The Langlands Program - Numberphile

Um language program yes indeed I would say it has been one of the main themes of my research since like forever it's a vast subject and fascinating fascinating subject so I want to tell you a little bit about it I want to start out with this quote I wrote this book it was published in 2007. language correspondence for Loop groups now correspondence here doesn't mean correspondence in the sense of sending Letters to each other although like once and I have done that a lot over the years here correspondence means a kind of a relation like

one-to-one correspondence or something like that so in the preface I say the language program has emerged in recent years as a blueprint for a grand unified theory of mathematics and I have to confess I said it a little bit a tongue-in-cheek you know because in mathematics mathematics is not like quantum physics or you know Where one can talk about the grand unified theory because you have certain interactions with certain forces and you want to find one underlying Theory describing all of them mathematics is just too vast to have one all encompassing Theory you know but

if there were one the language program would probably come closest so interesting enough this this phrase Grand unified theory of mathematics has Kind of cut on and in fact if you look in the Wikipedia article you will see it quoted as well so now let's try to live up to the expectation I have one question first okay why is it called langlands language yeah language is a person is a person this this guy look he is a very interesting man Canadian born he is professor emeritus at the famous Institute for advanced study in Princeton New

Jersey The Institute which you know had as his professors people Like Albert Einstein in fact langlands has Einstein's office and in this picture he is at his office which is formerly Albert Einstein's office a very nice office at my dad you know I I I have collaborated with him so privilege I had a chance to to spend some time in that office language program is named after Robert langlands because he was the one who came up with this ideas or the first set of ideas that we now call the language program in The late 60s

now what happened is that this idea is actually propagated to other areas of mathematics and that was due to research by other people not necessarily England's himself and I think that's why so many mathematicians are and Quantum physicists these days as well uh so excited about uh about this project about this Theory the fact that there are certain patterns which one can observe in many different areas of Mathematics such as number Theory harmonic analysis geometry and now even quantum physics to to set it up I want to say that to me mathematics is something like

a kind of like a giant jigsaw puzzle suppose that you are solving a jigsaw puzzle but you don't know what the final picture is going to look like it's kind of harder but more interesting because then the picture will gradually emerge but usually people start you know Finding pieces that fit Together and so on and so next thing you know you have several islands or continents of sorts on the on the big table you just throw the pieces on big table and several people start working on this that's kind of like different areas of mathematics

I also sometimes I use another analogy I say the continents of mathematics so what are they for instance number Theory that was a study of numbers and equations Um like the ones which we'll talk about in a moment there is also another field called harmonic analysis which is like trying to decompose a sound of a symphony into the kind of Elementary harmonics and to the nodes played by different instruments and that idea has found sort of a great following in mathematics where you kind of generalize it to a much more General situations rather than sound

uh there is also geometry obviously we all study Euclidean geometry in school but in fact this is just one small example there is also no euclidean geometry there are all kinds of geometries and higher Dimensions there are different shapes and manifolds that we call them and so on so there are all these islands as you know when you are working on a puzzle the most exciting moment is when you can connect those islands that you have built and that's what the language program does it connects different Branches and different Islands or different continents of mathematics

and specifically he started out with some difficult questions in number theory of the kind that we'll talk about in a moment and he was able to connect these questions to uh questions in harmonic analysis that have to do with analysis functions with um kind of like things that we study in calculus so kind of totally different Domain right and uh the beauty of it is that the question original questions seem to be interactable seem to be extremely hard unsolvable almost but when you translate them into this other domain harmonic analysis they have much nicer structure

and there are tools of harmonic analysis which enable us to solve them so there is a practical advantage of this type of you know approach because problems that up up until then could not be solved suddenly You find a way to solve them and not just one specific problem but a whole host a whole class of problems of that nature you see so that's why people got so excited about it and then they got even more excited when um this idea is propagated to other fields like geometry and then more recently quantum physics so but

historically language came up with this when he was a he was a young man in the late 60s and in fact he formulated his Ideas for the first time in a letter to another mathematician a sort of a a maestro Meister of mathematics Andre V who was a professor in Princeton at the time I will read it for you he says Professor Bay in response to your invitation to come and talk I wrote the enclosed letter after I wrote it I realized there was hardly a statement in it of which I was certain if you

are willing to read it as pure speculation I would appreciate this if not I'm sure You have a waste basket handy a kind of a understatement of the century you know so is he being humble or do you think he was not confident I think he was being humble I think he knew the importance of it but he kind of wanted it it's a bit of a show off I guess it's like nah it's not that important you know we call that a humble break now don't we I guess so okay so that's how the

English program was born so in the set of those notes and then soon enough you know he wrote a Paper and so on and a lot of people found out about it and those letters you know this is late 60s so it's before the internet so people were making zero scopies of the of this letter and sending to their colleagues and friends around the world and within 10 years hundreds of people working in this area so now after all this build up right now finally I have to show you what it is about and uh

usually you know like time's up Okay Saved by the Bell but you can have all the time you want yes I will I will explain I will explain it so I would like to give an example of this uh of of what appears in the language program that type of connection between uh problems in two different fields number Theory and harmonic analysis and the example I'm going to give is interesting for various reasons one of which is that actually it is precisely the the key result that was proved by Andrew Wiles and Richard Taylor to

provide the final step in the proof of probably the greatest uh unsolved problem in mathematics which had been solved by those papers namely firma's Last Theorem you look at the equation a to the N plus b to the N equals c to the N where n is an integer greater than two and a B and C are relatively prime non-zero integers now if n were two then we know many examples of solutions of this equation like 3 squared plus 4 Squared is equal to 5 squared believed that this equation for n greater than 2 does

not have any integer Solutions so an example would be a cube plus b Cube equals c Cube does not have a solution ABC in non-zero integers obviously if you allow one of them to be zero you could have a is equal to C and B is equal to zero so it'd have to be non-zero to make it interesting right so and so the theory what did You can't do it so if you're miles of course Therma famously wrote about it on the margin of uh theophanto's book about the integer equations saying that he found a

beautiful proof of this ethereum but the margin of the book is too small to contain it but because he claimed that uh Pierre Verma had a great French mathematician in the 17th century for three for about 350 years people were searching for the proof even though you know I guess after one or two hundred Years it became clear they probably didn't have it either he was just being provocative or he was mistaken we don't know already got it already got it maybe he got it who knows right but in any case it the fact that

he claimed that to have found it I think was a great accelerator of the host of the whole field because people really wanted to and produce his proof So eventually it was actually proved by uh in a very different in a very complicated way not By you know there isn't nobody was able to find a kind of Elementary proof where basically you take c b to the end to the right hand side and then like write as a product or something it doesn't work it doesn't seem to work so instead uh mathematicians were able to

relate this statement to another statement which is much harder to to formulate and that statement is called shimura tanyama vacancy after three great mathematicians So very andreway is the one to whom a language letter was addressed he had an important part in this project as well and shimurtayama were this great Japanese mathematicians so they had this conjecture which has to do with mathematologics which are called elliptic curves and so now so you have two different conjectures there is fermas Last Theorem well we call it thermosla's theorem because he claimed to have proved it Um these

guys did not claim to have problems they were honest about it they would put it out as an open question and in mathematics we call open questions conjectures those were two conjectures firma's Last Theorem and shimurtanayama very conjecture and so the decisive step in the proof of this was made by my colleague here at Berkeley Ken ribbett who I believe has been interviewed by a number file as well he showed that conjecture implies sharma's last year so Then the search was on for proof of this conjecture because people knew that as soon as you proved

this one you get the big prize right it's not one of those Millennium prices for the simple reason that Millennium prices were created after the stadium was proved but if Millennium prices were created before then Andrew Wells and Richard Taylor would collect a million dollars I'm sure it would be would have been included on that list but it was a Big prize and it is a result which is probably the most famous of all mathematical theorems um it stood uh for 300 more than 350 years you know so that's quite something these days I would

say human hypothesis is the the big prize but it has it was formulated by Riemann at the end of the 19th century so it's a much younger problem compared to verma's Last Theorem so can rivets accomplishment was basically showing there was a bridge Between the two that's right and so the race was on to do this one because you got that one for free exactly because he rivet established this already and that was a big deal this was in the 80s so then now what's the connection to the language program the connection is that this

is this can be viewed this as a special case of language now when we say language program it we are referring to the whole set of ideas but one of the central ideas of the language program is What's called the language correspondence so it is about correspondence that is to say relation like one-to-one correspondence between objects of different kind and simultaneous can be seen as an example of that language correspondence so let's talk about simultaneous conjecture I want to give an example of what this statement of this conjecture says and to Formulate it I need

to remind you something which is called a cloak arithmetic normally we do you know addition multiplication of integers in the usual way but uh for instance when we talk about time we uh you know in here in North America we identify 30 we don't say 13 hours we say 1 pm right and uh so they don't say 14 but we say 2 p.m so in other words we identify numbers which are related by 12 or a multiple of 12. that's why it's called Clock arithmetic uh normally we have 12 hours well in Europe it would

be 24 I suppose so but still you do module 24 it's called module 24 clock arithmetic but instead of 24 uh we could take any other whole number and uh especially it's interesting is to consider the case when that number is a prime number so here's an example where you have a clock which has seven hours right so you identify minus seven zero seven fourteen you Identify one eight fifteen and so on right and so on so this is more this is a this can be done for any prime number 11 13 and so on

so now suppose you have an equation here's a prime example a prime no pun intended example of an equation that is relevant to the story remember I was talking about elliptic curves so elliptic curves are is kind of a fancy name for equations of this type this type of equations in two variables X and Y for which one of the variables Has degree appears in degree two and the other variable appears in degree three like here on the left hand side you have y squared as the highest power of Y on the right hand side

you have the cube of X as the highest power of X so this equation defines what's called an elliptic curve now why why is it called it to curve well if you look and this probably I should I should say it kind of slow down a little bit and say it more precisely one of the main ideas That emerged in you know in mathematics in the last hundred years is that in some sense equations are more important than Solutions you know it's kind of interesting because I think a lot of our teachers will should learn

from that because the emphasis in our schools I think is too much on solving things and solving means the easiest way to solve is to memorize the formula and that's why so few people actually have good experience in their math classes But if you realize that professional mathematicians actually care about equations a lot more than Solutions then this would help you to kind of refocus completely and reorient the teaching and learning process now what do I mean by equations equations matter more than Solutions because once you write an equation like this you can talk about

its Solutions but you can talk about its Solutions in Different domains for instance you can ask what other values of X and Y in whole numbers so that the left hand side is equal to the right hand side that means you look for Solutions in integers we say or whole numbers right but you can also look at Solutions in real numbers you can also look for Solutions in complex numbers and in fact if you look at the solutions of such an equation and complex numbers You will find that the the set of all solutions can

be presented as a Taurus as a kind of surface of a donut and that's what it's an example of what's called a Riemann surface and you know I always say that it's a Homer Simpson's favorite treatment surface this Human Services the torai they're called elliptic curves so that's where the name comes from it comes from the solutions of this equation over complex numbers But for this story we are going to be interested in Solutions of this equation uh modulo a prime number you see so in other words we're looking for X and Y which are

integers it's whole numbers such that left hand side is equal to the right hand side but not on the nose but in the clock arithmetic with respect to a given prime number so for instance if we're doing it for prime number five it Would mean that we only need the left hand side to be equal to the right hand side up to a multiple of five so for instance if they're equal up to 20 or up to 15 or negative 5 and so on we will call it a solution in that numerical system are you

saying that we would find the solutions take them out convert them into modulo 5 and that's what we're interested in or are we only allowed to take the modulo 5 numbers to start with as our contenders that's right so first Of all is contenders let's see if it's modular five we should only consider number zero one two three four both four x and for y so you're really narrowing your options so there's only 25 options and then you can just calculate on a piece of paper uh what happens when you substitute you know x equals

0 y equals zero x equals one and all equals zero and so on and you go through all of these 25 cases and in some cases the left hand side may be equal to the Right hand side on the nose like for instance if you say x equals zero y equals zero you see that both sides are equal to zero so that's a solution for sure not only modulo five but in fact for integer solution right or you could take for instance a four on the left hand side and then if you calculate you get

4 squared which is 16 plus 4 that's 20. and 24 hour purposes right now is the same as good as zero so in fact I have listed here all the solutions out Of 25 options out of the 25 possibilities namely X and Y being between zero and four we find actually there are four pairs X and Y for which the left hand side is equal to the right hand side sometimes on the nose and sometimes up to a multiple five so for instance x equals zero y equals zero both sides are equal to zero x

equals one y equals zero both sides equal to zero so these are amnesty Solutions x equals zero y equals four that's the one Which I mentioned where you have left hand side is 20 and the right hand side is zero but that that also qualifies as a solution in our current framework 20 is the same as 0 according to a clock arithmetic module 5 right and finally we also have x equals one y equals zero where we also get the left hand side is 20 and the right hand side is zero again it is a

solution so we have four Solutions module five so even though we had limited contenders it also Was becoming easier because we we were becoming a little bit looser as what equals means that's right that's right so but you see that it's really a kind of a problem that uh uh uh Elementary School student can can solve you know it takes it takes time so here you would have to check uh 25 options and see for which of those options the left hand side is equal to the right hand side up to a multiple five if

you go to the next prime number which is seven you would Have 49 options which is seven times seven you would have seven possibilities for X seven possibilities or for y then it's more difficult so it will take you longer but you can do it on a calculator or you can program it and you can have a simple program which can do it for you know relatively High prime numbers fairly quickly so here I have compiled the list on the left colon you have prime numbers 2 3 5 7 11 13. that's a clock these

are how many hours we have In our clock we live in a strange kind of like Wonderland where we have not 12 but let's say seven hours or I would prefer 13 so today they would be longer right so and then how many solutions so we just saw that for prime number five there are four Solutions it turns out that for prime numbers two and three there are also four solutions for each for prime number seven there are nine solutions for prime number 11 or 10 Solutions thirteen nine Solutions and so On remember we are

talking about the same equation but in accordance with what I said earlier you can have just one equation but it can have Solutions in different domains and in different numerical systems here we have one equation which can be viewed as an equation with respect to the clock arithmetic module or any prime number so there are infinitely many possibilities of what we could mean in this context by a solution so if you say solve this Equation that's not a correctly uh phrased question because then you can ask back in which domain and so in fact there

are infinitely many possibilities here and they correspond to the choices of prime numbers but for each of those primes there is a well-defined number the number of solutions and you can see that it kind of grows so the prime numbers grow and the number of solutions grows so in fact it turns out I dropped Again like a 13 it dropped it dropped a little bit but if you if it's possible to compile a much longer list and you will see that it kind of grows it's almost equal to P itself but with some small defect

so that's why it's actually useful to consider not the actual numbers of solutions but the the kind of a defect the difference between the prime number itself p and the number of solutions that you have module p so then you see for instance if Prime is 5 we Found that there are four Solutions so this number is equal to one because five minus four is one likewise for seven you have nine Solutions so suddenly it's a little bit larger than the prime itself so that defect is actually going to be a negative number so this

defects could be both positive and negative and but they're going to be fairly close to zero they're not going to grow as fast as P whereas this numbers grow as fast as P so it's kind of a useful in Normalization if you will relabeling of things so now the question is what are these numbers and from what we have said it they look completely random and this sequence looks completely chaotic there is no if you well this is too short a sample to actually make any conclusions but obviously we can write a short computer program

so I could put them on the screen because you could put them on the screen uh the first 100 or whatever Right oh but wait if you do it directly right so if you do it directly you see that it's a hard problem because each time you have to set up a new problem because for instance you go from five to seven the results about prime number five have no bearing on the results for prime number seven unless you have found some solution which is for which the left hand side and right hand side are

coincide on the nose and not up to Multiple of a prime so then of course such solutions would be valid for every Prime but there are very few of those so in fact oh yes because every equation will have a different fingerprint like here that's right because when we talk about Solutions module five we saw that there was uh there were there were four Solutions only two of them were Solutions where the left hand side is actually literally equal to the right hand side and for two other Solutions The the two sides differed by 20

which was a multiple of five but but 20 has not means nothing for modulus seven it's not equal to zero module seven obviously if you have something divisible by five it's not going to be divisible by seven in principle because they're both prime numbers right so for seven you have to redo the whole calculation from scratch then just as when you kind of like you're tired and you're sweaty from that you say okay now do it for prime number 11. so again you have to do it from scratch right so you have instantly many computational

problems which have which are priority looks like have nothing in common and the numbers that you get look completely random so what if we could but what if we could find another way to solve it what if I told you that there is actually a totally different approach to this problem where all of these numbers could be written sort of in one stroke That's what the name conjecture says and that's what the language program in general the language correspondence in general says it enables you to express the solution of this seemingly intractable problem in number

Theory by tools of harmonic analysis I realized that the two columns are very very linked to one another but are you talking about trying to spit out column two or column three column three so column three is somehow is a little Better because it's kind of uh you have things which are kind of close average to zeros at speak as opposed to growing as as a prime number grows so column three column three is where the where the meat is the question is to express to find a way to express all of these numbers for

all primes all at once by some Magic by some miracle so now I want to describe this alternative solution which pops out Completely out of the blue so there is no nothing in the original problem that could suggest that there is a way to solve it in this way so this is a kind of insight this is something which to me I still think we don't really understand why we can prove it now but I still consider it as a sort of a miracle you know as a kind of black magic honestly okay so here's

how it works the claim is that you can write a certain infinite series and I will explain what I mean by this in this variable queue Q is like X but you know we already have X in the equation so we just need a different letter and so traditionally mathematicians use letter q but it's just a notation if you can use X or what a z or whatever you like so there is this thing which is called infinite series which is written as this process infinite product and I will explain what it means which has

the following property that if you open it has a very Regular uh Behavior I'll explain what I mean by that and then you can start opening the brackets using the traditional rules of how you multiply you know various expressions of this of this sort when you start opening the brackets you're going to get an expression which involves various powers of Q and there will be some coefficients appearing in front which will be a whole number so positive or negative and I claim that if you look at the power for Example the P the fifth power

the coefficient will be equal to exactly that number in the third column associate to five if you look at the coefficient in seventh power you will get number negative two which was the number that we had in the third column Associated to seven and if you look at the eleventh power you get coefficient one which is exactly what you get from the counting problem right and if you look at number 13 Let's hope that this is correct yes it is four and so on so this one line of code if you will if we take

the coefficient of Q to the power p p is a prime so like Q to the five Q to the seven Q to the 11 Q to the 13. if you take the coefficient in front because each of these Powers gets multiplied by something and call it B of P then a of P which I recall is the number which appears in the third column earlier it will be equal to B of P for All prime numbers so as you see it's like forever forever and there are instantly many primes as we know right yeah

so it is a solution not to one not to five not a billion problems it's a solution to infinitely many problems in one line it's a kind of a one-liner it's like if it were if it were a stand-up comedy act you know so it could be like that would be like the the one liner of all one liners which would contain all the jokes That could that was ever told and that will ever be told forever and ever as the prime numbers get really really big as prime numbers do to those coefficients to those

numbers in column three stay around the vicinity of you know one and three and or do they start getting they grow they grow they will grow but they will grow as a square root of P not as B so it's a kind of the growth is much slower but they do get bigger they Do get bigger yes and and they can be both positive and negative so they kind of they are average they're more or less average out to zero but occasionally they can become large positive numbers large negative numbers but always within the square

root of P where p is the prime that we're talking about right okay so colossal compression of information it's like you know fine it's like we're really literally finding order in what Appeared to be a completely random sequence of numbers a completely chaotic sequence of numbers and so another way I like to say is like it's like one line of code gives us a simple rule which kind of generates the DNA of the accounting problem it feels magical to me except I don't know where you got that original algorithm from that's right yeah that's right

so now I want to explain uh what I mean by by this by the series first of all so to explain to as A warm-up let me talk about this concept which we call in mathematics uh generating function so if you have a sequence of numbers so this would be number n natural numbers one two three four five six and so on suppose that you have a certain number assigned to each the F of n so the simplest sequence could be just one one let's say one to all of them that's the first example so

then mathem I just have this trick where they say I don't want To look at this at this table instead I convert it into what's called a generating function or generating series and so what this is is I will take F of one and multiply by Q Plus F of 2 multiplied by Q squared F of 3 multiplied by Q Cube and so on or you know we like this notation Sigma for the summation sign and we'll say that we just go from one to infinity and we put F of n times Q to the

N so what will this generating function be in our first Example where we assign to every natural number just number one it will look like this a cube plus Q squared plus Q Cube plus Q to the force and so on right so you see well there's not much of an improvement in some sense because here you look vertical kind of you go down vertically here you go horizontally to the right but you can play with this we can rewrite this generating function in the following way as Q times another Generating function which now starts

with one one plus Q Plus Q squared and so on clearly if you open the brackets this term will give you Q this term will give you Q squared this term will give you Q Cube and so on each number in the sum is equal to Q times the previous number and we know that if absolute value of Q is less than one this series actually converges to 1 divided by y 1 minus Q so that means that if in this original series Q was just a Bookkeeping device it was just a variable kind of

indeterminate we don't want to initially assign any value to it but then we notice that in fact this expression doesn't think some makes sense if instead of Q we substitute an actual number let's say a real number from negative one to one or it could be a complex number with absolute value less than one then the series actually converges converges for real not like one plus two plus three plus four Where it actually diverges but then there are ways to extract sort of meaningful value here it actually it actually converges according to all the rules

and so to this function 1 divided by 1 minus Q so that this expression becomes Q divided by 1 minus Q so you see instead of this infinite table we end up with this very cute looking function so it contains information about those numbers F of n in the following sense that if you expand it in Powers of q that is to say reverse this process and go from 1 divided by y minus Q to this infinite series then you get this series and from this you can read off all the coefficients F of n

which in this case that trivial kind of the O equal to one so that's the first example let me give you one more example suppose that you assign numbers not just once but one two three four five six and so on so then you're going to have a Q Plus 2 Q squared plus 3 Q Cube plus four Q to The fourth so it's a little bit more interesting now and so you say okay is it possible to also convert it into something like this and the answer is yes in fact you can because you

can also rewrite it as Q times 1 plus 2q plus 3 Q squared and so on I claim that in fact this green Series in Brackets is the derivative by respect to Q of this series because when you take and when we take derivatives we are just doing it term by Term so one is a constant it's derivative is zero so it will give nothing Q will give you one the derivative of Q is one with respect to Q right this is dgq okay so Q will give you one Q squared will give you 2q

Q Cube the next term will give you three Q squared and you see that it's exactly the pattern that you get each coefficient is going to be an integer one two three four because when you take the derivative of Q to the N you get n Times Q to the N minus 1. now what it suggests is that this sums up to the derivative of this and for those who remember their calculus course they will immediately see that it is one divided by 1 minus Q squared so you see the result here so the result

in a previous example when all the numbers were equal to one now we'll be so in that case it was Q divided by one minus q but now it's going to be Q divided by 1 minus Q squared So again you have a very compact expression which contains information about those numbers which in this case if you if you reverse the process and you expand this back into a generating function you will see one two three four and so on popping up skefficients in front of the corresponding powers of Q so that these are two

basic examples of the generating function and now the function that we have here which appears in the in the solution of the counting Problem is kind of like that except it is written well you see here we in our basic examples we were able to write things in terms of kind of some simple factors where we used actually geometric progression so in fact here it was essential that absolute value of Q was less than one in some sense this is very similar because except now we have only positive powers of things like 1 minus Q

or 1 minus Q to the 11. but the meaning of this expression uh is very similar to What we have just discussed it is something that would be converted into into a series of the form F of 1 times Q Plus F of 2 times Q squared plus F of 3 times Q Cube and so on by simply opening the brackets so for instance in this example F of 1 is is equal to one F of 2 is equal to negative two F of 3 is negative 1 and so on now you may ask how

do I go from this line to this line simply by opening brackets so you see first of all there is some regularity in This what is the what is the regularity so there is a coefficient Q which is kind of familiar to us now because in our basic examples we also had a q as a factor next it comes 1 minus U squared we know how to open brackets 1 minus Q squared is simply 1 minus two Q Plus Q squared right so then we replace this expression by 1 minus two Q Plus Q squared

likewise 1 minus Q to the 11 squared is 1 minus 2 Q to the 11 plus Q to the power 22. so we replace That square Pi by by this expression then again comes something similar minus Q squared so you see the regularity is twofold so so to write down this generating function uh first factor is Q after that I write factors 1 minus Q squared and I leave some space then I write 1 minus Q squared squared and I leave some space then I write 1 minus q q squared do you get the idea

yeah and then in this basis I write something similar I write 1 minus Q to the 11. Squared then here I write 1 minus Q to the 22. 22 being of course twice 11 squared then I write 1 minus Q to the 33 always squared and there are two sequences one is where you have one minus q1 is Q squared when Q Cube and so on so clearly the next one is one minus Q so Square Cube fourth squared and after that the red one is going to be 1 minus Q to the 44. squared

and so on right so that's clear now when you open the brackets you may be wondering Whether it is legitimate because there are infinitely many terms but because the powers of Q are increasing as we throw in more and more factors you you see you can see clearly that when you open the brackets the only finitely manufacturers will affect the specific power of Q for instance if you're interested in Q squared it will come from here it will come from here but this is already Q to the 11 it's not going to to contribute to

Q squared Likewise if you want you know Q to the power fourth or fifth this Q This terms will not affect it so only finite in many factors of this infinite product will contribute to a specific power of Cube that's why it is well defined so the only thing that you need to do here is just open the brackets and here I give the expression for the first few terms up to Q to the 13 and so the statement of the name of a conjecture in this particular case is that if you Focus not on

all coefficients not on all powers of Q the way we discussed in our basic example but only the prime Powers so you look just at the coefficients in front of Q squared Q Cube Q to the fifth Q to the seventh Q to the eleven Q to the 13 and so on where what do I what do they all of them have in common they are all Q to the power P where p is a prime number then each of these monomials each of these Powers will come with the coefficient so The equation I call

this B of two I will call this B of three bo5 bo7 B of 11 B of 13. and the statement is that this for every p and p being a prime number B of p is equal to a of P which gives the solution of the counting problem for that prime number namely I recall that we are talking about the third colon so a of p is not exactly the number of solutions but it is p minus the number of solutions that difference mod P so all of these numbers which well if You know

this guys you know obviously numbers of solutions because simply take P minus that number right to get to get the number of solutions so all of these numbers can be found in one in one stroke so to speak from the coefficients of this one generating function now what what what is the meaning of this function where does it come from right so and it turns out that this is an example of what is called a modular form So in fact this example illustrates the general principle that we can link elliptic curves remember the this equation

cubic equation that we talked about corresponds to the mathematical object we call an elliptic curve and so this elliptic curve like more precisely counting problem counting points on an elliptic curve modulo prime Numbers can be linked can be linked to a modular form in this way that the coefficients of the modular form give us more or less number of solutions module prime number up to some simple replacement and that statement is the statement of the name of a conjecture which is a special case of the language correspondence and that is a statement which gives us

The proof of verma's Last Theorem okay nice speaking about elliptic curves I want to mention these two books so this this one called elliptic Tails curves counting and number Theory so which talks about a lot of the things that we discussed today what this elliptic curves look like and their Solutions in different numerical systems and so on so it's a really nice nice book explained in Fairly you know understandable way okay and I think will be interesting to Both mathematicians non-mathematicians alike we'll link to it we'll link to it we'll have a link in there

and then the uh the other book is about cryptography using elliptic curves so the elliptic curves have recently found a real world applications to cryptography and also I want to mention in this regard that my very first video for number file was about an algorithm actually it was not an encryption Algorithm but it was an algorithm for generating student random numbers that this algorithm was proposed by the National Institute for standards and technology in the United States and then when Snowden revealed you know various documents of NSA then people found out that actually that algorithm

was had a back door installed by NSA and this was important because those pseudo-random numbers were used to generate passwords and things like that various various Numbers which were used in other algorithms so whoever had a back door to that algorithm it had like a master key to open all doors you know that's actually the that Infamous publication by the National Institute for standards and Technology nist recommendation for random number generation and then uh then describes this Infamous dual elliptic curve random bit generator dual ecd-rbg here's the elliptic curve that they used so in fact

you can go back to My old video and then I talk about in more detail but you see this is an equation very much like the one we have discussed you have a second order polynomial in y on the left and the third order polynomial of degree three cubic polynomial in X on the right hand side and here is the prime number here's how many hours the US government has in on their clock see how many digits there are so anyway sometimes people ask you know this is All very interesting but it looks very abstract

what are the real world applications and So my answer to that is oh oh there are applications you can be sure of that or if there aren't yet applications for example of the type of questions we discussed regard regarding simultaneous conjunction and so on there will be applications mathematics always finds real world applications the question is whether we will be wise enough not to Use it for evil purposes you know which unfortunately has been the case with some of the applications of mathematics and science in general and let's now go back and discuss modular forms

so I explained that this infinite product represents What's called the modular form and this is an object of harmonic analysis so it's a totally different continent of mathematics from number Theory where our accounting problem originated so what are these modular Forms it turns out that as I explained in my sort of basic example of generating series sometimes something which appears as just a formal sum of various powers of Q some coefficients like this one can be converted in in something which looks much nicer it's kind of a more compact expression namely one divided one by

one minus Q so we have to be mindful of the fact that this equality holds only provided the Q the absolute value of Q is less than one That's when the series converges right and then it converges to this expression with modular forms the situation is very similar uh here we also have an expression of this of this kind the coefficient of the first part of Q is always going to be one it's kind of a normalization for this modular form then there will be F of 2 Q squared F of 3 Q Cube plus

Etc right and it turns out that this Expressions will also converge when Q Absolute value of Q is less than one but in this case it is important to take not just real numbers but complex numbers now as you know complex numbers can be represented by points on a plane where the x-coordinate is the real part and the y coordinate is the imaginary part of the number so if you're interested in numbers whose absolute value in the sense of complex numbers less than one they correspond to a disk of radius one you've got to be

inside that so these Numbers all have to be can be viewed as points inside the disk so now we get some geometry out of this so what starts out as a as an infinite expression as a kind of a summation of uh various powers of Q with some coefficients even though initially we write it as a product maybe but you know eventually we can open the brackets and find out all the coefficients and so on but now we also realize that actually this infinite expression has a well-defined value for Any point inside the disk and

every time something like this happens you ask whether this function that you get because what we get is really a function it's a rule which assigns to every Point inside the unit disk on the complex plane some context number whether this function has any symmetries and the the modular forms are defined precisely by their symmetry properties now before I explain what the Symmetry property is I want to Discuss kind of a a baby version of this kind of a simplified version of this which has to do with trigonometric functions so in the case of trigonometric

functions the trigonometric function we consider as functions on the real line so you have for example sine of x important property of trigonometric function is that it is periodic so if you add 2 pi to the argument you will get the same thing right but in fact Then it means that if you add 4 Pi you will also get the same thing if you add 6 Pi you get the same thing and so on so in fact there is a whole bunch of symmetries of this function which are labeled by integers so you could add

2 pi times natural number and the resulting function will be equal to the original for any natural number or natural number or actually any integer so that's what I mean by symmetry that you modify the argument in some precise Way but as a result the function does not change so in this case the symmetries are labeled by integers and in fact symmetries form what's called a group so we can talk about the group of symmetries and in this case the group of symmetries is a group of integers with respect to addition the word group here

means that symmetries are not disjoint that we can always take the sum or the difference of Any two symmetries and we will get a new one and this addition amount and subtraction is just the usual addition and subtraction of natural numbers or integers and the mathematics is love to call it Z which is sort of like a double line that's an addition we typically use for the group of integers uh the the cosine function has the same symmetries the cosine function has the same property that if you shift the argument by any multiple of 2

pi you get back There is no function and this is a prototype a blueprint for studying uh symmetries of various functions so now let's go back to the context of modular forms uh the difference now is that a modular form is defined not on the real line but rather it is defined on the unit disk on a complex plane however it turns out that it also enjoys this kind of symmetry property the corresponding group of symmetries is much more sophisticated it is what is called sl2z Instead of explaining what it is I will approach it

in a more geometric way so here we can encode the Symmetry property of the trigonometric functions by marking these points 2 pi n here the point zero then this will be minus 2 pi then there will be minus 4 pi and so on and then here will be 2 pi and here will be 4 pi and so on so uh the symmetries correspond to shift lifting the real line either to the right or to the left by an integral multiple of 2 pi right And we can try to represent uh the action of this group

or more precisely it could be some subgroup it's called the concurrent subgroup of this group but it's a detail so let's ignore this for now instead of trying to describe how it acts I want to produce a picture which is analogous to this picture where the Symmetry the action of the symmetries of the real line which with respect to which the telemetric functions remain unchanged is Represented only now on the unit disk and for this I will consult a very interesting book that I came across recently actually actually the purpose of this book which came

out you know nine years ago was to give a kind of an exposition of the language program to know for non mathematicians so if a lot of pretty much everything that I discussed today is discussed in more detail in this book so if you're interested in this in the Subject and that's what's one of the sources you know also various talks that I have given like the the slides that we have seen which are available on YouTube and so which probably you can link also but I want to show you this uh this picture now

because it's hard for me to draw it but I will show you in so see this it talks about the terminal as theorem and the connection to the chimurtan Yama and here is the equation that we discussed today and the Co-charismatic and generating functions and all that and so there's a picture I wanted to show you so this is the this is the unit disk on the complex plane and this sort of triangles are the analogs of the intervals that you see on the real line under the action of symmetries of the trigonometric functions this

kind of a curved kind of hyperbolic triangles are the analogues of the what matters is called fundamental domains for the Symmetry Group of modular forms or in in one of the possible examples it could be a picture a little bit different from that but very similar to that so it's kind of triangles but core of triangles the triangles with respect to the hyperbolic geometry non-euclidean geometry which one can set up on the unit disk so that's the story of modular forms okay how are we going are we getting there yeah we're almost there all right

and so now we have at least some sense Of what uh elliptic curves are what the counting problem for elliptic curves is and uh that's on one side of the language correspondence or in this particular case uh we're talking about what's called the schmurt I am away conjecture and on the other side we have modular forms and now we have a sense of what what those are to put it simply the small time away conjecture is about the existence of a one-to-one correspondence Between elliptic curves on one side and modular forms on the other side

and this correspondence satisfies this special property that the solution to the counting problem for for a given elliptic curve can be read off of the coefficients of the modular forms module form when you write it as a generating function in the variable queue when I say to you then what is the language program it is the Toolkit and the methods and the conjectures and the algorithms to join these two parts of mathematics that's right that probably didn't seem like they were joined at all it's fine that's right it's finding the hidden tunnels this thread or

Bridges or Building Bridges you know so but you're right a kind of hidden tunnel is maybe a better way to put it because I think that even though we could now prove a lot of examples of this type of Correspondence I still think it is mysterious I still think we don't understand the underlying reasons so in other words there is something beneath the surface something hidden there's some hidden we still don't understand mathematicians today don't understand which I think maybe in 100 years 200 years we will have a much clearer understanding and people would you

know do a number of our video 100 years from now and they say you know 100 years ago people didn't understand what Was the reason for this correspondence but actually it's so simple it is so obvious you know so in high school yes but now we don't see it yet but it's important so it's important to emphasize that there is this there are disconnections and that there probably is a good explanation for this but it's important to how do we how can we come across those explanations we can come across by studying them more deeply

in more detail And considering the same kind of patterns uh in other contexts as well you see so in fact the time away conjecture which we which I have used to illustrate the general ideas of the language program I was known before the language language came up with his idea so he knew that this kind of an example so in fact the language program talks about much more General things than what we have Discussed so far and I wanted to just indicate what they are so so far the correspondence that we have seen is between

elliptic curves on one side modular forms and the correspondence is such that it matches certain numbers attached to objects of both kinds on this side you have this counting problem and you get these numbers AP which is basically number of solutions mod P more precisely P minus a number of solutions whereas on this side these are the Coefficients of the module form confusion in front of Q to the power P so that's the Prototype that's a blueprint of how language program language correspondence is formulated in in general in general in the general context so what

will be the objects in general so the idea is that actually what appears on this side in general is not going to be necessarily a curve elliptic curve or it's not necessarily going to Be some equation what is relevant to the side is something that has to do with a great French mathematician if that is golua so he was this French genius who died in a duel at the age of 20 but just before he died he made these profound contributions to mathematics he came up with what we call today the gala group group is

a group of symmetries which you see we have already encountered symmetries on the side of modular forms when we talked about symmetries of Modular forms being similar to the symmetries of trigonometric functions and so on but actually symmetries also naturally appear on this side we can call this side number theory side and this side harmonic analysis side right so symmetries are actually essential on both sides and on the side of what we see here as elliptic curves or on the side of number theory in general what we get is what's called representations of the gala

groups now it will probably Take us another bunch of videos to um to explain what this means but I discussed this I discussed this here so this representation of Gala groups in chapter 8 of love and math so that these are the more General things which in the case of simultaneama reduced to elliptic curves can be connected to ellipticals so it gives a much more a kind of simplified way of describing these objects in general we may not have a specific equation which is behind this Object yet they can be defined rigorously in number Theory

so these are the objects on the left hand side so to speak of the language correspondence the number theory side and the objects on the other side on the right hand side are called automorphic functions or automorphic forms these are the generalizations of the modular forums and again the idea is that there is a one-to-one respondents between the two or sometimes could be mainly to one Because sometimes it can be depending on how you phrase it sometimes it gets it gets kind of more murky but in the clearest examples that have been understood so far

one actually gets one-to-one correspondence getting one-to-one correspondence between objects of two kinds is not so interesting ultimately because you know if you have both two countable sets so you get a one-to-one correspondence between two countable sets so what But what really makes language correspondence interesting and exciting and what makes you know allows one to use it as in applications is the fact that under this correspondence certain certain information from number Theory which has to do with Galway groups gets translated to analytic information that has to do with Automotive functions in the case of the that

we have discussed I have explained what that information Is on the side of elliptic curves these are the numbers of solutions of the cubic equation underlying this elliptic curve and on the harmonic analysis side these are the coefficients of the module form in general this data can also be expressed more complex but they can be stated you know precisely what they are and the beauty and the power of the Linux correspondence is that under this Correspondence those numbers match just like they match in the keys that we have discussed if any link is found between

any two continents of mathematics that doesn't automatically mean it's language is more specifically links between the Galway groups and the automorphic functions right so you know I want to say I want to say something that um links between different continents are it's a quite common occurrence in Mathematics and it's something very which is prized precisely because it indicates that there are some things which you don't see yet which are beneath the surface and if you look in the history of mathematics that a lot of the great discoveries were made when mathematics somehow stumbled upon this

kind of links without understanding what they are but eventually were able to find an explanation and that was always fueled progress in mathematics and so This is an idea which I like to call unification so there's a certain sense of unity in mathematics that even you know subjects which appear to be far apart in fact are not so far apart and so you know one of the simplest examples just to illustrate what I mean is the following if we just talk about numbers natural numbers then of course uh we know the uh kind of the

simplest way to introduce numbers by counting so you have one pencil two pencil three Pencils but that's not the only way let me show you another way and for that I'm going to use a probe you know my dentist always complains that I don't I don't floss enough but you see so my dance will be proud of me if if you watches this video what you can do is you can also wrap the flaws this thread on the finger you do it once you do it twice it is three times four times and so on

you see so that's another way To introduce numbers and the advantage of this is that whereas all these pencils are different so in fact when you explain to Children numbers they say but how can you all consider them as objects of the same kind they have different colors but here it's clearly something of the same kind just kind of what we call winding number how many times the you know the floss has wound up on my finger and the second Advantage is that I can do it clockwise or Counterclockwise and this way I can represent

both uh positive and negative numbers whereas I don't know what is a negative one pencil but I know what is a negative one a number in this realization I just go clockwise instead of counterclockwise you see so this is for me is basic illustration of unity of mathematics that the same concept the same idea namely in this case a very basic idea of natural numbers arises from two areas of mathematics this we Could School counting is number Theory and this is what's called topology the study of these windings and so on is you know one

of the illustrations of of the field of mathematics or continent of mathematics called topology and so ultimately I think mathematicians always find it extremely exciting and and fruitful to find those connections to find different ways and you can think of this as different ways of looking at the same thing so it's Like you know uh if I have this I can look it from above I can look from the side and so on so ultimately the myth ranks is a kind of a mysterious Beast which we can observe from different sides and to me that

those connections that we find like the ones we have talked about today are in indications that we are looking from different angles at the same object and you tell me like the langlands programs finding all these exciting Connections between representation of galawar groups and automorphic functions I don't know what either of those two things are and I don't want you to tell me because I won't understand but you're telling me that those are two continents that are important enough for language to be getting all these prizes and that's right in 2018 he received the Apple

price which is considered as a kind of equivalent of Nobel Prize for mathematics is given out by the king of Norway I was actually asked to give a public lecture about his work in Oslo during the festivities one of the things that I want to mention is that yes today we talked about these two continents number Theory and harmonic analysis and how this link between them enables us to translate seemingly inattractable problems in number Theory to much more feasible much more tractable uh you know problems in harmonic analysis but in fact that's not that's just

the Beginning of the story because that was the original setup the original a way that how language thought about this ideas at the outset right so the in the 80s these ideas of a language program propagated to geometry and so mathematicians were able to find the similar patterns but in the study of of uh things for complex numbers so we talked about ellipticals for example um in a more in the clock arithmetic the solutions modular primes but you could Also look at Solutions in complex numbers and in that case elliptical appeared to us as a

surface of a donut as a as a Taurus so it turns out that the kind of things that we observe in the language in the original classical language program can also be kind of carried over to that domain where you look at equations but now over complex numbers And it's very exciting and what is even more exciting is that physicists also observed similar patterns in the study of what they call dualities of gauge theories uh gauge theories are the is a mathematical apparatus of uh of the standard model of the best theories of interaction of

of subatomic particles that we have although in this case we are considering what's called supersymmetric versions of those theories which do not necessarily have Direct bearing on our universe but maybe have bearing on some other universes who knows in any case they are beautiful mathematical gadgets and in those mathematical gadgets we can observe similar correspondences like the one which we have discussed in other words this is just the beginning of a story there are many other things many other places in mathematics and quantum physics where similar patterns arise so perhaps in another video we can

Discuss discuss this but perhaps that's that's plentiful for today well you made it this far there's a good chance you'd like to see even more from Edward Frenkel I'm going to link to a playlist of videos he's made here on number file and a podcast he did with us as well as a link to his book love and math which is definitely worth checking out all those links everything you need will be in the video description