 So welcome and such a beautiful day to the last lecture of dawn on quantum modular forms and quantum invariance of knots So perfect. Yeah, everything else. I can do it. Perfect. Perfect. So The numbers are decreasing partly because of the glorious weather but I thank those of you who are still here for your faithful attendance for five days and for coming despite the Other conditions outside so today is the is the last of the five lectures and Again, I don't I won't use this for the lecture at all But since I have a couple of pictures they need a computer and the board here. It's quite disjoint So it's not a problem. So again, I pirated I don't know how to make these slides, but I pirate an earlier talk where somebody helped me So there will be some pictures But not for a while so the theme of today's last lectures is written there is Quantum modular forms and quantum invariance of nuts So that might suggest to some of you a connection with quantum theory and of course, that's the intention of these words So quantum modular forms is not an object which exists in the mathematical literature. I Invented the name is the title of the paper. I published four or five years ago and most of which I'll Tell here some things are new and It was an idea that that paper is essentially no theorems But it was something that I observed in various different contexts both from number theory some from That were from quantum theory, but I won't say that at least indirectly and from not theory and they had a common property and the common property was that there were objects which somehow Had a modernarity property, but very different from the usual one and it involved Some kind of perturbative thing it had the feel of the things that you see in perturbative quantum field theory where typically you go to Infinity or to some cusp of your situation. There's a parameter, which usually you call h bar So I'll follow that tradition and call that h bar, which is going to zero and you have little power series in h bar but What you also often have is a variable q q for quantum But it's also the q that we've been having all week e to the 2 pi I tell and that q if tau if you think of tau in the upper half plane as being I Times some little h bar, which is going to zero it shouldn't really be h bar It should be h because I haven't divided by 2 pi But everyone calls it h bar because there are too many agents in mathematics And there's only one h bar. So if you have this thing going down that means that Q will tend to 1 But if you think of how modular form behaves an actual modular form prints on the full motor group So a true modular form, which is not what we're going to be looking at today Then we have that such a function is either invariant under the group That would be a modular function or more generally c tau plus d to the k and sometimes there might be Maybe a root of unity Might be there sometimes and we'll have examples in a minute. So in that case as tau goes to infinity then Q goes to zero. So that's not what we're doing here We're doing the other thing and then of course f has an expansion as we've been talking about all week as a power series in Q Q is now very small and so a Zero may be zero but some term will not be zero for instance a three and then if that's the leading term Then f will have that asymptotics. So the asymptotics at infinity are completely clear It's just the leading term of this Fourier expansion But then from this thing you see that as tau goes to infinity a tau plus b over c tau plus d Goes to a over c and since every rational number can be written as a over c with a and c co-prime and every such pair Co-prime can be completed to a matrix a DB a b cd In fact tau is not only approaching zero tell might be approaching any other rational number so in a very important point about modularity is that if you Have your function in the upper half plane and you let tau when it goes to infinity, you know everything It's this expansion But as tau goes to zero you also know everything by using the modularity, but if tau tends to any other Rational point you'll also know everything because you can again move to infinity So if I draw the picture instead, I'll well I've written it already ten times and you've seen it many times if I write instead Things in the Q variable then of course tau in the upper half plane Corresponds exactly to Q less than one or more precisely Q between zero and one and absolute value This Q is not zero that would be the point at infinity So we're in the punctured disc and then what I'm saying here is that Q might tend to one that would be if alpha were Zero, but Q might also tend to minus one So Q tends to minus one that would be alphas half So here this alpha might be any rational number so somewhere we have one But alpha might be half then eat the two pie I tell if tell tends to half we tend to minus one and Similarly it might tend to I so Q tends to I this would be alphas a quarter and here you'd have somewhere You know alphas a third and so on so From a different point of view we could say that Q tends to a root of unity and let me fix a notation Which to the number of this is fairly standard. I'll write mu. I don't know why it's called mu I've no idea but already kind of a double mu and you write it here to be the set of all roots of unity And so that's the set of all complex numbers and I'll typically write them as zeta zeta in C Which of the property that zeta n is equal to one for some natural number n? So I mean sometimes I might want to fix the order But you will be the set of all of these so here it would have just been Q But here it becomes the dense set which is no longer Q It's isomorphic of course the Q mod Z the isomorphism by sending a point alpha to zeta equals e to the 2 pi I alpha Okay, so already if you have a true multireform Then it has as well as being a holomorphic function in the interior of the upper half plane It also has an asymptotic expansion at infinity which the Q series those were the two aspects that I've been emphasizing all week But if you want to combine them remember the key idea of the whole course why multireforms are so beautiful is that they have a dual Existence it's a little like mirror symmetry from one point of view There's a towel in the upper half plane so a physical point tell somewhere in the interior You can think of a point somewhere here in the interior of the disk Then that function of towel has this infinite group of symmetries Which is responsible for the depth the fact that we have an infinite non-nebulant group and for the connects with algebraic Geometry and differential equations and all of those things but at the same time we have a Fourier expansion at infinity Which is what Ramanujan love that gives us coefficients and those coefficients are numbers and offer their interesting numbers And that gives all the applications to number theory that I talked about especially the first day, but also actually every day Now if those two aspects were completely disjoint you could never put it together And so one way to combine them is to say aha We know they have an expansion infinity, but we have an expansion near every point And so we do this and now the quantum idea is why don't we now forget the interior? So I keep only the second aspect, which is the Q expansion aspect, but I keep more Well, not even the Q expansion. I just want the leading term. So there'll just be a leading asymptotics, but I keep that Asymptotics at every rational point So I say if I had a true model of form if I had a function of a complex variable in the upper half plane Which satisfied some transformation equation of this sort then it would have completely predictable Asymptotics at all rational points for the reason I explained all rational points go to infinity and infinity of this Now Imagine that you have just if you've read Alice in Wonderland the smile on the Cheshire cat The cat is gone the function in the interior is no longer there. There's no functioning word. There's no interior There's only the boundary and you don't even have the whole circle. You only have the countable set So the new function is going to be a function just on roots of unity Well, I can put capital F Some letter on roots of unity, but I want to think as I already said we have a map Q mod z by e to the 2 pi i alpha and I really want to think of q So I really want to think of this as a function on the rational numbers which will send Alpha to e to the 2 pi i alpha and then some function there, which would be kind of the limiting behavior If there is a limiting behavior So if you had that you would have something that only keeps the limiting behavior as q tends to one or the roots of unity and this is really a quantum phenomenon also in the mathematical sense of quantum For instance in the set in the theory of quantum groups You also have groups that have a q and they're called you know you q of sl2 and so on and the q often You're interested in what happens in the degeneration when q tends to 1 but also minus 1 i or any other root of unity So the idea is q will be tending to a zeta But actually the q won't be tending if we had a true model of form We would have a q in the interior which tends to a zeta Which is one of these roots of unity but in a quantum model of form We won't have the q and we won't have the interior will only have a function on roots of unity So that's the basic idea and it turns out there many nice examples So I described some in this paper which was completely expository as I said no no theorems And it's already been used by a few other authors and I don't know if we'll ever be a useful tool But at least it's appearing in other contexts So let me remind again. Let me write this equation slightly differently The usual true multidorm form Has two properties one So it's a function somewhere which happens to be H to C and it is two properties one is it's got good analytic behavior Good function theoretic behavior in this case the case we've been talking about It was simply holomorphic functions an analytic function in the strongest sense in the complex sense But you also have things like mouse waveforms and we have more multidorms which are not holomorphic But they're least real analytics, so they've they're much more than continuous or even see infinity They have local power series expansion. They're very good things and secondly f of but now I want to write it slightly differently I want to write that This is a group action, and so I want to put the c-tell plus d To the k on the other side so it's easy to check and I think I mentioned earlier in the week that if you have a fixed function and gamma the group, you know gamma So abcd is some element of tau and this is in gamma which is equal or maybe contained in SL2z If you have such a thing then this is an action I mean we already know that the action sending tau to a tau plus b over c tau plus d is a group action if you compose group elements You compose that action, but also this action on functions that sends a function f of tau to this new function Which in the literature is often called f slash gamma of tau So of course the the slash depends on the way So you put a vertical line called slash with an index k if you want to be clear So the usual this is a group action and the usual meaning of a modular form is simply that f equals f slash gamma for all gamma So these are the usual properties. Let's call them one and two So now a quantum of the reform obviously can't do both of those properties But in fact it can't have either one so a quantum of the reform one fails and two fails So as a definition that leaves a lot to be desired if I say a quantum of the reform So now it's no longer a function on h, but a function on q or it might be a partially defined function It might be on q Minus, you know finite set or something like that or a finite set multiple multiple one So it would be q mod z or something It's something like that Well, if I just say it's a function that is not required to be continuous and is also not required to be modular Then it would just be nothing it would just be any old function and there wouldn't be a definition But the point is each of these things fails, but they fail in a specific way the failure of modularity so the failure of two satisfies Property one this is a little mysterious, but I'll write in one second what I mean and the failure of property one satisfies two Kind of a dual thing So it doesn't have either property and it's easy to see that it wouldn't be reasonable So now I'll call the function of use x because x looks more real than tell which looks somehow complex So x for me is going to be a rational number and this will go to f of x and typically my function will be periodic so f of x plus one will be f of x or it might be f of x plus 24 is f of x in which case I might have Something like that. Maybe you remember the notation z to n is the standard nth root of unity in this group You and so yeah, or it could be any other say to so I'll have functions which possibly up to an a root of unity in the examples Will be translation invariant so essentially although it's a function on q it's a function on q multiple of z or q multiple of 24 z or something but if you think of that if I required this property exactly if I required that f of x is Equal to f of ax plus b over cx plus d even with the other morphe factor Then that'd be completely idiotic because if you just tell me what f of you know your favorite number is seven Then that would imply the value of seven a plus b over seven c plus d But the action of sl2z on the rational numbers is transitive So you would get every number would be completely fixed by one number So even if I had one such function it would just be fixed by one value So I would just be talking about numbers So it's completely unreasonable to have a function that has such a strong invariance property on P1 of q on the rational numbers because sl2z Promutes that all the points are the same Promutes it to a transitive thing so we can't have either of that. So what do I mean by saying that the failure? Satisfies ie said now is the definition But I'll put it's a definition in quotes because you'll see it's a little psychological You can decide how good things you want to be and call them all quantum of the forms in stricter and stricter senses It's not a completely well-defined object It's like saying a function is in a nicely behaved. It's not good. It depends on the context So it means that the following that for all gamma in my group The failure of modularity the failure of gamma invariance So if it were a modular function that would just be f of x minus f of so gamma is some You know a bcd as usual If it were just a modular function It would be the failure of invariance. It would be this difference and in general there might be you know again Cx plus d to the weight and there might be some root of unity which depends on gamma I don't want to go into details But those things that if it had been truly modular this difference would have been zero Well, even if it's not zero. This is still some function. So let's call this h gamma of x is the failure of modularity if this failure is analytic on Let's say Q mod z minus a finite set so my function I don't care about finitely many values. So I'll allow myself to throw away a few so this would be a piece white. Sorry There is no analytic on I mean analytic on arm on z minus the same find I'd said in other words Where before we just had a random function which jumps around any way it wants now I want a continuous function on the reels. So f itself has no kind of continuity requirement let alone analytic f can be completely it just jumps from one point to the other or to say it in a Technical language Q has the discrete topology and so it's continuous But it's meaningless in the discrete topology, but this difference should suddenly become analytic or as I said Maybe just see infinity or maybe just see six some given number of different or maybe just continuous or Even less you'll see examples of each of these there shortly where it's even less So it's not even continuous, but it's still nearly continuous and very very much better than F So that's the idea that the failure of modularity the difference between f of x and what f of x should be equal to if it were Multi, which is f of gamma x maybe with the automorphic factor This difference is not zero then it would be multi there but the difference at least is Holographic and you can say that differently if you work with the Distributions or something if you think of F as an element of the space of functions Multi the space of analytic functions So that's doing it the other way then that function becomes modular because modular analytic functions This is zero and so F multiple analytic functions becomes modular That's what I meant by saying that also the failure of modularity is modular just as the failure of modularity The failure of analyticity is modular just as the failure of You'll see many examples No, you don't have a function That's what I've been saying for 10 minutes Just as before z or half z you fix the k It's a property of the form you know it because you've got a modular form It has some weight just like if I have a vector space it has some dimension it depends on the example But if I have a modular form of weight 3 then f of x minus cx plus d to the minus 3 times f of gamma x Will be analytic but then if you change that to 2 it won't be so of course the weight is is well defined And you know it because well you wouldn't know it was modular if you didn't know for which way You have to check this property and to check it you certainly need to know k It's it's just a property let the dimension of a space Sometimes you know how to compute it and sometimes well you always know how to compute in fact In this case So now I wanted to give you several examples and some of them I'll have on the slides So I have three and a half examples that were more in the original paper So the first one I'll do briefly because it's not quite an example So in the paper also I called it example zero, but it's very classical So if C and D are co-prime integers then you have The dedicated sum this is a famous object in number theory If you've never seen it don't worry, I'm about to write it down And if the definition tells you very little again, don't worry It's just here for a for a minute. I'll use the classical definition with a double bracket so When I put For any lambda any real number lambda is the fractional part of lambda Which is therefore lambda minus integer part if lambda is not an integer and if lambda is an integer then it's just sorry minus a half and If lambda is an integer It's zero So this is the famous sawtooth function that looks like this and then takes on the value zero between it's completely unimportant I won't use it for anything just for completeness I'm going to write down this definition and I'll just mention where this came from This was discovered as the name says by dedicated in the same paper that I talked about his comment on a fragment of Riemann when he introduced the dedicated a function and described the transformation equation of the a function So in fact there is a true modular form buried behind this one It's the dedicated a to function or rather its logarithm Now let me it's very easy to see that you can also do this of DNC or not co-prime And this only depends on the ratio So it's actually a function of D over C and for convenience I'll put in a 12 to simplify the formula slightly so this gives me a completely well-defined function from Q to Q and If I do a graph it would just be a mess it jumps all over the place There's no kind of continuity, but it has two very nice properties Both of which are classical one is trivial Namely if you change X to X plus one nothing changes because if you change D over C to D over C plus one You change D by C, but then you change KD by a multiple of C and this function here is period one So he doesn't change anything so trivially this function is invariant So S of I wrote it's a function on Q is actually a function Q mod Z But I don't want to write that because it's only as a function on Q that I can write the other property Which is what happens when you apply the other generator vessel to see here The weight is zero to answer your question if I needed an X squared it would be weight minus two, but here it's weight zero So here the formula is very nice And I'll first write a wrong formula So the basic formula is that S of minus one over X minus S of X is X plus one over X plus three if X is positive and minus three if X is negative So three times the sign so certainly this you can graph this the original function S would just jump all around But this is a perfectly nice function because a pole at zero and a negative pole and then there's a psi correction term Which depends whether X is positive or negative. I want trying to draw it Now this actually isn't right quite true That's if this were true that would be exactly an example of what I said This would be quantum-multidiform because this difference is zero for one generator of the group a piecewise Analytic function even piecewise rational function For the other generator of the group and therefore piecewise Analytic for every element of the group, but unfortunately it isn't quite true. There's a correction term I think there's a 12. I forgot the formula Is it no, it's just one It's you mug X remembers the rational numbers it has a numerator and a denominator both to find up to sign But their product is well-defined and so usually the numerator and the denominator are very big and this correction will be extremely small But still there is a very small correction So this function is almost a continuous function on the reels this part is continuous on the positive and on the negative reels And then there's a tiny little correction term. So that's just this was more for number theorists because the dedicants say Some is a particularly famous function, but it's it's not a good example and I continue so I want to give you three real examples and the first one is Very similar to some of the things I was talking about yesterday. We have two But Ramana John remember called Eulerian or what I called Q hypergeometric. I won't write it out series and I'll write them both down this one was actually found by Ramana John himself. It wasn't published, but in the so-called lost Notebook, which as I told you was never actually lost but kind of mislead a bit I Forget the formulas right I'll just copy them because they're complicated. So sigma of Q is a sum and it's exactly the form of the ones we had yesterday Remember yesterday we had things like Q to the n squared. Well a basic one the g of Q of Rob's Ramana John was This and then one of the ones I had you had to change all the minus signs to the plus signs and square them here I don't square them so it's to the first part, but the n squared becomes n squared plus n over 2 so a triangular number Okay, so if you multiply this out. This is just some Q series. It starts in some nice way So this is a true Q series. This actually does converge Minus 2 Q to the fourth and so on and the second one which is completely analogous It's the partners you'll see was discovered many years later by Andrews and the form that's got a 2 in front for some reason And then it's minus 1 to the n Q to the n squared this time if I'm copying correctly and now it's 1 minus Q 1 minus Q cubed Up to 1 minus Q to the 2n minus 1 Okay, and so this also some expansion it starts minus 2 Q minus 2 Q squared minus 2 Q cubed plus 2 Q to the 7th Eventually the coefficient to get big of course They're all even because it is too but eventually they go rapidly to and well They don't go rapidly to infinity, but they're they're not bounded actually that was the big surprise and there's a famous and Absolutely beautiful paper by three very well-known mathematicians Andrews actually I mentioned them all already Andrews Freeman Dyson and Hickerson I Forget when maybe the late 80 something like that at the reference here And they proved that if you multiply out sigma of Q Then it turns out that just like in many of our money chance examples to make it. It's not much or anyway And it's not more moderate. It's a new kind of thing. It's related To yet another kind of multiple form called mouse, which I won't talk about But just as in the other examples I had including even the aid of function You have to multiply by power of Q to make it a good function It turns out to be Q to the 124th and then the function that they formed that they found is Here a and b are integers, but a is bigger than six times the absolute value of b and Then 12 over a don't worry if you don't know what the symbol means It's equal to 0 1 or minus 1 depending on a mod 12 and it's some standard definition. That's completely boring Maybe I skip it and I somebody wants to see it and Here it's a typical theta series. So we have a quadratic form, but it's an indefinite theta series We have Q to the a squared over 24 minus b squared rather than a square of 24 plus b squared Then it would be you wouldn't need this in that case and it would be multi there But here you see if I didn't put any restriction Then a might be zero and B might be a million and I'd have Q to the minus a million squared It would diverge but because a is bigger than 6b a squared over 24 is bigger than three halves b squared And so we still have a nice convergence So this is a perfectly well-defined function. In fact, it's exactly this function. You see that this symbol is One or minus one when a is not visible by two or three then a squared is one mod 24 And so this whole thing will be Q to the one 24 times a power series and believe me It's this power series that was their theorem. So that's an a wonderful theorem and just for comparison Let's take an actual modular form of weight one Which is sorry eight of tau cubed over eight of tau and then this is an easy formula I mean in the sense that all multiple formulas remember are easy. This is actually not that easy and the formula is Identical in all respects. It's the same exponent the same minus one to the people here It's minus 12 over a which is another standard number theoretical thing which is again zero one or minus one Depending on a mod 12, but and it's also zero when a is not prime to six But it has different values different signs and now this thing is monitor So this is not monitor and this one here is a monitor form of weight in fact one As you can see on the left, but if you're an expert, you can also see on the right So this was made sense of these equations and there was a similar one that I want right down for Sigma star and Henri Cohen, maybe I won't write down his name since we're not using it Maybe I will a bit later. He Showed that that's equivalent to saying that these are connected in a beautiful way with certain mass forms Which are hecke eigenformed, but not holomorphic. So there is something actually multiple going around But then that same Henri Cohen gave identities Well actually one of them was already due to Andrews the other one he found but he found the consequence so Cohen and Andrews Gave closed formulas that you can write alternative I won't even write the second but the second one is for Sigma star It's similar to the first, but there's an explicit completely different formula for the same q series And it's the sum minus one to the end well, I can just put Minus q to the n plus one The minus sign and then here one minus q one minus q squared up to one minus q to the end So if you just multiply out you can do it on your right by hand the first few terms if you start multiplying that out You'll find that the first few terms are indeed one plus q minus q squared plus two q q It's not a difficult identity in fact, but this and there's a similar one that Cohen found for the other So this implies that Sigma of q can be defined See before q is less than one So in the unit circle it was in the strict interior of the unit circle Because this thing makes no kind of sense when q is on the unit circle. It doesn't converge in any sense at all This also doesn't converge if q is some random point on the unit circle either the i theta, but if q Happens to have the value zeta in my notation with just some root of unity So some element of this group of all roots of unity then of course There's no problem because if q is for instance a tenth root of unity Then here you have one minus q up to one minus q to the end as soon as n is ten or bigger One of those factors is one minus q to the tenth and it's zero and so the product is there and so this whole sum is simply terminating so we have no convergence problem and in fact and Sigma of q in that case will even be a simply a polynomial with integer coefficient in q But of course every time you change q it's a different polynomial So it's a function sigma. It's not at all the polynomial So now we're beginning to see a quantum thing this new sigma doesn't make sense on arm of those e But it does make sense on q motor z Well the way I've written it on q motor z because it's a function only of q and Similarly, there's a formula in sigma star of q and there's something very very nice that comes out That when you do a sigma star in sigma star turns what the negative Sigma of one over q so q inverse again if q is a root of unity So normally both of these things make sense at roots of unity, but up to sine and reflection. They're the same function So now we can make the quantum multiform and so now the first example so example one of The quantum multiform the true quantum multiform is the following function We already saw that to have any kind of reasonable behavior You have to multiply sigma by q to the 124 so that's the function I'm going to take q to the 124th sigma of q in the sense just explained q is e to the 2 pi i x But x is no longer in the upper half plane It's now a rational number and so q is e to the 2 pi times rational It's a root of unity and because of what I just told you you can make sense of it And because of what I also just told you it's also equal to minus q to the 124th Times sigma star of one over q so we have two different formulas for it Now I can finally go to the slides and Here's what that function looks like well It's a complex valued function so f goes from q or q mod z actually now It's no longer q mod z because now we obviously have if I change f by one I don't change q, but I do change this so actually it's not quite a function on f mod z It's a function on q mod 24 z because if you just translate by one you pick up a stupid factor But that's a detail so we now have a bit forget even that periodicity. It's a function in C I can't draw a graph a function in C because I would need a three-dimensional screen, but here's the real part It's just a mess. This is a you know a few hundred points But I sure if you take a few thousand points It just gets denser and denser and if you take the imaginary part. It's just as bad So it's a complete mess, but If I'm right that this is an example of a holomorphic form then we have the following proposition So here it turns out the group is the group called gamma 0 of 2 that already mentioned and it's generated by the translation and By this element the lower translation sort of doubled So that means that if it were a true modular form Then here it's got weight one as it happens as we already saw so I have to divide by the weight So it's 1 over 2x plus 1 and if it were a true modular form This would be f of x maybe times the root of unity and turns out again say to 24 and I write h of x. This is just a definition. This is completely Empty because I'm just defining h of x, but H well, I could just show you now. This is f so pay attention That's what f looks like and now comes h Which is a combination of two values of f f of x and f of x over 2x plus 1 with some study factors And now here are its real and imaginary part the gap is simply because I'm taking rational points And I didn't have enough to get the intermediate points, but you can fill it in Mentally and so you see now it's a completely well-defined holomorphic function except that it's actually got a singularity at I Actually forget where but somewhere it has one singularity. So h is now analytic So h is now a function that extends Not just to well over you see it of course in one of the functions a pole and the poll is at half, right? The one of the things the real part I think is that way, but the imaginary part is a pole. So maybe it's so You have to subtract the point half. I don't quite remember. This is now analytic So this is the best kind of quantum of forms I said we might at ask for analytic or merely see infinity or merely see six or merely They're continuous or even less than that but here we even have analytic So this really I hope shows you so the meaning of this quantum effect is that again? I'll go up. This is what we had before we took the difference There's no modularity of any sort and there's no well There's no continuity of any sort and here the modularity is that if if we're truly modular this h Really zero and you would see the zero graph. Well, it isn't that good But this age at least is continuous and in this case even analytic so that I hope sort of brings a little bit to life How these things look? Now the second example I want to go into little less detail and I've no picture for it. I think I've one little table so Let me take this is a very pretty formula very elementary you can show this to Good high school students But it's still extremely surprising and the proof is not quite trivial and the deep reason for the theorem is actually Multi-reforms although you can't see it in the theorem at all So here's a theorem that I discovered many years ago. I'll give a very special case. It will look very strange So I'm going to fix a number and I'm going to fix five for my example So you can take any positive integer, which is zero or one mod four. I take five. It's the first interesting case So let X be any real number and then I'm going to look at the following very very strange function I'm going to look at quadratic polynomials of X. So a x squared plus B x plus C and Here a B and C are supposed to be integers. So it's a quadratic form with integer coefficients. I want a to be negative But I want the value of this form at this particular point that I'm looking at be positive Okay, so if I draw a picture of the graph of this function, here's my X And here's infinity so to say that a is naked means that infinity the probabilities turn downwards rather than upwards But at the particular point of X that I'm looking at my parabola has a positive value So this is the value that I'm summing But if I took that to the infinite I have to fix the discriminant and since I chose five I fixed the discriminant B five But if you put eight or thirteen or anything you like you will have a similar theorem So this is not very promising you should presumably think Whatever I now write you're bored because why on earth should you care about this sun? And the reason you should care is only that the answer is so surprising this sun converges and it's constant You always get two So that to me was when I found this many years ago It's quite elementary but not trivial adult to proof and then the deep reason turns out to be deeply Connect with multiple forms and if I will see a bit of it in a moment So that's that's one theorem and as I said you can put here eight or twelve or thirteen any let's say non-square because there's a slight Change any non-square which is zero or one about four and you will not get to but you'll always get a constant So let's call it alpha D. So here if I put D Then I would get some constant which will be a rational number, but even an integer But will not necessarily be To so let me just give an example so that you at least slight slightly believe me let X be zero Then I have said a is negative because I'm assuming that the value at zero which is C is positive But B squared minus 4 AC is five But this thing is positive and so it's at least four because it's divisible by four and the sum is five So B has to be plus or minus one so therefore the form ABC I won't write out the excellent is always minus one plus or minus one and One in this case so here. They're exactly two forms And so we see that a X squared plus B X plus C when when X is zero is Either one or it's again one and the sum is indeed two so indeed this sum Equals two if X is zero so I've just checked one value that it's two And I'll do one more and then you can check for yourself if you take X equals a third Then you find that there are exactly four forms that contribute first the same two we just had so minus X squared Plus X plus one minus X squared minus X minus one, but the other two I don't remember them exactly minus seven X squared no minus five X squared This is an amusing exercise. Please don't do it right now But you know when you're falling asleep tonight check that if you choose X to be a third and you require That a B and C satisfied that a over nine plus B over three plus C is positive But a is negative and the discriminates fight that it's a finite set and if at this finite set exactly four values and Oh, I forgot to write down the values anyway It's something that you can compute in your head something something when you add them all up Believe me you'll get to and if you take any other rational number It'll be a finite sum, but if you take for instance pi then this is not a finite sum It's an infinite sum, but the terms the non-zero terms are very very very rare and the few you get Converged extremely fast, and if you take ten non-zero terms you get 20 digits 2.000 So this is really a theorem It certainly doesn't look true, but it is true And it also doesn't look like it has anything to do with multiple forms, but it does have to do with multiple forms So let me continue a little with this example. So this is my second example But I haven't yet come to the quantum model of form. So Theorem, so let's call this theorem one theorem two and theorem three, but there'll be a theorem n for every n But I won't write them. So the theorem two is you do exactly the same thing all of this You still take x squared plus B x plus C, but you know a qubit Don't ask me why it doesn't work with squares There's a reason but you don't want to know then the theorem is not very surprising in this sense Or very surprising in some other sense. It's again two when you take the cubes you still get two It's completely different numbers, but the sum of their cubes happens to be the same But it's a coincidence that it's the same because here if I put again B squared minus 4ac is not 5 But D this would be some rational number actually some integer But it will not be always the same as alpha D But it happens that alpha 5 is 2 and beta 5 is also 2 so in that case they're the same So now if you're a mathematician you say well, of course, I've understood you can generalize so if I take the next power To the fifth then I will get again a constant, but you don't now you get a function phi of x Well, actually the way that I do it you get a constant Gamma D just like we had before and then another constant times phi of x if D is 5 It's just a new function 5x, but the point is that there's one phi of x which works for every D and all needs you get a differently in your combination So suddenly we have something very new Instead of getting constants we have some strange function and this function now is to find well It's to find Everyone the real numbers, but it turns out that this function is exactly The sort of the quantum effect of one of these quantum model forms I think I've a slide of a few values, but it won't Tell you much But in this case this phi if you interpret it correctly I'm not going to write it out. This is exactly the h of x that I was talking about before well h gamma where Again x goes to x plus 1 will give you 0 and x here. It's the full group So 1 over x will give you this h which is 5 So again, this is exactly the function and so this function phi I won't draw a graph because it's completely boring if you draw a graph Nobody with the naked eye can distinguish from a cosine curve. So there's no point in my showing you the graph It looks extremely smooth. This phi is periodic It's no longer constant as it was before but it's periodic and it's smooth But now it's not so phi is one of our ages and in this case phi is not only smooth It's four times continuously differentiable, but it's not see infinity. In fact, it's not even six times continuously differentiable You can say that now I don't want to say anything more about it But I'll say just one word to convince you that this is something to do with multiple forms I'll give you the formula for phi Phi of x is a certain constant, which I know I'm in a numerically so some constant some scalar factor And this is the sum n for 1 to infinity. It's periodic. So it has a cosine Furia expansion and it's even so there's a cosine expansion It is no constant term because I've put that in this part and so it's just like that But the coefficients are not just any old number. They're exactly tell to the end of 11 where The famous Ramanujan delta function, which was our first interesting model reform has the coefficients tau of n So the fact that we didn't see this thing here and we didn't see it here is not because you know The powers one and three are somehow boring. It's because we're on SL 2z and the first cost form and SL 2z occurs at weight 12 And so 12 over 2 is 6 and 6 minus 1 is 5 and that's this 5 So this is intimately related to a true model reform but this function for itself is exactly one of these defects for modularity, but it's and it's not analytic It's four times differentiable Coming from a quantum of the form. So that's my second example and you see the second example So the first the 0th example. They didn't quite work at all. So the 0th example H was not even continuous But it was very near a smooth function. The first example The H was piecewise analytic The second example H was C4 so four times differentiable, but not C infinity If that not even C6 and the third example, which we'll see is H is not even Continuous, but it is C minus 1 or C minus I have some holder, you know some weaker continuity So it has something you'll see the pictures very soon So now I want to come to my last example And that's the one that will relate to the quantum theory of knots and that's also why I call these things quantum Modern forms because they really come actually there are several other examples of quite a different nature that I'm skipping Also coming from three-dimensional topology and quantum invariance of three manifolds and of knots So I now come to the last example So the third example comes from knot theory if if K is a knot So I mind you what a knot I can't ever draw the figure eight knot that lots of people can anyway My example be the the figure eight knot Which I can't draw I meant to but I forgot to copy a picture if K is a knot that means that it's an embedding Well, I think you've all seen a picture of a knot you imagine the three sphere over that matter Just are three and then you embed a circle in it in some way But as the circle moves around it can cross itself in many places and then You get something and then eventually closes up. That's a probably really stupid knot So it's an embedding of s1 to s3 but up to deformation and the quick key question of knot theory is can we find enough Interesting invariance or calculative invariance to distinguish different knots for instance Can I just can I recognize the trivial knot? Can I tell the forgiven picture like this can be unnotted just by some computable numerical invariant computable means There's a computer program which will take such a picture and make a number and that number will be the invariant It will not depend which picture you take because of course if you look from some other point If you or move it around you can have a completely different picture in the plane Describing the same knot so in this in the theory there are many many invariance There are old ones like the Alexander polynomial and much more recent ones of which the most famous is the Jones polynomial Which is a polynomial except it's not quite a polynomial It's a Laurent polynomial the variable is again called Q for some people It's a polynomial in one of a Q to the one half or even in Q to the one quarter for me It'll just be Q and Q inverse so this is the famous Jones polynomial that Jones found in connection with von Neumann algebras And it was a Sensational piece of work. He got the Fields Medal This was a big big breakthrough many years ago when such invariance were found that really come from quantum groups Or from well, there are many points of view, but they're highly sophisticated invariance Although there is an elementary definition Inductive definition and I should put the knot somewhere else. I'll put it in the air just to remind you But usually people put JFK, but I don't like that So if you fix a knot you get a polynomial or more precisely Laurent polynomial I'm going to skip the definition it plays no role at all except that in a given example It's always calculable and in the particular example Well, if I took the figure 8 not which will be my example. It'd be some very simple boring polynomial Which won't tell us anything but now This has to do in fact with some group, you know UQ of SL2 and SL2 has the two-dimensional representation. I mean it's 2 by 2 matrices But of course it also has 3 by 3 matrices as a representation and 4 by 4 because SL2 Sends X and Y to AX plus B over CX plus D So if you have a form say cubic form Then it sends cubic forms to cubic forms and that's a four-dimensional space So for each dimension, this is a famous n-dimensional representation the n minus first symmetric power of this one And if you do something that I'm not going to explain either You get what are called colored Jones polynomials. So you have an extra invariant Kind of decoration which the physicists and the mathematicians call the color and that depends Also on an integer n and it's again a polynomial in Q and Q inverse But now n is one two or three and the original Jones polynomial is simply j2k, j1k is somehow always trivial It's just Q or something I forget but it's it contains no information about the knot So the Jones polynomial turned out to there are many many other generalizations the home flight polynomial It goes way beyond that. I don't want to talk about any of them. So we have a sequence of polynomials, but now Let me give an example So I'll write it down explicitly for the figure eight knot I can actually give the formula jn of Q is Going to be the following formula m goes from zero to n minus now In my node side little n so maybe for safety if you're taking notes called it either one you want It's the same but here you can put or infinity. You'll see why in a second. It doesn't matter what I put here So you go from zero to n minus one all further terms would be zero anyway And then you take Q to the power minus mn times the product J from one to m so m is growing when m is here. It's the empty product. So it's one and then inside the product sign You have a Q to the one minus Q to the n minus j and one minus Q to the n plus j So you'll notice that if J took on the value n Which would happen if m were n or bigger That's why I could stop if m is greater than or equal to n Then in this product you have a term J equals n But then you've Q to the zero which is one one minus one zero and so those terms all vanish So I could equally make an infinite sum and it would be just as good So now something very beautiful happens Let me instead of thinking of n as for instance a fixed number like J7 and think of this as a polynomial or the wrong polynomial Let me think of Q as a fixed Number and then I have the function which sends n to Jn of that given Q I won't write the thing each time it's the figure 8 not in my example from now on It works for all other nots conjecturally for this one. It's actually proved But it's been checked for many other nots numerically Like Garouf Alidis and myself and by the way I should apologize some of you were here for the 50th anniversary celebration of the ICTP and I use this example there too But emphasizing some of different aspects, but the pictures you're about to see were shown then too So if you were there you'll either be bored or happy to see all friends Okay, so let's turn it upside down. Let's say for instance. I take J to be I so that's a fourth root of unity. Well, then it's completely easy to see That this thing as a function of n just this period for and the reason is remember I already said I can go to infinity and Anyway, it'll be fine. That's enough. I to the minus mn I but the only place this is independent of n the only place where you see the end is in the Combination q to the n q to the n q to the n to the power m So you only see I to the n So if you change n by 4 you won't change I to the n so it's periodic So it means that if I make a table here n will be One two three four. I'll have some value here some value here some value here some value here But then it will simply repeat It'll be a periodic function and because it repeats I can extrapolate backwards and go to zero And so that now gives me a function J zero so J zero of I would be equal by definition To J four of I which is also J a divide since it's periodic any multiple of four So in general, I'll define J zero of q if q is a root of unity By definition to be J n of q where any n Such that q to the n is one Well any positive n and then it doesn't matter which end you take for the reason I just said so in that way We suddenly have made out of what were a polynomial is as continuous as you can get But now we've suddenly made a new function which normally isn't continuous It's not even defined on a thing with the topologies just defined on rational numbers are here on roots of unity So this is the function J zero. Maybe I show you Now I won't say one thing more Namely my picture was a little bit misleading. I don't remember the values of J zero Let me make the picture again. What I said here is that if I took here n One two three four, I would have some values But then I would have some other value and then it would repeat periodically and I could repeat but actually What well, of course, there's no curve. It's just values. I'm just showing you where they are But what will actually happen at least before this was the fourth root of unity force replaced by bigger number This value will be much much higher So that's what I'll show you on the next slide. So here are a few values of this function So in the first row, I show you the values of J zero of Q When Q is a root of unity of order one then it's one order two then it's minus one order three Then it's either zeta three which is here or zeta three inverse, but the function is even so it's the same It's 13 or it's plus so first I took one and here it's one then I took minus one and here It's five then I took the cupboards of unity and it's 13 then I took I and minus I and it's 27 But then when I come to fifth roots of unity you have four fifth roots of unity And they aren't just in complex conjugate pairs you have these two and they're complex conjugates So you get two different values forty six plus two squared of five and forty six minus two squared of five And when you come to the sixth root of unity there now There's only one pair zeta six and zeta six inverse then the values already 89 and you see that these values are quite big But they're only big because I'm taking this particular value J zero so in the second table It's a table, but just look at the third column. It's it's instructive enough. I've picked and be 300 And so I take zeta to be zeta 300 so remember that's the standard 300th root of unity where this angle here is very small It's 2 pi over 300 so a little more than 1 degree and if I take this Then if I look at the value of the Jones polynomial at Zeta, well, there are only 300 values Because after I go from 1 to 300 then it repeats periodically as I just told you but all of these are less in absolute value Then 25 but this one is 10 to the 45 So the picture is not at all like I drew, you know like this. It's sort of more like this and Then a peak just a n or just a zero which is much much much much bigger Which is exponentially big and this is a completely well-defined Phenomony comes from the churn Simons theory with a compact and a non-compact gauge group And I only have understand and I won't say anything even if you ask So this is the phenomenon so what you see again in this table if I take 300 or 100 or 200 Then the value of Jn of this nth root of unity for smaller n is very small. It's just polynomial here It's like 25 so it's much smaller even the 300 but J0 Which is the same as Jn is certainly 10 to the 45 So you have exponentially big growth and in fact, there's a very famous conjecture due certainly to Kashaev, maybe Kashaev and Fadiyev I'm not quite sure so it's called the volume conjecture. It's still unproved It's proved for a few knots certainly proved for this one, but only for two or three others. I think so I'll just put Kashaev I'll just put it for this not where it's not a conjecture, but I'll write it in a way which is arbitrary Well, I'll write it Okay, so I have a knot and I take this J0 of z to n so remember z to n is the standard nth root of unity Not any nth root of unity, but e to the 2 pi i over n. So this thing is actually called the Kashaev invariant and It's not called that because he had the idea of Extrapolating backwards he actually had a completely different definition He had his own invariant called the Kashaev invariant and a few couple of years after he defined it It was discovered by the brothers Murakami and Murakami that his invariant was equal to the backwards Extrapolated value that I'm calling J0 and since that's definition is easier. That's the one I've chosen So it's equal, but it's not that's not the definition, but in fact But it doesn't matter since you haven't seen that this is called the Kashaev invariant and the conjecture is that it grows In this case like a constant I'll make it first in the vaguest form and this constant will always be The volume of s3 minus k where k is the knot Which in our case will be zero point three two zero three nine. I think I can look up the actual value now I've wrote it down and of course I Don't know where I put it here zero point three two Three oh six five nine Now I'm not going to go into this, but this is Thurston theory That tells you that most hyperbolic manifolds is now completed by Perlman That most three dimensional manifolds and in particular most knot complements s3 minus a knot have a hyperbolic structure Which means they have a completely well-defined rigid Structure as a differential manifold which is curved just like a sphere, but negatively so negative curvature Normalized to be minus one and that structure is completely unique and rigid So in particular you can talk about the diameter of the thing the volume. I mean it's it's it's completely rigid It's like a block. It's no longer only up the topology It suddenly has a metric structure and in particular if it were two-dimensional in the area since it's three-dimensional volume And this constant you need is one over two pi times that volume So his conjectures that for any knot at least any hyperbolic knot you will always have well Maybe I should put the absolute value in this case. It's real So it doesn't really matter very much, but it should look like that in general So that's true for any knot, but it's not known to be true for more than three or four knots But you can check it numerically for lots of knots that it's true up to very large n But there are only a few knots where it's actually known So in this particular knot I can be much more precise with the C which I just wrote this is in general and This particular value is is for the figure eight knot and you can give this number in terms of the dialogue with the function If somebody wants to see the form I'm happy to write it. So actually we have much more here We know more I won't say all of the names because they're like seven of them But Anderson and his then student Hansen proved an asymptotic formula that this J zero zeta and in this case is Not just e to the C n plus little o of n which means something less than exponential But the it's actually polynomial growth and it's in fact end of the three halves And it's even exactly asymptotic that up to a constant and they also found the constant It's not particularly hard in this case because of the very explicit formulas like the one I showed you So they found this formula and then both kind of Garoufalides in two different unpublished papers with different collaborators and in another paper Compute of the entire asymptotic expansion. It's as I say not difficult here because there's the explicit form is quite simple It's essentially just an exercise used in the Euler-McLaurin summation formula in one of the methods So here are the coefficients So the theorem is not only it's e to the C n as the volume conjecture predicts Not only it's that times n to a power which is three halves not only it's that times a constant But it's all of that times an entire Power series not convergent but asymptotic power series in 1 over n and in fact in pi over Square to 3 times n with rational coefficients. So the first I'll just write down two terms 697 over 7776 I squared over n squared Plus etc. So that says you see here. We normally know the volume conjecture. We actually know much much more We seem to have completely I seem to have lost the thread Where is the quantum modularity and now you'll suddenly see it appear? So this was a well-known factual as I said this expansion was not known that to Garoufalides and I found with various Other people but that's an easy expansion But at one point I thought well, why don't we let zeta remember this z to n Means angle 1 over n 2 pi over n. So it's approaching 1. So I said, you know why 1? Let's look at some other root of unity. So I started with minus 1 of course later I did experiments at all kinds of other rational points and what I found experimentally now. It's proved is The following I mean all of the third things are proved if I look at minus z to n I'll first write something slightly wrong and then I'll correct it So if you're taking notes leave a little bit of space as I just did you again get it's very very similar So it's a complete bore instead of c and it's cn over 4. That's because this is a second root of unity If you had a fifth root of unity, it would be c over 25 So you have to divide by the square of the order of this number. It's always into the three halves There's always some constant which now the three to the one-quarters upstairs and downstairs There's a two to the three halves. So that's not at all surprising and again. There's an asymptotic. Oh, I forgot that I want to leave some space and Then again, there's an asymptotic expansion. It looks exactly the same in fact, it even is the same denominators Only the numerators have changed a bit. So this is no in no way interesting. I find It's exactly what you'd expect after all of this an expansion like that when you approach one Then why shouldn't there be a similar expansion when you approach some other point? But this is not what you find on the computer. What you find is this is true When n is 2 mod 4 so twice an odd number, but what you actually find is kappa n It's a constant and kappa n takes on only three values It's either 27 if n is odd It's one as I just said if n is 2 mod 4. It's a twice an odd number and it's 5 if n is 0 mod 4 So suddenly we have something new and very very surprising We have an asymptotics that even though this is a well-defined function and given by a formula You would think it is uniform asymptotics suddenly some number theory is coming in this asymptotics knows Whether my point which is now approaching minus 1 with an angle again 2 pi over n It knows whether that n is an odd number twice an odd number or four times something so if you Turn that on its head a while well first you can ask Where is this number 27 1 and 5 occur and if you have if you're observant you'll have seen aha I just saw 27 and 1 and 5 namely. They're right there 15 and 27 are the value of J 0 the original j the Kashyap invariant or j 0 evaluated at 1 minus 1 and plus or minus i So in all cases this is simply i to the power n plus 2 if n is 2 mod 4 then n plus 2 is 0 mod 4 That's j 0 of 1 so it's 1 then a 0 mod 4 its j 0 of i squared Which is minus 1 so it's 5 and if n is odd it's j 0 plus or minus i so it's 27 So that's very very nice because suddenly we're seeing modularity namely And now I'm coming to the end The grind grand climax. Let me just define. Well, it's meant to be a dark j But I'll just make the afterwards because it's too much trouble remember. I wanted to use I want sl2z acting and sl2z does not act on roots of unity But it does act on q or more could correctly q in infinity, but let's forget it So this more or less acts Remember we had a map From the roots of unity all roots of unity in in C and the map simply send a rational number Joint I called it alpha for it doesn't matter e to the 2 pi i x so then me define I'll take this extrapolated j 0 and here I'm taking the figure 8 But the same kind of story works for any other not and here I'll take e to the 2 pi i x So because x is rational, this is a root of unity and this makes sense So the above table that I that you still see there would say here's x and here's j of x and the table now If it's 0 mod 1 so an integer Then you get 1 if it's half mod 1 so denominator 2 you get 5 if it's plus or minus a third mod 1 It's an even function It could also be 7 thirds, but it's mod 1 so every number with denominator 3 mod 2 to 1 is either 1 third or 2 thirds then you will get 13 if it's plus or minus a quarter you'll get 27 if it's plus or minus a fifth You'll get what was it 46 plus 2 squared of 5 but if it's plus or minus 2 fifths You'll get something else 46 minus 2 squared of 5 And if it's plus or minus 1 6th Then you'll get 89 so the same table that I gave you now have written in terms of rational numbers and we now suddenly see a function which is You know moving around and so one can wonder now what's happening and now what I just showed you is this we trivially have That this function is translation invariant because if I change x again by the integer I don't don't change either the 2 pi ix at all So if it's already said here whether x here is a quarter minus a quarter of seven quarters a hundred and thirteen quarters of Multi-dor one and up to sign. It's the same. I'll always get 27 So it has a periodicity now it does not have it's also an even function as I've also mentioned and as you see in this table So in this particular case, it's real and even in general It wouldn't be either one that's because this particular not is its own mirror image. It's a symmetric not Okay, so that's a detail that not an important thing But the interesting one is minus one of rex the other generator of SL 2z But since it's even I can drop the minus just put j of one of rex This is somehow in some very mysterious way connected with j of x But you can't see how it works, but you can see some part of it because here Let's look what I've just written here This number that I call j zero of minus 2 pi over n is this new function j Minus one is e to the 2 pi times a half and say the n is 2 pi i times 1 over m so now I'm approaching The point a half I'm very near a half, but I'm approaching it by half plus one over n But then the answer is that this thing depends on That but now you can do an easy Check which is I hope I can Quickly find it here Otherwise, I'll just tell you that it's true Okay, I can't find it quickly if I take a half plus one over n There are three cases if n is 4k plus 2 or let's say 4k minus 2 Let's say I have a half plus 1 over 4k minus 2 but put that over common denominator It's 2k minus 1 plus 1 which is 2k so it's k over 2k minus 1 But that's equivalent if I applied the matrix 1 0 2 Minus 1 will remember I can put a minus sign because it's even so this isn't an SL 2z But in gl 2z this is equivalent to k So all numbers that are 4k minus 2 if n is Convert to 2 multiple of 4 then a half plus 1 over n is equivalent under SL 2z to k So therefore j 0 of k remember it's periodic So j 0 of k is just j 0 of 0 which is 1 but that's what my table says once you have a k It plays no role whether k is big small even or odd. It's just always one So in that case you see that there's a symbol multi-transformation which takes this argument a half plus one over n and Sends it to k, but j 0 of k is just one on the other hand the values aren't equal There's this whole asymptotic thing But still the interesting part is somehow reproducing itself. So we see some kind of shade of modularity Let me do the other case is just for completeness. So if n is 4k I Didn't prepare this but I hope it comes out then a half plus 1 over n will be Again over denominator 4k it's 2k plus 1 and now this one is not Equivalent to To but if I think of this I can't do it in my head unfortunately. I'm too dumb It should be if k goes to infinity. I'm applying 0 1 2 so to have a matrix in SL 2z I should put minus 1 and 1 minus 1 And so I should write this as some you know Y minus 1 over 2 y minus 1 and if you solve that I hope that it's correct. This would be 2k plus 2 Minus 1 over 4k plus 2. I obviously can't do this in my head. I'm not nearly good enough But anyway, I think you get something like 2k plus 1 and again j's there if this will again be sorry It shouldn't be it should be k plus a half So why here should be k plus a half Does that come out by any chance? The heck with it. I didn't prepare. I can't do it. You can do it as an exercise You'll find that this thing is equivalent to k plus a half or maybe k minus a half And so if you believe some kind of modularity then the asymptotics of j zero near this value a half plus 1 over n If there's any kind of modularity should have to do with j zero of k plus a half But again by period is lead that's just a zero of a half Which is five which is what we found and similarly in the last case if you do it You'll find that when n is odd Then your thing is always equivalent Modular SL2z to an integer plus or minus a quarter and so then you get this 27 So we can now write out what the theorem says and I'll write it now as a conjecture But it's actually a theorem I proved that in August it'll be in the joint paper with Garafalidis with whom we did many other examples So this particular case is not actually a theorem Cycle can remove the word conjecture put theorem. I better get the statement exactly right. It's complicated I wrote it down, but I think it's safer if I copy it from the paper because it's full of little details The theorem is this fix an element gamma equals abcd. I don't think I'll need the name in SL2z Okay, and then and So I'm going to call alpha is gamma of infinity. It's though It's the value of a x plus b over c x plus d at infinity. So it's a over c. I'll just call it alpha So I don't have to keep writing it then there exist Actually, you don't even have to fix out. You only have to fix this there could be different gammas for the same alpha the numbers That I'm about to write will simply depend on Alpha these will be real numbers such that Let x now go to infinity Well before I write it, let me give you a very simple case But I'm about to remember that we had the original j and its polynomial at z to n had this asymptotic expansion a Constant 3 to the minus a quarter times n to the 3 halves times e to the pi of some cn and then further terms so if I go to my j that would be 1 over n and so this would look then like x of Well, there'd be an n to the 3 halves And a constant and then it would be x of cn plus a constant Plus some constant over n and so on Okay, so we had an asymptotic expansion like that now this was if n Is equal to let's say a thousand. I mean it should be big, but it's an integer Okay, but actually the same modularity that I showed you I discovered it originally with this experiment with the half But it works perfectly well also at zero namely if you take n To be a thousand and a third A thousand and one and a third in all of which you still let n go to infinity But no longer through integers, but interest with a fixed denominator say an integer shifted by a third Then you'll get exactly the same asymptotic expansion to all orders Cn plus all coefficients are exactly the same n is not the same because it's now a thousand and a third but it will be multiplied by 13 and 13 is exactly J zero of a third So you actually see the modularity even at zero you don't have to take the minus one Even here if you change the volume conjecture and let the n no longer be an integer but an integer plus a third Then you'll have the entire same asymptotic expansion with the j zero of a third So what that tells you is that you should look at j of ax plus b over cx plus d So in this case this would have been one over x But since n is no longer an integer. I'm going to call it x just to make things Remind you that it's not necessarily an integer So I fix a matrix in sl2z or gl2z like one over x or minus one over x and then this factor j zero of a third Well, because it's periodic all of these x's have the same j zero and is that j zero of a third? So here this is just a fixed number because x is going to infinity with a fixed Fractional part and that's just j of the fractional part But to make it look multi-rided write it like that. Remember a quantum model of form I'd f of gamma x minus f of x is something nice or nicer But my function is exponentially big as you saw. So now my f will be log j And so if I took the log of this I would have you know, I mean it says j of It's the ratio So now I can write the the complete theorem And the theorem is that this is a power series expansion It's always got something with x to the three halves But it's actually better to make a slight shift. So I'll call that h bar I'm going to set h bar To be pi over the square root of three which we had before it's for this particular example over x minus gamma inverse of infinity Which is therefore minus a so it's actually one over x plus a over c So it's roughly one over x but slightly shifted. It's a detail So that's the end of the three halves we saw before and here you'll have x to the power s zero divided by h bar That's the big term plus s one plus s two times h bar and so on in these s i so these coefficients So the claim is a big refinement of the volume conjecture. You have an entire asymptotic expansion But the interesting thing is not that you have this asymptotic expansion That's just the form everybody expected But that you don't have it you have that Times a factor and that factor is j of x and that's the modularity So in the next slide, I'm nearly finished. I'll show you some of these numbers So you probably can't read them certainly the lower ones, but just that you see that they exist So here on the left my alpha, which should be a rational number It starts again zero a half a third two thirds a six five six plus or minus a quarter And the last row is the fifth roots of unity and already the the numbers become extremely complicated But these are algebraic Well, this is if you exponentiate s one it's algebraic. This is algebraic. This is algebraic There's a complete statement conjectural in this case proved in what number fields they lie So suddenly extremely sophisticated number theory is coming out and I didn't tell you with the leading term this e to the s one I mean the sort of the constant term So this one it turns out to be a number closely related to algebraic k theory and with frank calligari We have many discussions and he discovered a new invariant in algebraic k theory simply based on this There had to be an invariant which predicted those numbers And so he found a way to construct something and indeed the numbers it predicts agree with those numbers So if this well, this is Nice, but it's very very complicated But now I can make it very simple by giving you pictures and that'll be fun again just for the last two minutes Just like the pictures I showed you for example one Here's how I now take my f my quantum multidiform which is the quantum multidiform to be the log of j As I told you j remember j of one over 300. We had that Long ago. I gave you the example that j zero Of zeta 300 was approximately 10 to the 45 So that means that j of 300 of one over 300 is that number Which means that even its log is still, you know 45 times 2.3. So whatever that is 98 or something So you have to take the log otherwise. I can't draw a picture that goes to 10 to the 48 So it's of course on a logarithmic scale, but that's what I want anyway As I said because in this theorem you're taking the ratio of j values But for my quantum multiforms, I want the difference So indeed the quantum multidiform here is the log which I'll call f is I've been doing And now you see it's not as random if you remember the picture that we had Here the original picture was completely random. There was no structure at all Only when I took the difference did it become smooth But here it's not completely random. You see some clear structure Let those little Prevalence and those structures are exactly the kind of thing I was saying here It's this asymptotic expansion multiplied by various constants. So we can analyze it a little but still it's a mess That's the function itself, but now Because the group sl2z Has only two generators x plus 1 and minus 1 over x For one of them, it's trivial as I said and so the real one we care about is the ratio j of 1 over x divided by j of x or Equivalently if I take the log f of 1 over x minus 1 over x and now the picture looks like this So that's the h belonging to our F and you see it's a completely different behavior. It's not analytic. It's not even continue It's it's clearly jumping at various points in the next two slides I'll show you a close-up of the jumps, but at least it's a it's a well-defined function It's more or less monotone decrease and it's actually not quite But it's kind of a well-defined If you take that graph and take its closure You actually get a well-defined function on the real numbers Except that there are jumps finite jumps at every rational point But the but the jumps get smaller and smaller So if you have a rush if you have an irrational number, you get a completely well-defined function Whereas the original function if I go back there's absolutely no way to interpolate I'm doing one thing if I go back to this actually there are only finitely many dots below any given line So if I take x to be you know 1 over pi some irrational number Then it's your proximate by closer and closer rational to go to infinity The only limiting value you'll get will be infinity. There's no kind of a limit But as soon as I take this difference h Then suddenly at a point like 1 over pi you get a completely well-defined Then it's a suddenly h even though it's Not quite a function on r But it is at least a function on r minus q a completely well-defined real function by by closure at least experimented That's not completely proved And here. I'll just show you blow-ups of two points if I take a typical rational point here. I took the point 3 8 So there's a pointer somewhere, but maybe it doesn't matter The 3 8 is in the middle of this picture Two-thirds of the way to the right and you see a jump and you see that near the jump It's quite smooth and when I large it here's a big blow-up of the Little tiny neighborhood of the point 3 8 then you see it's actually very smooth So actually what happens is that if elf is a rational number then there's a jump So the function has a different value to the left or the right so it really jumps But It's actually c infinity on both sides So this function although it's not continuous rational numbers is c infinity It's infinitely differentiable But it's infinitely differentiable to the left and again to the right with completely different derivatives including even the value So it's not even continuous let alone continuously differential at the point But it's c infinity in the limit to both sides It is a tailor expansion Which gives the right hand and the left hand to arbitrary accuracy And finally if I take an irrational point like the golden ratio Then you don't see that now it's the opposite the function is now continuous Because as you can see here in the middle right in the middle is one over five one over the golden ratio And at that point you're we're within one thousandth of the original value It zeroes in on a well-defined value. So now it is continuous At a at an irrational point. There's no jump But the function is sort of oscillating around so you do have a limiting value But you never quite get there. So the functions, you know holder one half continuous or whatever it's called Okay, that's the last slide the last example the last lecture and the last word. So thank you Any questions? I had in mind to make in any case this question, but the last graph is even more Suggestive alias The john's polynomial has to do with churn simon as you say, right? Yes A churn simon has to do with quantum all effect at the end of the day So where you have really carve which are As plateau or jump depends on you interpret the conductivity or the other So is there any application this analyzer modular function? As far as you know to quantum all effect To the quantum all effect. I don't know anything. So let me say a few words about that question It's a nice question. There's a lot of background on this Uh, actually when I did the experiment and first discovered this which is almost 10 years ago this You know at minus a to n I was talking to two friends both of whom are mathematical physicists also very good mathematicians Werner nam who's Also been at the ictb often and sergey guckhoff who's now affiliated also with the max plunk institute partially And we were discussing this. I say they're both physicists And then a couple of years later. I wrote a big paper. It's on my website It's on the archive, of course with four people. Well, I won't write all the names because it's too many de moff de gunnels not hands lennels gunnels another friend guckhoff and myself And it's a big like a 60 or 70 page paper. It's called uncomplex turn simons theory and I mean with a complex gauge with a non-compact gauge group and there we talk a little about this kind not the modularity But just the asymptotic expansions Near so there we already found and we stated this part as a well here It's not a conjecture as I said this particular case is easy But for other knots of course such an expansion is conjectural because even the volume conjectures are known But we conjectured in carof alitas independently that you will always have an asymptotic expansion in powers of pi over n where the coefficients are algebraic numbers and we said in what field And then I discussed with guckhoff very much this modularity behavior Which I'd found partly even as I say experiments that I did when we were talking about all of this And he has been convinced for years that he can find using quantum field theory An explanation that there is an Underline sl2z symmetry of some quantum field theory But he's so far has not been able to make precise statements So he cannot prove so far as I know any part of this modularity Using even waving his hands using a quantum field theoretical model He know he thinks completely that way he uses very much path integrals But he does not have even a non rigorous argument that produces this extremely precise statement and predicts the numbers and so on We're far from that That's not quite what you asked because you told me something I didn't know I did I know about quantum Paul de mando about sure and simons. I didn't even know they're connected He of course would know he's a top physicist and I could ask him whether If his ideas ever work or if some of those with a joint paper with witton and so on If those ideas ever work and can be Made to give a quantum field theoretical approach because there are witton wrote down the famous integral for the Jones polynomial of the kashaev. Of course, it's not rigorous. It's a path integral But even using that you cannot prove this modularity yet But in our paper we describe four approaches one is with the path integral So if you could get out of that the modularity then you could ask if that's related to the quantum halt effect and presumably also Fractional quantum halt and those things should have to do with this number theory But I've never heard until your question. I've never heard so there's a strong connection that I mentioned But I'm not an expert on although I co-signed the paper with experts Between this story and churn simons. I don't know about quantum about the hall of the quantum hall effect Wait a moment for the microphone. It's easier for them and for me Are these uh rational numbers six nine seven over seven seven seven six Appeared in some other known sequences. Well this number. I hope that everybody here recognized No Everyone should know these numbers by heart. That's just six to the fifth. I mean just like that sixth grade That's nothing. This is just some normalization The interesting part of so this is essentially integers up to some renormalization, you know 36 or something that's that's nothing But this number is interesting Let me tell the short story about this number I don't know what they are But I think you asked have these numbers for instance 11 697 and then later 41 and 1 2 6 2 5 have the same numbers occurred elsewhere And the answer is yes, beautiful question in the work with the same person. I mentioned his name several times He works on the I mean he's a serious not theorist and topologist But he works on many types of quantum invariance Also of things called spin networks and quantum spin networks So spin network. I won't say what it is. He told me once it doesn't matter It again has invariance, which are complicated combinatorial sums very similar to the joints polynomial But there's no topology now. It's I'm purely combinatorial quantum combinatorial thing And he was looking at a particular one called the quantum cube And he could compute the thing and he asked whether I could find the asymptotics numerically because you have to interpolate to find these values numerically is very hard because you know, you can't take n to be a million You can take n to be 200 and so to find many values You need a very good interpolation method which I'd found that that worked very well numerically So he asked me we did the experiment together at my house And we spent a long evening trying to find these numbers they oscillated So it was very difficult and we finally got a well-defined power series And this power series involved some coefficients, which were rational and a transcendental number And then it was very amusing. He recognized the rational number the real number. He said wait, that's 0.32 9 3 the one I just erased a while ago 0.3 2 0 5 9 3 That's the constant for the Jones polynomial It was a completely different thing of this Jones polynomial It was the c that I had in in this asymptotics this c which was some you know 0.3 2 3 0 6 5 9 and I recognized the 697, but I didn't remember where I'd seen it And actually it wasn't 697 because we had the log of this. It was some other number But it was one I think 1156. I said I've seen that number. I've seen that number. Where did I see it? So in the old days you had to ask a priest or an oracle Or a psychotherapist or hypnotist if you lost something with these days, of course I went to my computer and I went to different directories and I put grep 1156 and then I found a file that contained the number 1156 And it was this coefficient. Well, it was let's say 697 right take the log And then guess what the next coefficient and the next and the next were the next ones we had found And so in this completely different expansion of a quantum invariant of something not coming from not theory Not coming from topology. No three dimensional manifolds some purely combinatorial quantum spin networks And some q series of the sort of insane and their asymptotics I could write it down It's very pretty the actual q series So this was I think I even remember it So I'll write it down that might be slightly wrong But it's something like minus 1 to the n q to the 3 n squared plus n over 2 So that would be Euler's series and then you divide by 1 minus q up to 1 minus q to the n Maybe squared or cubed or something. Let's say it's like that something like that And then you look at the asymptotics again now this is a function the upper half then it's not quantum But you look at the asymptotics as q tends to a point on the boundary And you look at the asymptotics you get power series It's quite complicated how they work and the coefficients of those power series at zero But then at minus one at every point we're the same as these And a completely mysterious connection. So the answer is yes, we have seen the numbers twice, but we still don't really know why So that's that's very puzzling. So there's no easy description of these numbers But they have at least occurred twice in mathematics and not just the first two but the whole Infinite collection of infinite sequences. They all reappear So that was a fun question. Thanks One more before your time is up Because you know once he says it's finished then only the students can remain Okay, in that sense. So thank you very much Don for great lectures. Yeah, well, thank you ICTP and Fernando squared for having me. Let's thank Don again for all the lectures