
{
"end_time": 4776.254,
"index": 190,
"start_time": 4748.797,
"text": " Fun fact, Corrado de Cancini's father, Ennio de Cancini, was a famous Italian filmmaker. He actually got an Oscar. He co-wrote a screenplay of the film Divorce Italian Style 1961 with Marcello Mastroianni. And he received an Academy Award with his co-authors for the screenplay. Anyway, it's just kind of a"
},
{
"end_time": 4800.981,
"index": 191,
"start_time": 4777.073,
"text": " Interesting connection to what was his son's contribution to this. So there was a there was this paper appeared in a volume in a in a issue of a journal, pure and applied mathematics in honor of his 70th birthday. So we dedicate dedicate the paper to him with admiration. He's a great mathematician, great human being as well. So"
},
{
"end_time": 4830.811,
"index": 192,
"start_time": 4802.602,
"text": " So that so there is this now this other this other thread. In addition to geometric Langlands correspondence, there is now this analytical Langlands correspondence. It doesn't mean that it supersedes. Just to be clear, it doesn't mean that it's better than the other one is different. And the two are connected. So they complement each other. And I will comment more on that as we as we proceed. All right. Sure. The Langlands would probably say this one's better. So the Langlands would say is better. Yes. So you know, this is what happened is like,"
},
{
"end_time": 4855.06,
"index": 193,
"start_time": 4831.305,
"text": " You know, I mentioned in part one that I collaborated with Langlands. We wrote a paper together and this was about from 2008 to 2010. And I spent a lot of time talking to him. And then he, our work was not in the geometrical language because it was more like a traditional one, but for curves over finite fields mostly."
},
{
"end_time": 4875.043,
"index": 194,
"start_time": 4855.435,
"text": " as well as number field case. With another mathematician, by the way, Ngo Bao Chau, who is a professor at the 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 is about? What is this about? What is this? What are these shifts? How do they fit? And I tried to explain"
},
{
"end_time": 4893.336,
"index": 195,
"start_time": 4875.879,
"text": " And the result of it was that he hated it. Okay, so sorry to say 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 are not have no place. There has to be a function theoretic version. He even went"
},
{
"end_time": 4920.776,
"index": 196,
"start_time": 4893.712,
"text": " As far as saying during his talk at the Abel prize symposium, you know, not symposium, but it was kind of acceptance speech after the, he said that they might as well remove my name from geometric correspondence. I don't want it to be, I think it's a little too much because it is really, to me, geometric is right. It has a place in this general overarching theory."
},
{
"end_time": 4951.032,
"index": 197,
"start_time": 4921.067,
"text": " but different matters have different tastes and they have different styles. This is the reason why I'm sharing with you this stories is to show you what a robust dialogue we have with each other that even though we respect each other tremendously, you know, 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 than other and so on. So in this case, here is Langlands."
},
{
"end_time": 4976.63,
"index": 198,
"start_time": 4951.596,
"text": " He's in his eighties, right? And so, but he's still very full of energy and he's like, no, I don't like, uh, I don't like this shift formulation. There has to be a function theoretic formulation, which by the way, at that time in the subject, nobody believed 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 coauthors to, to try to do it."
},
{
"end_time": 5004.002,
"index": 199,
"start_time": 4977.193,
"text": " And it turns out indeed that it was possible to do. I should mention, as I mentioned, there was, there was a York Tashner was another mathematician in Hamburg, but Michael physicist who tried to do something similar. There was another Russian born mathematician who works in France, uh, 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 and as a kind of a,"
},
{
"end_time": 5031.92,
"index": 200,
"start_time": 5004.326,
"text": " really a branch of the Langlands program, which is what we called the Analytical Langlands program. Now is that the same Koncevich who generalized mirror symmetry to homological mirror symmetry? So Koncevich is a very brilliant mathematician who has worked in many different areas and made very important contributions, but at some point he got interested in the Langlands program and so he tried to do it 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"
},
{
"end_time": 5060.708,
"index": 201,
"start_time": 5032.346,
"text": " We kind of see now from our perspective, where we have this general theory that this is a special case of what we are just we are considering as the analytical language correspondence. Got it. Yeah. So 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 that who have influenced the subject. One of them is Israel Gelfand, who died in 2009."
},
{
"end_time": 5090.913,
"index": 202,
"start_time": 5061.459,
"text": " 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 Gelfand's, but they were closely connected and they were extremely influential, not only in the Soviet Union, but all over the world. And I want to mention them because"
},
{
"end_time": 5114.514,
"index": 203,
"start_time": 5091.664,
"text": " In many ways, all of what all these ideas that I'm going to talk about that have to do with the Hellenic correspondence for human surfaces, both the geometric formulation and analytic formulation, in some ways are can be traced to these two giants, Kelfand and and Manin. So, for example, Bellinson and Drenfeld, these two mathematicians who are perhaps"
},
{
"end_time": 5141.34,
"index": 204,
"start_time": 5115.52,
"text": " most responsible for these developments were his students, you see, of Eurimania. They were also very closely connected to Gelfand. So Gelfand had this famous seminar. By the way, in my, 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. And, you know, it's really hard to, to, to capture it in a few words, but I really wanted to, to mention our debt to these two people, because"
},
{
"end_time": 5167.568,
"index": 205,
"start_time": 5141.749,
"text": " 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, I have been greatly influenced by his, not only his ideas, but his general approach to mathematics."
},
{
"end_time": 5175.845,
"index": 206,
"start_time": 5167.91,
"text": " one of the things"
},
{
"end_time": 5203.387,
"index": 207,
"start_time": 5176.288,
"text": " 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 Neumann, you know, so there is this famous Guelph-Neimark theorem about sister algebras and so on. So he really, but he worked in so many different fields. He also worked with computer scientists. He worked with biologists. He was really in the Renaissance man."
},
{
"end_time": 5232.09,
"index": 208,
"start_time": 5204.445,
"text": " 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 Gelfand. For instance, David Cashdown, my co-author and collaborator on the analytic language correspondence was, one could say, his favorite student, favorite student of Gelfand. And in some ways, he's perhaps the most connected to this whole circle of ideas. My other co-author Pavel Letingov was a student of Gelfand as well, to some extent."
},
{
"end_time": 5261.101,
"index": 209,
"start_time": 5232.619,
"text": " but also Yuriy 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 Yuriy 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."
},
{
"end_time": 5287.449,
"index": 210,
"start_time": 5261.596,
"text": " 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."
},
{
"end_time": 5303.865,
"index": 211,
"start_time": 5287.91,
"text": " Interesting."
},
{
"end_time": 5334.053,
"index": 212,
"start_time": 5305.265,
"text": " This episode is brought to you by State Farm. Listening to this podcast? Smart move. Being financially savvy? Smart move. Another smart move? Having State Farm help you create a competitive price when you choose to bundle home and auto. Bundling. Just another way to save with a personal price plan. Like a good neighbor, State Farm is there. Prices are based on rating plans that vary by state. Coverage options are selected by the customer. Availability, amount of discounts and savings, and eligibility vary by state."
},
{
"end_time": 5365.094,
"index": 213,
"start_time": 5335.674,
"text": " Alright, 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"
},
{
"end_time": 5394.019,
"index": 214,
"start_time": 5366.049,
"text": " 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."
},
{
"end_time": 5423.951,
"index": 215,
"start_time": 5394.497,
"text": " 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, in the, I'm talking here about the Andre Wey, Rosetta Stone."
},
{
"end_time": 5453.677,
"index": 216,
"start_time": 5426.647,
"text": " 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."
},
{
"end_time": 5482.79,
"index": 217,
"start_time": 5454.019,
"text": " 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 was p was p taking us back to zero, then p plus one is one and so on. Right? We have addition and multiplication here, modular p"
},
{
"end_time": 5512.398,
"index": 218,
"start_time": 5483.029,
"text": " 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,"
},
{
"end_time": 5541.681,
"index": 219,
"start_time": 5512.927,
"text": " Let me do kind of a divided this into two parts and explain it as a kind of a translation between two languages. Uh, the highlighting the objects, which are analogous to each other. Uh, the first thing to start with on the left hand side left has left hand side has to do with number theory. We have the set of integers, right? So you have like, you know, minus one, zero, one, two, and so on. Right."
},
{
"end_time": 5571.715,
"index": 220,
"start_time": 5542.585,
"text": " So the analog of this is what's called FP of T, which is where the T is with square brackets, which is all polynomials, polynomials over FP. So what does it look like? So for example, you can have polynomial. So you have a zero plus a one T plus a two T squared and so on plus some a N"
},
{
"end_time": 5597.005,
"index": 221,
"start_time": 5572.039,
"text": " 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 school polynomials with real coefficients right so a typical such polynomial would be would have this form where a zero a one and so on is a real number but now"
},
{
"end_time": 5625.896,
"index": 222,
"start_time": 5597.5,
"text": " We're considering these expressions where each coefficient is an element of this finite field Fp. You see? So now what is it? 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 zero. Sorry, p is equal to two. Let's consider a special case when p is equal to two. In this case, ai"
},
{
"end_time": 5654.923,
"index": 223,
"start_time": 5626.442,
"text": " is either zero or one because the field F2 actually has only two elements zero and one because two is already back to takes us back to zero right so then a polynomial would look like this for example it would be like one plus zero times t plus one times t squared and so on for example here's an example one plus t squared is an example of a polynomial over F2 right"
},
{
"end_time": 5678.251,
"index": 224,
"start_time": 5655.725,
"text": " so what you see is that you have a sequence of zeros and ones really what 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 the n of them where n is a degree of the polynomial"
},
{
"end_time": 5705.828,
"index": 225,
"start_time": 5678.933,
"text": " 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 at least 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"
},
{
"end_time": 5723.643,
"index": 226,
"start_time": 5706.698,
"text": " you can kind of say all right well here you have one and here you have one here you have zero okay and here you have one the difference is that here we we take two and then two squared and two cube and so on and here we have this additional variable t"
},
{
"end_time": 5753.251,
"index": 227,
"start_time": 5724.155,
"text": " 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 two squared plus two squared is two cubed, right? Can we do that with any number if you just change the base?"
},
{
"end_time": 5781.22,
"index": 228,
"start_time": 5754.087,
"text": " 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 zero and one. 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're both are described"
},
{
"end_time": 5801.288,
"index": 229,
"start_time": 5781.869,
"text": " Objects are described as sequences of zeros and ones. Okay. But with different arithmetic. So this is just to indicate why we believe why mathematicians 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."
},
{
"end_time": 5828.848,
"index": 230,
"start_time": 5801.954,
"text": " Oh, yes. The first of many, by the way. Okay. Yes. Yes. This is the very first observation. And 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, the conjectures, where you actually say this set and this set are in one to one correspondence, for instance, she more than I am way conjecture, what does it say? It says,"
},
{
"end_time": 5856.766,
"index": 231,
"start_time": 5829.002,
"text": " that there is a one-to-one correspondence between cubic equations of certain kind and these modular forms of weight 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."
},
{
"end_time": 5886.169,
"index": 232,
"start_time": 5857.961,
"text": " 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 have polynomials, but now we also have rational functions or"
},
{
"end_time": 5915.265,
"index": 233,
"start_time": 5887.261,
"text": " It consists of ratios of 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."
},
{
"end_time": 5939.445,
"index": 234,
"start_time": 5915.794,
"text": " All right, so that's 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 a 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."
},
{
"end_time": 5963.763,
"index": 235,
"start_time": 5940.998,
"text": " But now you can appreciate also why Riemann surfaces appear. Because you once since you know, 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 of two with two elements zero and one."
},
{
"end_time": 5992.329,
"index": 236,
"start_time": 5964.445,
"text": " Right. Then if we do this, do we not lose the translation between the number field and the remote surface? Of course we do. Of course it's very far away, but you see, you see that you can appreciate on the one side that yes, there is an analogy on the other side, how far away it is, because in one case we talk about finite field and in that 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 always finite coefficients."
},
{
"end_time": 6011.459,
"index": 237,
"start_time": 5993.08,
"text": " In the other case, it's polynomials with complex coefficients, but it does look like a very tenuous analogy at first glance."
},
{
"end_time": 6041.886,
"index": 238,
"start_time": 6012.671,
"text": " 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 the hard theorems, and then we prove them, that's the beauty of mathematics. It combines both the intuition, which sometimes looks like"
},
{
"end_time": 6069.411,
"index": 239,
"start_time": 6042.312,
"text": " 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 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."
},
{
"end_time": 6092.108,
"index": 240,
"start_time": 6070.486,
"text": " once again you know who's just pushing the envelope so that's what we're doing here we are like a child we're being a child and the child has learned the number fields okay a over b where a and b are integers and just oh but look numbers can be written binary and in binary form they look very much like polynomials with coefficients and f2"
},
{
"end_time": 6122.551,
"index": 241,
"start_time": 6092.654,
"text": " 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, 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,"
},
{
"end_time": 6148.848,
"index": 242,
"start_time": 6122.892,
"text": " and not being afraid to push this envelope we come up with some ideas then 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"
},
{
"end_time": 6172.927,
"index": 243,
"start_time": 6149.411,
"text": " But then there is more because, for instance, square root of two is not rational, right? So square root of two cannot be written as a fraction of two 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."
},
{
"end_time": 6202.381,
"index": 244,
"start_time": 6174.531,
"text": " 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 agreement surface, guess what it is? It is actually sphere. The"
},
{
"end_time": 6232.346,
"index": 245,
"start_time": 6202.756,
"text": " 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, if it can be a surface, Riemann 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"
},
{
"end_time": 6255.998,
"index": 246,
"start_time": 6232.824,
"text": " 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 so I can see there's something similar, but very different because now we fix the fix the ground field, we fix for example, f five or f seven. And we're considering this equation only over five or only over f seven."
},
{
"end_time": 6286.049,
"index": 247,
"start_time": 6256.715,
"text": " 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 on the 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"
},
{
"end_time": 6316.203,
"index": 248,
"start_time": 6286.51,
"text": " 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."
},
{
"end_time": 6343.763,
"index": 249,
"start_time": 6316.732,
"text": " Okay. So that's the fields in general. These are the parallel things. So the theory should develop in 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 over Riemann surface. Okay. All right. Next, we have Galois groups."
},
{
"end_time": 6373.592,
"index": 250,
"start_time": 6344.633,
"text": " 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 are joined square root of 2 and so on. In the case of curves over a 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."
},
{
"end_time": 6401.527,
"index": 251,
"start_time": 6374.121,
"text": " namely, we interpret it as what's called the fundamental group of the streaming surface, which is denoted by pi one of X. So let's talk about the fundamental group and how it is connected to 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."
},
{
"end_time": 6430.316,
"index": 252,
"start_time": 6402.09,
"text": " Great. So it's not something advanced at all. It's kind of like bread and butter. So this is like, this is this is an analogy, which is well established by now. Granted, in 1940, that's what Andre we explained, but 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, more, more clearly what what the objects are. So"
},
{
"end_time": 6459.77,
"index": 253,
"start_time": 6430.828,
"text": " 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, what are the elements here? They have the form alpha plus"
},
{
"end_time": 6487.722,
"index": 254,
"start_time": 6460.435,
"text": " Beta times square root of two where alpha and beta are rational numbers, right? So then you notice that there is a symmetry There is a symmetry of this of this field Let's call a sigma which sends square root of two to minus square root of two and vice versa under the symmetry this element goes to"
},
{
"end_time": 6515.196,
"index": 255,
"start_time": 6489.957,
"text": " Alpha minus beta times square root of two. You see, so the reason why this is a symmetry is that the square root of two is a solution of the equation. X squared minus two equals zero. We can all agree on this in real numbers. But so is minus square root of two."
},
{
"end_time": 6536.032,
"index": 256,
"start_time": 6515.52,
"text": " 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 that was minus square root of two, which shows you that square root of two 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."
},
{
"end_time": 6565.708,
"index": 257,
"start_time": 6537.432,
"text": " 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 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"
},
{
"end_time": 6594.565,
"index": 258,
"start_time": 6566.783,
"text": " to the case of function fields here instead of Q. Now we will have something like, um, FP of T, right? So that's what we discussed. Uh, and so what's the analog of this type of symmetry and the symmetry, by the way, is an example of a, of a, an element of a Galois group is Galois group is appears as a group, as a group of symmetries of this field, which preserve, preserve, uh, the, the structure of the field."
},
{
"end_time": 6616.869,
"index": 259,
"start_time": 6595.316,
"text": " 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 two 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 it's not present"
},
{
"end_time": 6636.544,
"index": 260,
"start_time": 6617.244,
"text": " 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."
},
{
"end_time": 6663.319,
"index": 261,
"start_time": 6638.456,
"text": " 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 of symmetries of this larger field. So now do the same for complex in the complex case again you can take a square root of t and again you have a symmetry exchanging"
},
{
"end_time": 6680.776,
"index": 262,
"start_time": 6663.473,
"text": " 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."
},
{
"end_time": 6711.869,
"index": 263,
"start_time": 6681.971,
"text": " It's"
},
{
"end_time": 6741.118,
"index": 264,
"start_time": 6712.363,
"text": " The group is what's called a fundamental group. And here's how to explain it. You see, you can think of geometrically that one curve here is below is kind of a cover of two curves. So you have you have a curve, which I, which is actually the 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 CP one."
},
{
"end_time": 6772.159,
"index": 265,
"start_time": 6742.346,
"text": " but I will approximately"
},
{
"end_time": 6799.155,
"index": 266,
"start_time": 6772.79,
"text": " And so now you see geometrically what you're doing is you're just exchanging the two branches. Right. So that's a, that's a, it's a very nice example of how things become geometric. As you move across this Rosetta stone of Andre V things that were algebraic, like here, you know, um, here this, we're talking about some algebraic equation and we have two solutions, square root of two minus square root of two. It's not clear what kind of geometry."
},
{
"end_time": 6826.169,
"index": 267,
"start_time": 6800.009,
"text": " 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 was called CP1 projective line, complex projective line."
},
{
"end_time": 6843.473,
"index": 268,
"start_time": 6826.578,
"text": " And this extension of the field actually corresponds to a cover, a covering of this project flying by which is a double cover. That's right. Because for every point, for every point here, except zero,"
},
{
"end_time": 6868.882,
"index": 269,
"start_time": 6843.78,
"text": " okay now the reason is that in the middle there's 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 so this point is special you see it's a marked it's a special point because over it there is only one point but for all non-zero points"
},
{
"end_time": 6895.077,
"index": 270,
"start_time": 6869.36,
"text": " Well, actually, to be honest, if we talk about CP infinity, there is also a point at infinity, which is also where you also have ramification. Right. So, but I'm only showing you the part outside of infinity, so to speak, so that then we can see one of the zero point as a point ramification for all non zero points, there will be two points above in the cover. And that's that that corresponds to square root of T and minus square root of T."
},
{
"end_time": 6924.684,
"index": 271,
"start_time": 6896.032,
"text": " If t if number is non zero, that 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. Okay. 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."
},
{
"end_time": 6953.729,
"index": 272,
"start_time": 6925.384,
"text": " So which will show us that if we go to Galois groups, not of like this, the second 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,"
},
{
"end_time": 6983.712,
"index": 273,
"start_time": 6954.428,
"text": " 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 unranified situation for human surfaces. This is something that we'll perhaps need to talk about in more detail next time."
},
{
"end_time": 7011.493,
"index": 274,
"start_time": 6984.497,
"text": " But now let me just make a few more steps so that we get to kind of a good place. Okay. What other objects are involved? Remember how we talked about the group and its language dual? So first of all, we are now positioned at the beginning of this correspondence of explaining this correspondence correspondence always has two sides, right? So it's one to one correspondence between this and that."
},
{
"end_time": 7040.077,
"index": 275,
"start_time": 7012.381,
"text": " 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 the left on the left hand side, we have this homomorphisms from the Galois group to the Langlands dual. So remember we had this discussion about the Langlands dual group."
},
{
"end_time": 7066.903,
"index": 276,
"start_time": 7043.183,
"text": " 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."
},
{
"end_time": 7095.247,
"index": 277,
"start_time": 7067.295,
"text": " So we're going to consider homomorphisms from this fundamental group to the language dual group. These are the left hand side of the language correspondence, because remember you have to have two sides to talk about correspondence. Okay. And this is, 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?"
},
{
"end_time": 7124.974,
"index": 278,
"start_time": 7096.169,
"text": " 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 the Langlands dual group. In other words, the objects on the left hand side, which I have presented as numbers of solutions of a cubic equation,"
},
{
"end_time": 7155.23,
"index": 279,
"start_time": 7126.34,
"text": " 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."
},
{
"end_time": 7186.323,
"index": 280,
"start_time": 7156.698,
"text": " 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. So the special case, that's a special case. In this case,"
},
{
"end_time": 7215.947,
"index": 281,
"start_time": 7186.613,
"text": " F is Q, so you have the Galois group of Q, and it maps to GL2. And such a map, such an object, 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."
},
{
"end_time": 7245.776,
"index": 282,
"start_time": 7216.459,
"text": " 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."
},
{
"end_time": 7273.797,
"index": 283,
"start_time": 7246.203,
"text": " And the result is precisely this number AP that we talked about. So in interesting, I could get by with some, with more classical notions in this case, namely, I could, I could talk without ever mentioning function field or actually in this case, there's no function field, but number field, a field of rational numbers or the Galois group or the language do group. I could completely put this aside and speak about cubic equations."
},
{
"end_time": 7302.244,
"index": 284,
"start_time": 7274.104,
"text": " 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,"
},
{
"end_time": 7331.442,
"index": 285,
"start_time": 7302.585,
"text": " 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"
},
{
"end_time": 7354.804,
"index": 286,
"start_time": 7331.988,
"text": " 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 conjugate classes."
},
{
"end_time": 7383.507,
"index": 287,
"start_time": 7355.179,
"text": " And briefly, what dictates the group on the reductive group on the right hand side? So the GL two in this case, is it the elliptic curve or the field? So these are two separate parameters. Oh, sorry. So elliptic curve gives GL two, why elliptic curve gives GL two? Oh, yes, that's very good question. Okay, let's talk about this briefly. So here is the here is the explanation is the following that there are certain structure in this elliptic curve, which is two dimensional."
},
{
"end_time": 7412.517,
"index": 288,
"start_time": 7384.77,
"text": " And this structure is not easy, but is useful to explain in terms of the analogy we talked about. The idea is that 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."
},
{
"end_time": 7442.756,
"index": 289,
"start_time": 7414.036,
"text": " This 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 by analogy with complex Riemann surfaces."
},
{
"end_time": 7466.749,
"index": 290,
"start_time": 7443.507,
"text": " 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?"
},
{
"end_time": 7495.418,
"index": 291,
"start_time": 7467.79,
"text": " So you ask me why the Galois representations associated to elliptic curve are two dimensional, which is to say, why do we get homomorphous to GL two and not to GL three? Yes. And the answer is, is because the elliptic curve, 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? Yes, absolutely. We would get so for a, for a,"
},
{
"end_time": 7524.138,
"index": 292,
"start_time": 7496.459,
"text": " Higher genus curves, you will have a higher dimensional 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,"
},
{
"end_time": 7540.384,
"index": 293,
"start_time": 7524.343,
"text": " whether there is a link between the next program and the Moonstyne. What's it called? 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. I think after this long conversation,"
},
{
"end_time": 7568.729,
"index": 294,
"start_time": 7541.101,
"text": " 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, 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."
},
{
"end_time": 7596.118,
"index": 295,
"start_time": 7569.121,
"text": " 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 human surface, which is a sphere. And the sphere does not have"
},
{
"end_time": 7626.067,
"index": 296,
"start_time": 7596.92,
"text": " Any useful color representations. They only have trivial, basically trivial representation. Gallo representation associated to it. That's why the so-called helped module, 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 Gallo group. And then for higher genus,"
},
{
"end_time": 7650.845,
"index": 297,
"start_time": 7626.749,
"text": " You see,"
},
{
"end_time": 7679.343,
"index": 298,
"start_time": 7651.63,
"text": " For the CP one for the project flying for the sphere, there are there are no noncontractable cycles. They can all be contracted to a point. Therefore, there is no homology. There is no first homology or a coma. Right. But for a for a service of a donut, you have two cycles, two independent cycles, which are noncontractable, the one which goes across and one which goes along the the Taurus, and they give rise to two dimensional representation of the Galois group."
},
{
"end_time": 7699.241,
"index": 299,
"start_time": 7680.094,
"text": " That's why we get, we get a non-trivial example of the language 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. For the diehards, where we will explain actually. If you've watched this far, it's for you."
},
{
"end_time": 7728.865,
"index": 300,
"start_time": 7699.599,
"text": " 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 a kind of a teaser. So in the number field number theory context, these are the water morphic functions, which are the generalizations of modular forms. In our example, in our basic example, right? In our basic example, what corresponds to"
},
{
"end_time": 7758.968,
"index": 301,
"start_time": 7729.633,
"text": " 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."
},
{
"end_time": 7780.862,
"index": 302,
"start_time": 7759.292,
"text": " Here already, they will have a nice geometric interpretation as functions on what's called Bungie. And this I will explain next time. But now the punchline is that for a Riemann surface, now there are two versions in the analytic Langlands, 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."
},
{
"end_time": 7806.032,
"index": 303,
"start_time": 7781.305,
"text": " of actually half what they call half densities on bungee and we have some computing operators which are the annals of the HECA operators of the classical theory as well as some differential operators which are actually very similar to the kind of kind of Schrodinger operators and higher and higher order differential operators and so there is a well-defined spectral problem where you can talk about the"
},
{
"end_time": 7834.633,
"index": 304,
"start_time": 7806.613,
"text": " Eigen functions and Eigen values of those operators, they commute with each other. So we can see the joint Eigen functions and the corresponding Eigen values, which are given in terms of the Langlands dual group. So that's one formulation, but the in the geometric Langlands corresponds instead of functions, you can see the sheaves. So instead of the Hilbert space of functions or more properly have densities on bungee, you have a category of sheaves of a certain kind on bungee."
},
{
"end_time": 7861.971,
"index": 305,
"start_time": 7835.077,
"text": " And it is what the object on the right hand side. So we will, next time I will explain what this modular space of G bundles is, what those, then what those are functions versus sheaves, vector spaces versus categories and Langlands correspondence as a Fourier transform. Okay. All right. So, so then we'll have a rough formulation of both, both stories."
},
{
"end_time": 7892.056,
"index": 306,
"start_time": 7862.722,
"text": " 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."
},
{
"end_time": 7920.759,
"index": 307,
"start_time": 7892.261,
"text": " A shift 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 shift assigns a vector space to every point. You see, this is a much more sophisticated object. 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."
},
{
"end_time": 7946.442,
"index": 308,
"start_time": 7921.425,
"text": " 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 neighborhood, in the formal neighborhood of this point. It's a much more sophisticated object. Even though the word functions is used,"
},
{
"end_time": 7975.452,
"index": 309,
"start_time": 7946.732,
"text": " 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 five dimensional vector space, you have how many vectors? You have continuum of vectors. And now on the other side, you have number five."
},
{
"end_time": 8005.469,
"index": 310,
"start_time": 7976.357,
"text": " a given function will have a value five at a given point. A given shift will have 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, I hope, I hope there 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"
},
{
"end_time": 8032.79,
"index": 311,
"start_time": 8006.22,
"text": " Also, thank you to our partner, The Economist. Firstly, thank you for watching, thank you for listening. There's now a website, kurtjymungle.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."
},
{
"end_time": 8058.234,
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"text": " That's just part of the terms of service. Now, a direct mailing list ensures that I have an untrammeled communication with you. Plus, soon I'll be releasing a one-page PDF of my top 10 toes. It's not as Quentin Tarantino as it sounds like. Secondly, if you haven't subscribed or clicked that like button, now is the time to do so. Why? Because each subscribe, each like helps YouTube push this content to more people"
},
{
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"text": " like yourself, plus it helps out Kurt directly, aka me. I also found out last year that external links count plenty toward the algorithm, which means that whenever you share on Twitter, say on Facebook or even on Reddit, etc., it shows YouTube, hey, people are talking about this content outside of YouTube,"
},
{
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"text": " which in turn greatly aids the distribution on YouTube. Thirdly, there's a remarkably active Discord and subreddit for theories of everything where people explicate toes, they disagree respectfully about theories and build as a community our own toe. Links to both are in the description. Fourthly, you should know this podcast is on iTunes, it's on Spotify, it's on all of the audio platforms. All you have to do is type in theories of everything and you'll find it. Personally, I gained from rewatching lectures and podcasts"
},
{
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"text": " I also read in the comments"
},
{
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"text": " And donating with whatever you like. There's also PayPal. There's also crypto. There's also just joining on YouTube. Again, keep in mind it's support from the sponsors and you that allow me to work on toe full time. You also get early access to ad free episodes, whether it's audio or video. It's audio in the case of Patreon video in the case of YouTube. For instance, this episode that you're listening to right now was released a few days earlier."
},
{
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"text": " Every dollar helps far more than you think. Either way, your viewership is generosity enough. Thank you so much. 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 level, high school level."
},
{
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"text": " 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 in how deep we can go i think it's important to um 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 i think so it'll be interesting to see the results absolutely"
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"text": " Think Verizon, the best 5G network, is expensive? Think again. Bring in your AT&T or T-Mobile bill to a Verizon store today and we'll give you a better deal. Now what to do with your unwanted bills? Ever seen an origami version of the Miami Bull?"
},
{
"end_time": 8237.585,
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"text": " Jokes aside, Verizon has the most ways to save on phones and plans where you can get a single line with everything you need. So bring in your bill to your local Miami Verizon store today and we'll give you a better deal."
}
]
}No transcript available.