 The only lecture in this series that is specifically related to physics in the title, although many of them are in thought, including, of course, the word quantum into tomorrow's lecture, but the one lecture that specifically mentions physics should coincide with Abdul Salam's birthday, so that I didn't even remember, although I'd been told that it was today. So I changed my mind this afternoon a little of what I would do to make it, I hope, more fun and more elementary. So I will still talk, of course, about multimodular forms and a little bit about the string theory of black holes, about which I understand absolutely nothing, but one of my co-authors is in the room and can help me out if I get stuck. But I decided to begin by pirating kind of a popular lecture that I gave twice, once in France in the Bibliothèque Nationale de France, before a big audience of mixed from researchers to school students, and one sort of a repeat of the same lecture in Oberwollfach a year later. And so I have put the PDF on this computer, which I usually don't use, and it will contain also the text, not in English or in Italian, but in French or German, French and German, but it doesn't matter, you're not supposed to read it, it's a completely different lecture. Simply, the pictures are nice and they're there and I have no idea of the technology, how to transfer pictures from one thing to another. So you'll see it and if you read bits and pieces, you can practice your German and French if you get bored with my English, but it's also my German and French, so you'll be equally bored. So if you can tell me how to switch it on, then apparently I just press a button to go up and down, I hope it works. I'm very poor with this technology, I basically think mathematics should never be projected on the board because it goes too fast and I already go too fast without it. So, as I say, I'm going to begin by kind of a repeat of some of the ideas of that lecture in a completely different order, especially because later in this lecture, I try to give a general idea what multiple forms are and I've been doing that now for three days, but still I can show you one or two pictures that I could have showed you on Monday if I decided to use the machine then. So the picture, as you can read in the opening sentences if you can see them and if you know French or German, is about what I consider one of the most romantic stories in the history of mathematics. And I tell here that it's the story of a poor, unknown scientist from and highly at that time, underdeveloped country, now rather developed, namely from India, who wrote a letter to a very prominent scientist in a very developed country who responded magnanimously and very, very well, which sounds good for the West, unless you know that the poor first person who was of course Ramanujan sent off several such letters and all the other recipients threw them in the waste paper basket and only Hardy actually responded. So on the left you see from an Indian postage stamp a picture of Ramanujan, he's quite young on that picture but that's normal because he died when he was 33 so he was always rather young. The other picture shows Hardy when he was young and he lived to a much riper age, although not particularly old, but that's when he was a young man smoking a pipe and looking very deep but both of them were very deep. So the story is kind of a wonderful one as I say, very romantic one. So in the talk, maybe it's already written on the next page but I don't want you reading at the same time so I'll keep it secret for a bit. Hardy, one day out of the blue he's in Cambridge, he gets a letter from an unknown Indian clerk and well I do have to show you the letter. If I can remember just up, no down. Funny if I want to go down, I go down. Okay, it's probably illegible in that size but I can read it to you, at least you see it. Somewhere I even had a facsimile of the original but not here. It begins, dear sir, I beg to introduce myself to you as a clerk in the accounts department of the Port Trust Office at Madras on a salary of only 20 pounds per annum. I am now about 23 years of age. That was completely truthful, he was 25. I have had no university education but I've undergone the ordinary school course and then he said I've not trodden through the conventional regular course which is followed in a university course but I am striking out a new path for myself. I've made a special investigation of divergence series in general and the results I get are termed by the local mathematicians as startling. And then follow pages and pages of, I have a couple of them here both in the original handwriting which is more or less illegible and then is this only the handwriting versus the tech version? They're both more or less illegible so one is printed, one is handwritten. You aren't supposed to read them anyway just to see very dense formulas. There were 100 formulas roughly. Pages and pages of formulas and all of them looked completely wild. So Hardy looked at this thing and in the talk I say imagine yourself in this situation. You're a very, very well-known mathematician. You get a letter from somebody you've never heard of with 100 theorems, all of which look rather crazy. So normally we all professional mathematicians get letters from cranks all the time. I've proved Hermann's last theorem, I've proved the Riemann hypothesis and usually they go into the waste paper basket. But somehow Hardy kept looking and these formulas looked so strange after a while he called in his colleague Littlewood, the famous Hardy and Littlewood collaboration. At the back of his mind says C.P. Snow in his famous book, at the back of his mind getting in the way of his complete pleasure in his game he was thinking about a cricket game the Indian manuscript nagged away while theorems, theorems such as he had never seen before nor imagined. A fraud or genius? A question was forming in itself in his mind. And the question was, is it more likely to have a fraud who's capable of inventing such plausible looking theorems or a genius who can just write them down? And after an entire night studying the manuscript with Littlewood they stayed up till deep in the night studying they decided it had to be genius. Because some of the theorems, a few of them they even recognized, they knew. A few they could prove with great difficulty, these were the two of the top European mathematicians of the time. A few they couldn't prove at all but could imagine how you'd go about and could imagine proofing and many they couldn't imagine at all. But by that time they were convinced that this man was something extraordinary. And as I say they behaved very well. Hardy wrote to him, invited him to come with a scholarship to Cambridge then they sent actually another well-known person all the way to India to bring him. It was all very complicated. He had dietary restricts there were. He didn't really want to come but they brought him. He got a very fancy scholarship. He became a fellow of this Royal Society. I mean he was actually rather well treated in England. He hated England all of the years he lived there. I mean he loved working with Hardy. He hated the country, partly the food. Well everyone hates English food but his problem was of course not that. But that you couldn't get food that he was allowed to eat. At the time it was all very strongly meat oriented. And his health got bad and he actually got very depressed. This was this initial letter was in 1912. Wait no that's, I know these dates by heart but something I'd block on it. What was the date? 1913, January 1913. By the time he came to England the war broke out and he would have normally gone back in 1915 or 1916 but he couldn't go back till 1918, actually 1919 because of the war. And actually in the last year of the war he made a suicide attempt. He tried to throw himself under a streetcar and was just stopped in time by somebody. He was very, very unhappy. And when he did go back to India he felt very sick and died just over a year later in 1920. So in that first letter he sent all of these amazing theorems and actually many of them the really amazing thing from the point of view of a modern mathematician is not just that he could find them. That of course once you've accepted that somebody's an extraordinary genius well then he's an extraordinary genius but that so many of them were based on multiple forms. But he had no idea what a multiple form was. He sometimes for him, remember in each of my lectures I've emphasized this duality between the two phases of multiple forms. You have a function of a variable tau which is in the upper half plane and this function is equal maybe up to some other morphy factor but it transforms under this infant group of symmetries. And that infant group actually I can page ahead because in this popular lecture later I try to explain in words to a very general audience what multiple forms was. And I said if you look at these pictures on the left you see a parabola which you learn in school and that is a symmetry you know x squared is equal to minus x squared. So you see a two fold symmetry. Then I show a picture of a cosine curve or a sine curve and that is an infinite group of symmetries of translation by two pi and also a reflection. So this is a group Z or even the semi-direct product of Z with Z two is slightly non-abelian. But I said when you come to multiple forms you have an infinite group and since it was a non-technical lecture I didn't talk about SL2Z and so on but I showed probably illegally a picture by Escher one of the famous Escher drawings and this Devils and Angels and they're filling up the disc which is actually holomorphic, it's the disc but it's holomorphic equivalent to the plane and if you think of each angel and devil together as making a fundamental domain for some group like SL2Z this is actually exactly a picture of the way that a modular function repeats. So that's just because I never got to show you that picture before, now you've seen it but I don't want to stay with it. And also in that lecture I explained about partitions which is one of the things that made both hard and Ramanujan utterly famous because they found their famous exact formula for partitions practically exact or exact in some sense which is explained here but since I talked about partitions in detail yesterday I don't want to go back into that. So let me go back to the letter but what I was saying is this we know that there are the two phases of modular forms and that's been kind of the main theme of this lecture starting with the introduction on the first day that on the one hand it's a function with this infinite symmetry group and that's why it has deep properties. From a modern point of view it's a section of a bundle of the modular space of curves of one pointed curves of genus one but it has this infinite transformation group. But from the other point of view F becomes a function of Q just a power series and now there's no symmetry at all but the AN are interest often not necessarily but usually and very often interesting arithmetical functions and secondly identities so of the style AN equals BN in other words five two modular forms and I believe that they're equal they become well I don't have room to write trivial so I'll just write easy but there's a mechanical way to check any identity. So the amazing thing is that a normal human being can do wonderful things with modular forms because of this richness by going back and forth but only if you're Ramanujan can you live completely on this side only using the Q expansions he never used the modularity in any way and still discover very very deep relationships and identities some of which we still not really quite know how he found I can't say how he proved because he very often didn't prove things he found them and wrote down this is how it is so those hundred formulas were all true but they were not necessarily a hundred theorems we have no reason to think that he approves of all of them certainly he never published proofs of most of them but some of course he did. Later when he came to England of course Hardy was a top analyst Hardy did explain to him the basic theory of complex functions Cauchy's theorem and he learned sort of what a multiple form was but he never really felt comfortable with it and he didn't really need it because he already could do such wonderful things on the Q side but he did know very well with asymptotics so if you have this one of the things you have remember is f of tau is f of minus one over tau well up to an automorphi factor so typically that means that f of for instance i epsilon if you're very low in the half plane will be well up to a factor i epsilon to the k will be f of i over epsilon but at infinity we know exactly what this is it's roughly let's say a zero of a zero is non-zero so we have the asymptotics at zero and also the asymptotics not just if you approach zero but if you approach any rational point P over Q well Q is bad m over n then since every rational point can be brought to infinity by an element of SL2z you'll have asymptotics at every point and Roman Neutron was one of the great experts of history in using asymptotics intelligently and he certainly knew these asymptotic properties and that convinced him that certain identities were true because they were not just true at infinity but also true asymptotically which means that he was secretly using the modularity but he never used this infinite group or the notion of a group so far as I know and so on and it's a very, very interesting historical situation so that is sort of Ramanujan and Hardy in this romantic story and one of the early sets of contributors to the theory of multiforms neither one ever proved even Hardy a general theorem about multiforms but very, very many of the things that they studied more Ramanujan but both of them and he predicted the partitions that I showed you a minute ago and talked about yesterday are in fact deeply connected to the theory of multiforms and certainly Hardy did know what they were and was well aware and even had a statement which is actually wrong that you could never find their formula without using the modularity properties in fact you can do it just with asymptotic properties but you can't prove it rigorously but you can find it so that's what I wanted to say and these things explained a little bit what I've been telling you in the last two days in particular this magic principle and some applications to sums of two squares but now I want to well I might as well leave that up there for a minute but don't waste your time reading it you've heard it all I hope much more clearly presented I mean here it's just summarized a few names of special functions so now I want to come to the theme of today's lectures which are mock multiforms and this starts it's the other half of that story and we'll come to that slide in a moment it's the second half of that lecture that the title of the lecture was Ramanujan and Hardy from the first letter to the last so the first letter was January 1913 and the last letter so that was the one where Hardy that Ramanujan wrote as a completely unknown person Hardy who had never heard of him introduced himself and that's where their collaboration started then when he went back as I said after the war ended he went back to India and in 1920 he became quite ill he wrote in January so exactly seven years later and he died actually in April so this was the last letter well the last mathematical letter I think of his life and in that letter in 1920 he wrote and he was very very happy he was otherwise I said quite depressed tried to commit suicide by that time he knew he was sick I don't know if he knew that it was fatal he had I think he had tuberculosis I'm not sure but in that letter he's terribly happy and he writes that he's just made a wonderful discovery so I'll show you the letter and since it's undoubtedly illegible on the screen I'll read it to you at the same time so there it is 12th of January 1920 almost to the day in fact the other letter was the 13th of January so the next day would have been the 10th anniversary it was the 16th but practically to the day 10 years, 7 years later he writes to Hardy I am extremely sorry for not writing a single letter up to now and then there's some more that I'll leave out I discovered very interesting functions recently which I called Mock and he himself puts the Mock in quotation marks theta functions now I've told you about theta functions in these lectures and my theta functions were special functions like q to the n squared or more generally q to some quadratic form in some variable running over some lattice but for him, theta function he just met the Jacobi theta function that he had learned about already in India as explicit functions found by Jacobi and he meant functions that transform nicely and it's exactly what we would call multiple forms so in his terminology, theta functions are simply Mock multiple forms so that later, actually I introduced the word Mock multiple forms because it's a more general class that includes his examples so Mock theta functions was his great discovery I've discovered very interesting functions recently which I call Mock theta functions unlike the false theta functions of Rogers in his interesting paper I won't talk about them but they're in fact closely related these enter into mathematics as beautifully as the ordinary of theta functions in other words, he's claiming these are as beautiful as classical true multiple forms in a modern terminology they enter into the theory as beautifully I'm sending you with this letter some examples and then there are 20 examples sorry, how did I come to this, 2017? He had actually quite a few more in his so-called lost notebook which was never lost but hidden on the shelf by Waltz and rediscovered sometime later anyway, after his death so we know that actually he knew about 30 examples some of which were rediscovered before it was discovered that he had known them but he gave 17 examples of this thing but no definition and so this 17, well, 17 as you'll quickly see is four plus 10 plus three so he gave four Mock theta functions, I'll call them I don't want to write it out all the time Mock theta functions of order three and then he gave another 10 of order five and then he gave another three of order seven and this was also all very well except that there's no definition at all of what a Mock theta function is nor is there any definition at all of what order three, five or seven means and people puzzled over it for the best part of 100 years so that's also part of this very mysterious story like Fermat writing in the margin he left behind this text which was obviously there was a great deal behind it unlike Fermat's text which nobody seriously believes that he had approved but he had a beautiful discovery but here he had tumbled on something utterly wonderful he knew it was wonderful, he said that and he said they have wonderful properties but all he said in the letter was negative properties he said they're not modular forms but they're a little like them they have similar asymptotics but not the same and they don't have quite the asymptotic so it was all kind of some properties but they had properties if you simply define every function that has all of those properties he said you get a huge and meaningless class he wanted functions which had a very good property and so if you remember here the key thing this thing about identities I called it I think the magical principle I certainly did in that lecture and the magical principle remember in that case was that if you have ordinary modular forms so for ordinary modular forms the magical principle so I'm going to define it is if f and g are modular forms having first of all the same weight remember the weight is an absolutely key invariant which is the k from c to l plus d to the k but even without knowing a technical thing one can just say modular forms have certain invariants and the most crucial one is called the weight and it's an integer or half integer and if things have different weights they're certainly going to be different so if I want f and g to be the same which should be the conclusion then they should certainly have the same weight then they should have the same level so the weight is you know we have this f of a tau plus b over c tau plus d is c tau plus d to the k I'm just reminding you f of tau and the weight is this number k and this is for all a, b, c, d in some group gamma which may be contained in gamma one which is sl2z so it's either sl2z or subgroup and the level is which particular group so it might be a subgroup of index 30 and then you need a little more information than if it were on the whole group because it's got less symmetry and finally the same first few Fourier coefficients in other words f is the sum a and q to the n g is the sum b and q to the n and a n equals b n for n going from 0 to let's say 100 and the first few is the specific number it might be 2, it might be 100 it depends how big the weight is, how big the level is roughly it's the, if you know what the level is exactly if it's the product of the weight and the level divided by 12 roughly so anyway it's a specific number then I don't know if you can still see it here because of the things I'll put it here then so that's the end of that sentence then f equals g in other words to prove any identity among two arithmetic sequences that may be very interesting like numbers of representations of the numbers the sum of four squares and the sum of the divisors with some restriction of the number if you want to prove such an identity and if both sides happen by luck to be the coefficients of multiple forms then you have a mechanical way to prove it you just compute some invariance the weight, the level and the first 33 Fourier coefficients are usually the first one or two in practice and then you're done so you don't have to prove anything the theorem is a pure computer variable or hand or computer verification depending how big the numbers are so this is a wonderful principle and the question that one could ask is is there a way to make sense of Ramandachan's notion of multiple forms such that they become such a specific object these multiple forms I'll give you some of the 17 examples in a moment they still have Fourier experiments like this so they're still Q series but they no longer transform under any multiple group the right way but they're not completely off there was clearly a connection that's what Ramandachan saw and I'll show you in a minute how he knew that there was a true connection with the theory of usual multiple forms but they weren't multiple and therefore you didn't have this magic principle and so the really big question was can you make sense of this and make the whole thing come back to life and in particular I'm going to add if I replace ordinary by Mach then as we'll see in a moment there's one more invariant and I'm going to leave it blank for the moment the same something you can't see it anyway depending where you're sitting but believe me it says the same something that I haven't specified then they're equal and that something is again calculable so again we're in the good world that if you found some interesting arithmetic functions and it turns out now there are many more we already had many interesting arithmetic functions that came as cofuse of multiple forms now we have a much bigger class and it doesn't disappoint us there are many new interesting arithmetic functions like class numbers of imaginary quadratic fields that occur here but not here but not in the original world but that wouldn't help us unless we had a variant of the magical principle and that's exactly what we have so that problem was open for 82 years and during those years Watson and I'll just maybe mention three or four of the most famous names that were connected Selberg Hickerson is not such a famous mathematician but made one of the big contributions to this field Andrews is a very famous combinatorialist but many many other people there was a lot of work especially Watson wrote many papers it's the Watson of Whittaker and Watson and of the treatise on Bessel functions and he came to this puzzle you know the final he spoke of the final solution of course it was before World War II so he didn't know the final solution meaning the final enigma in the sense of Sherlock Holmes what was the meaning of Ramanujan's last letter and he gave very good partial answers and some part of the later theory is visible in Watson's work Selberg did something very interesting there were many papers in this paper where Ramanujan gave his 17 examples he gave also many formulas many identities there were no proofs and all of those were then proved during those years but for instance the last ones were proved by Hickerson I think in 1990 around then in a nearly 100 page paper in Inventiormus so you know the best part of the century later in a top journal it was very very difficult so even the identities which were completely well defined they were just formulas but they were much harder to prove than for multiple forms because you didn't have the magical principle now that we do have it as I've already hinted all of them become trivial that paper of Hickerson you can now just say theorem and then write trivial because you check that these invariants agree and you're finished but without that theory you have to do it using only combinatorial properties and it was very hard so in around 2000 I was I had a professorship in Holland part-time like the one here and I had a young student called Sanders Wehers and he was very good with the formulas and with the he liked Ramanujan so I suggested that he work on mock theta function maybe you know find some more identities like the ones Ramanujan had found and maybe some more examples there were 30 examples known I of course didn't suggest that he should try to solve a century old problem it was only supposed to be a PhD but in fact he did solve it actually I suggested two things he could work on quite separate I thought and both I said of course you can't solve them but maybe do something in that direction the other was indefinite theta series and he sort of solved the other one too at least in a huge special case and it showed that it was the same thing that the indefinite theta series gave examples and could be used to form the whole theory of mock theta functions which I hadn't even suspected so a very very impressive thesis that he wrote in 2002 in Utrecht and then he decided that he wasn't really cut out to do mathematics and quit and luckily two or three years later it was possible to talk him into changing his mind and he accepted it the math job began in Ireland and later came to Germany as now in Cologne so he came back to the world of mathematics but for some reason he thought oh I'm not sure if I'm good enough for this game you know people are strange anyway a fantastic thesis that opened up a whole new field and there are many many contributions since I'll mention some of the names maybe maybe I won't so we now know the answer to the puzzle there is an answer and I'll come to it but first I want to go back to Ramon's letter so here if you have links like us you can see some of those formulas but I'll give not all 17 of course it would take too long but I'll give a few of the formulas just to give you a feeling for how it works so let me go back now to Ramon and John's letter and the order three that he talked about remember I just erased that there were four order three functions you don't know what order three it means but nor does anyone else or these nor did anyone else until it's Vegas so he had four functions and he called them f, phi, psi and chi and to add to the confusion when he got to the order five he again called them f, phi, psi and chi and the fifth letter that I've forgotten and to make it worse there were ten but they came in pairs and he called both of them f and f rather than f1 and f2 so he and then again for order seven he used f so it's a slightly confusing notation I'm going to suppress chi because I want neither now I'll also change in his phi and I'll change the sign of q just to make the formulas better as I did in Geper-Boerbachy talk on Svekert's work several years ago and I'm copying formulas from that so let me give those three functions that will give you a feeling so maybe you remember from yesterday's lecture that I'd the example there was a pair g and h the Rogers-Ramanichan functions and they were defined by q to the n squared over 1 minus q up to 1 minus q to the n and h was a similar formula and those were modular functions true modular functions actually weight zero in this case if you multiply by suitable power you should actually put q to the minus the sixtieth here and here q to the eleven sixtieth times the power series with integral exponents those series of that kind Ramanichan called Eulerian because Euler studied them very much and I like calling everything after Euler that Euler did and not after later people but in this case nobody calls them that anymore they're called q hyper geometric series which simply means if you look at this nth term this is a sum n from zero to infinity but if you look at the nth term let's call this a n then you see that the ratio a n is the previous term and then downstairs you've just multiplied by single factor one minus q to the n and upstairs you've multiplied in this case by q to the two n minus one so this is a fixed rational function independent of n of q and q to the n I mean if n is fixed like five then rational function in q and q to the fifth is just rational function q but there's a fixed function n can vary and it's some function like here q to the n squared times one over q over one minus q to the n so that's what we call today a hyper geometric q hyper geometric function ordinary hyper geometric is the same but this would be a rational function let's say over q or c of n that would be usual hyper geometric that's the reason for this name okay so let's consider these q hyper geometric or Eulerian series and all of Ramanichan's examples were like that and that by the way is why it took so long it's a complete fluke when you have a formula like this this is q hyper geometric and it's also multiter but the two fields they're both huge fields and many many books on q hyper geometric series they have a third name called basic hyper geometric series many of the books are called that so there's a huge field of q hyper geometric series and of course a huge field of modern reforms has been convincing you but the overlap is tiny it's a sheer fluke when a q hyper geometric series is multiple and vice versa and so in the case of mock multiple forms since there was no definition people only at these examples and they were all q hyper geometrics everybody concentrated on that and what speakers did is he looked at the proofs of the identities in literature and he saw that they used one of three classes in each case people had shown that one of these q hyper geometric things could be by combinatorial tricks put into another class and that class was not q hyper geometric so he wrote three classes and he thought maybe I can figure out what one of them is but in fact he figured out what all three were each one had his special transformation law a modification of the usual symmetry of a modern form and to his surprise and my surprise too each of those three transformation laws was the same so each of the three classes of non-hybrid geometric forms that people had used to prove the identities turned out to have a common property and so that common property that was Staker's discovery to crystallize that out that's the one that I then dubbed moch multidiforms as I said theta functions to run on the John Nant multidiforms so moch multidiform is something with this mysterious property which I'll come to in a moment which turned out also very useful for physics actually many place in physics I'll only mention one or two so I'm coming back now to the examples as I said they're all going to be q hyper geometric so in the case of order three I've just erased it but I told you that I was going to have three functions and I'll just copy them from the paper because from my handwritten notes I can't read it so here we do exactly like this g of q it's exactly the same except that it isn't of course because if it were exactly the same then it would be the same function and it would be modular we make two changes first of all all the minuses become close and secondly we square the denominator okay so that's called f of q so this is all I'm doing order three and as I told you he gave four and I'm giving three of them but changing the sign in phi and psi phi of q is very similar for Ramanujan it was again q to the n squared but for me to make the formulas better it's minus n squared times one and this time it's like that but the square has descended inside so it's one plus q squared up to one plus q to the n squared and no square outside and finally psi of q again for Ramanujan the numerator had a q to the n squared but then he needed minuses later in his formulas I'm simplifying by this and this time you again put one plus q but instead of taking everything twice you take only the old ones but you go twice as far so they sort of have the feeling that they're all in the same ballpark but you certainly don't see what they have to do with each other and now as I told you Ramanujan had many identities so in this case let me write down the two identities that he had so Ramanujan claimed and that was certainly proved later I think by Watson already the order three ones the identities were not terribly hard it was the order seven that Hickerson worked so hard so if you take two times phi minus f it doesn't matter what the coefficient are but a linear combination of those two then it's going to be something very nice which I'll write in a second but if you also take f of q plus four psi of q well then it's also the same very nice thing so if I add these two functions before I write this if I add those two and divide by two I will also get this relationship so before I even write the right hand side of your second identity we already have that you have quite separate things but they satisfy identities well that should ring a bell with multiple forms we had for instance e4 squared and e8 and they were completely differently defined as q series but because the space was such a small dimension in that case dimension one they had to be equal and similarly with delta e4 cubed and e6 squared there were three functions but because the space was only two dimensional there had to be a relation well shades of the same thing here you have three functions that have nothing obvious to do with each other there's no obvious way to see that phi is f plus two psi but if it's true that there's a generalization of multiple forms some mock multiple forms which is a very specific space with this magic principle and if it happens to be two-dimensional well then there's going to be a relationship and of course Ramanujan will find it at least numerically whether he approved this again I have no idea but it's true so but it's even better not only are these two combinations equal and therefore these three functions actually span a two dimensional space that's the only relationship between them but also these two combinations which are equal by that and I'll write it exactly the way that he did so he wrote it exactly like this this is from his letter as a quotient where the numerator is the alternating sign of q to the n squared remember you take q to the n squared every square except 0 comes twice because 9 is the square of both 3 and minus 3 so here this is the sum the numerator would be the sum minus q to the n squared but it's in fact exactly what we have here anyway and the denominator as you see is an infinite product it's just like the dedicant eta function except the dedicant eta function remember that I talked about yesterday well there's q to the 124th but then it's one minus q to the n times one minus sorry one minus q one minus q squared and here it's one plus but if you simply remember that one plus x is one minus x squared over one minus x then you immediately see that if I double tau and then take the quotient I'll again get q to the one over 24 but now it's one plus q one plus q squared and so we immediately see from a modern point of view that this thing is q to the to make it multiplied I should multiply by one over 24 so this is q to the one over 24 times a modular function I think but it might be q to the minus one over so this on the right is a true modular function sorry it's not a modular function it's a modular form eta has way to half but this quotient with minus signs but the quote with plus signs it's a quotient a half minus a half to zero it is weight zero and so the denominator's weight zero the top is weight a half so this actually modular form which I always abbreviate mf of weight a half so now we see something really beautiful this was in of course Ramanujan didn't say it about it's being a modular form as I said he never really knew what a modular form was or if he knew he didn't terribly much care but this very much suggests that we have a bigger space which I'll write with a double you know mk is modular forms where maybe I don't specify the level for the moment modular forms of weight k and then there should be a notion of a bigger space of mock modular forms and these are true modular forms and we'll have a bigger space well depending on the level it might happen to be the same it might be bigger and in this case you can fantasize and of course it's exactly right that if you insert the q to the one 24 plus or minus I've probably got it wrong but if you put the same power it has to be the same power each time because we had the linear relation remember but if you put the correct power of q to the one 24th of q in front of each of these then they will belong to a three dimensional space a two-dimensional space here which will contain a one-dimensional space of modular forms that means since it's a two-dimensional space but there are three of them they have to be linearly dependent but since that two-dimensional space contains a one-dimensional space of mock modular forms given any two of them like phi and f or f and psi there will be a linear combination which is that unique modular form which here will be this one again with the power q to the one over 24 and that's exactly how it works so this example actually already tells us a little bit what's going on and it's already told you it's not like remanage understood nothing he about this that it was complete mystery and just writing down identities he did know as i said a great deal about asymptotics and he proved an asymptotic statement i'll just write it for f but he had similar statements probably for all of his funs certainly for some of them he wrote them out and he said this that if i take t going to zero and i look at f of minus e to the minus pi t so this is the q and so it's tending to minus one so in the unit circle q is going to minus one but if you think of q is e to the two pi t a tau then you see that tau would be simply one plus i t over two because then e to the two pi times a half is minus e to the two pi times i t over two is e to the minus pi t so in the upper half plane we have the point zero we have the point one here we have the point half and we're descending we're very very close to the point half so to find the asymptotics there if it were multidor would be easy you would use the modular behavior to send that point which is very near this rational point to some very high point and then you just use the q series and so if you did that this is i didn't finish the phrase it's 24 and that you see the 24 it's exactly the 24th already had that you have to take q to the minus one 24th yeah it's correct yeah this is the q to the one 24th so q to the minus one 24th f etc they're going to be the multidor forms and they have a negative power in front as it happens so okay so that means if these were multidor then you would just have an identity since they're not multidor something will happen and so if it were multidor you would simply use the modularity and you would push this thing this thing to find its asymptotics you would go from this point near a half to a point near infinity and you would get a formula and that that formula would include t to the weight minus the weight but the weight is a half so it would have a one over square root of t and then it would start with a constant which would be one except that our function doesn't start with one it starts q to the minus one 24 so it would actually start with a power pi over 24 over t because now we've inverted t using the modularity so if we had this to very high order let me say this very slowly if this were a multidor form of way to half then typically this thing on the left would have a formula like this some pure exponential in e to the pi over t t to the minus the weight which here is t to the minus a half and then the next term would be another exponential which might be for instance e to the minus 23 over 24 times pi over t because and so on so it'd be exponentially smaller than the first term well he also finds that the next term is exponentially smaller than the first term because t is very small so this is exponentially big but it's not four over the square root of t it would have to be if it were multidor instead he just writes it's four plus little o of one so he writes the difference looks like four but if it were multidor it would look like four over the square root of t or some other constant so it already doesn't look multidor and actually we know and he probably knew that actually it's a whole power series in t and that's completely different from a multidor form a multidor form the next term would be a constant but with a different exponential with the square root of t but then there would just be one term and the next would be again exponentially much smaller but here once you get to the four there are infinitely many terms which are not exponentially different they're just different by powers of t which is small but not exponentially small so it's a completely different asymptotic behavior and this he knew well he probably knew all of it but what he said is just this plus little o of one so that much he gave so he gave a very strong clue that these things were close to multidor forms in two ways first of all they have asymptotics which resembles very much the asymptotics of a multidor form there is this change of t goes to one over t but it's subtly different and secondly he showed that there was an actual connection with multidor forms because you could take two of his multidor forms actually in different ways and find a linear combination which was a true multidor form so that was the story before and maybe I won't give the other functions for the moment but I'll say something about order five and order seven so for order five it's clear already from the description he gives I told you that he gave five functions but actually the five and I even told you he only used five letters and used the same letter twice chi and he said here's another family chi we would call chi one and chi two so there's a chi one and chi two and then again an f and a phi and a psi and another Greek letter but then he showed that each vector each column of this matrix is equal to a linear combination I mean any two columns of this matrix some linear combination of them is truly multidor so again and also they're dependent again so again the full space is dimension less than five but it's now vector valued forms and there's a sub space of multiple forms and the co-dimension is one so there's really only one new one which I can call chi vector which would be chi one of tau chi two of time I might write the form this later I probably won't they're very similar again to the Hardy Ramon and John they look almost the same and similarly for the seven there he had three and here you notice these were one tuples there was no vector that's because three minus one over two is one five minus one over two is two so they're vectors of length two and seven minus one over two is three so in the third case there aren't any identities with multiple forms and so here's three functions he just calls them all f but I would number them now f1 f2 f3 and again well here no combination of this model this is just a single mock object and actually he gives no identity so in a sense he says nothing at all but nevertheless one easily checks that these are somehow related to multiple forms there are many ways to check that without understanding exactly what's happening so that was the mystery and I want to tell briefly as Vegas's resolution of it and then even more briefly we'll talk about the application in physics so so the resolution is this and I can put the mysterious word and if you can't see it I'll just say it very loud shadow so every modular form every mock modular form has the word is invented by me but the notion was from the Vegas thesis I called it the shadow if you have a mock modular form like one of these three of Ramanjan then it has associated to it a shadow and the shadow is a true modular form of weight two minus the original form so in this case these Ramanjan's examples all the way to half but there are many examples of other weights today and therefore the shadow would have had weight three halves so now what would happen and this thing is calculable if you have you cannot read it off just from the q series but that's like usual modular forms from the q series you see nothing you need a proof that your particular q series is a modular form it has to come from Eisenstein series Theta series L function elliptic curve it has to come from something of which you've proved the modularity nobody can read off the modularity but if you have a mock modular form and similarly you have some proof that it's mock modular that proof will include the shadow so the shadow is just as calculable as the weight the level the first few Fourier coefficients and well as those three things and so what you have is you have the shadow map and then the kernel if the shadow is zero or rather if it's modular the shadow is zero and if the shadow is zero it's modular so this is exact if you're not used to the level of exact same as this just says there's a shadow and the shadow of a form is zero if and only if it's true modular form and so we have a bigger space but you detect the non-modularity but the physicists would call the modular anomaly you detect that with the shadow which is not a number but itself a modular form but remember that a modular form although it's not a single number it's an infinite power series but it itself is described by finite amount of data its weight which is two minus k its level and the first few Fourier coefficients so in other words it's completely calculable and in that way as I've already said any identity like these three identities that I wrote down with the much harder ones for order seven that Hickerson became famous by proving all of them now become trivial you say well it's not quite trivial you'll have to show that Ramanjan's functions really were modular forms in the sense of the current definition but that's as Vakres did in his thesis for all all 17 cases and it's often he did it in three ways because as I told you each of these functions was in three different classes so that's not hard and once you know that then the identity is trivial you just compute by hand or by computer the weight the level the first few Fourier coefficients and the shadow and if they all agree well then the functions are equal and the proof is now clear if f and g have the same shadow then f minus g is shadow zero so it's modular but they have the same level the same weight and the same few coefficients and so I use the previous principle so now we can prove all of these identities in a marvelous way and I've already said I'll give one example actually it was the first example of a modular form ever and it's due to me but I never had any idea that it was a modular form I didn't see that but I wrote down a property which in Sveger's world turned out to be the definition of a modular form it was of course many years after Ramanjan but I wrote down the modular property and I won't give the the complete definition but I'll just say if you take the sum h of nq to the n and this is some class number it's called the Horvitz-Kroniker class number it's a very famous version it's roughly the class number of the quadratic field q of the squared of minus n and if n is the fundamental discriminant and not minus three or minus four and not three or four it actually is that otherwise it's a small modification anyway this thing I actually showed in a paper 20 years before Sveger's though that this is a modular form of weight three houses I simply wrote down a formula I had no idea that that was part of a general class and that so I wrote down a certain property and now I'll tell you what that certain property is so here's Sveger's property or here's the definition of a mock modular form as given by Sveger's work so as I said if you have a mock modular form then you have to find and this is a bit mysterious you have to write down in each case a shadow but the easiest way to do it certainly it works in the 17 cases is you just guess and once you guess then you write down what I'm going to write down and if it works then you guess right and if it didn't work you guess again some there's not a well-defined process how to find it it's unique but we don't know there's no algorithm just looking at the q series but g will be an ordinary modular form this is the shadow okay so this function is remember some sum a n q to the n and I'll simplify a little some slightly line but it's very close b n q to the n okay and now you make a new function out of g in a completely algorithmic way this is easy algorithmic and I'll just write it down this is called the non-holomorphic Eichler integral I'm not even going to write it down but you simply take the same function and remember this q to the n is e to the 2 pi i n tau but now that means it's a certain function e to the 2 pi i tau but you replace tau by n tau so now I'm going to take a slightly different function and that function roughly this is something of a lie but it's very similar to that I'll just put with some constants that don't matter at all n times y where y is the imaginary part of tau and erfc well erfc is the so-called complementary error function and it's not quite right there's a factor and it depends on the weight this is for weight three halves it might be a slightly different function but depending on the weight there's a specific function of one variable which I'll call the complementary error function because it's very close to that for those of you who don't know it that's roughly the function the integral from x to infinity e to the minus t square dt so it's the function that is the tail of the Gaussian well from the square root of x so it's roughly like e to the minus x and there's a constant in front and the result is very nice because I lied I forgot the most important thing it's e to the minus 2 pi i n t so if you didn't have this then this would diverge because tau is in the upper half plane so minus tau is in the lower half plane so e to the minus 2 pi tau is bigger than 1 in absolute value and so if you didn't have this factor this would diverge but actually what you have here you know there's some 4 pi n y I'll put in a little more and this error function in the version I want would be let's say from the square root of x to infinity e to the minus 2 square it's roughly like e to the minus x so though this diverges like e to the minus 2 pi n tau this goes to 0 like e to the plus the e to the minus 4 pi n y this is e to the plus 2 pi n y and so the whole thing converges just as quickly as q to the n just in a different way so it doesn't matter the actual formula it happens to be very nearly this this is as I said slightly oversimplified but you can find the formula you know it's written down in many places but it's completely algorithmic this is just a well-known function it's in every book on classical you know special functions like the gamma function and your computer has it so this is a completely well-defined series and now what do you do with it so far I've just said to f we associate this mysterious g which the holomorphic form g it's shadow and then g stars this non-holomorphic thing which is no longer modular but now I define the completion and this is exactly the definition the completion f hat of tau is you simply take f of tau and you add this non-holomorphic piece the non-holomorphic piece is non-holomorphic which is bad this is holomorphic which is good but this is has unknown properties but this is explicit because we started with the multi-form and I hope I've convinced you that we know multi-forms of a given weight and level kind of completely so g and therefore also g star are completely known so you had certainly non-holomorphic but completely explicit and after you make this correction this thing is modular it's not a multi-form in the sense of my definition of the last two days because it's not holomorphic but it does satisfy since I raised I'll have to write it again it satisfies on the nose the usual modularity property with the same weight as before so in other words Ramanjan was utterly right that these things are very close to multi-forms they are modular after you complete them by adding a non-holomorphic piece and this is something in physics happens all the time that you're required to do these some failure of gauge invariant somewhere forces you to add a correction term and then everything comes out right but you lose things like holomorphic so I mean let's say this is a familiar kind of an idea in mathematics and here it worked like a charm so for instance if you do that for this vector valued form I won't write it out I was going to take too long if you do it for this vector valued form of order five then although chi one and chi two actually will have level five that turns out to be more or less what Ramanjan meant but the vector valued thing will actually look very very similar to what I wrote yesterday for the rod Ramanjan I'm sure you've forgotten but I'll write the corresponding thing here that if you take if you I won't put the vector anymore I'll just put chi for chi one chi two and then complete that completed thing trivially will change by roots of unity sorry I left out the power to the minus one to the 49 simply because each of these chi ones I haven't written them down but this is q to the minus 120 times something with integer coefficients so therefore this is this part is trivial but what's not at all trivial is that you get something with the square root of tau and then some constant and then a two by two matrix is actually essentially the same as I wrote yesterday for rod to the Ramanjan sorry here minus one over tau and then again chi of tau so here the two individual functions have some level five but the whole vector value function which Ramanjan already clearly saw and you wrote them in pairs and here wrote as a triple that whole thing actually transforms under the full motor group so he had it's an absolutely beautiful discovery and he found this now now I do want to skip a little because time is running out and I want to say something about the physics story so just to say briefly first the history of it I was invited to a conference in Paris that I would have usually not gone because the title started black holes and I knew at that time exactly nothing about black holes now I know not exactly nothing but still nothing but not exactly nothing but the the title was irresistible it was called black holes and modular forms and I mean of course you couldn't keep me away with the 10-foot ball I was asked to give just an expository talking explaining multiple forms to physicists working on black hole theory and there were many wonderful lectures one was by Ashok Sen and he explained something the physicists had done which about counting certain quantum states of black holes in the string theory of black holes but there was a mystery these counting functions were related to modular forms to so-called zeal modular forms but they couldn't themselves be the coefficients of anything modular because they exhibited what was called in both physics and mathematics a wall crossing phenomenon so they depended on an auxiliary parameter some high-dimensional modular space and in that space there are co-dimensioned one sub-facial walls and when you cross them when your modulus crosses these numbers jump and of course you know modular forms are not supposed to jump they're supposed to be integers and be discrete and number theory and so that completely messed up there being modular and so there was a mystery what are these things so it took me you know 10 seconds or one second to solve the mystery not because I'm so smart but because Samus Vegas had written in his thesis exactly the same thing namely he showed that in one of the three approaches to mock modular forms which which was so called Fourier coefficients of Meromorphic Jacobi forms I don't want to go into it at all he showed that you've exactly the same phenomenon there's a wall crossing and what's left after you allow for it by adding this correction term suddenly no longer jumps when you cross the wall it's a well-defined power series and therefore there's coefficients that count something but that one is no longer modular it's mock modular so I went up to to send me it left the lecture I introduced myself we hadn't met before of course he's a well-known ictp person he's on the scientific council here in an old district and I said I know the solution I can't yet show it but it'll be easy it'll take a week it's in the thesis of my student that's not yet well known even in mathematics or in physics that will be the solution these things are going to be mock modular forms so he said great why don't you work with with my friend at each double car who's sitting there and is now like me permanently affiliated with the ictp but unlike me properly and not in a quantum state so he's actually here and quite recently and I hope you're happy but I'm very happy about it and we started working and after a few days it was clear that this was not going to be done in in one week and then he said do you mind if I bring I for bright postdoc if I remember correctly Samir Murti who became a very close friend of mine he's now in England and he's even more associated with the ictp he was in postdoc here he was here for four years and not only learned fluent Italian he's the only foreigner I know I mean I'm sure there are many but the only one I know who learned Tristino dialect which is kind of amazing he learned it because he gave tango lessons and when you give tango lessons you have to talk to the you know you go to the bar afterwards and talk and of course the students were not physicists and mathematicians they were the Estinian they spoke dialect so he speaks I don't know how well but anyway he knows Tristino dialect which impressed me no end so it's a completely Trist kind of a collaboration and the two weeks that I estimate it turned out to be four years which made both my collaborators furious with me but they didn't know you know what what slow meant until they met me anyway we we wrote a huge paper 150 pages that's going to come out very soon as a I hope as a book in the Cambridge University press and we worked out this connection using what's Vegas had done but applying to the physical model and I obviously can't go into details for time and for understanding and for probably your understanding since it's mostly mathematicians here and even if you're physicists not necessarily the string theorists of black holes because it's a fairly specialized domain but I'll say a few words about it maybe I read something from the abstract or actually there was an earlier abstract that says it even better in an earlier version of the paper no longer the one on the archive I mean one that is not not the one on the archive I'll read a sentence it's meant to be mysterious to you it certainly was to me although I think now I know what all the words mean we show that the generating function that's a word you've seen of quantum degeneracies of single centered black holes in n equals four string theories is that's the generating function is a mock model of form that was the one set in summary of the whole 150 page paper and then it went on the failure of modularity is governed by the shadow which itself is a true modular form so and then it went on to explain a little so let me just say a few words well first I have to also read a very well-known quote that we put it's kind of a half dedication in our paper it was a quote from Freeman Dyson at the Ramana John Centenary in 1987 since he was born in 1887 and he wrote and we felt we had done at least part of that and actually Atish wrote to me got a nice answer that thank you I'll read it someday when I'm have a few extra years my dream said Freeman Dyson very famous physicist mathematical physicist my dream is that I will live to see the day when our young physicists struggling to bring the predictions of super string theory into correspondence with the facts of nature something we'd all like to see happen someday will be led to enlarge their analytic machinery to include not only theta functions but mock theta functions but before this can happen the purely mathematical exploration of the mock modular form so he actually invented the word and their mock symmetries must be carried a great deal further and that's exactly what's done by this because by since this is a true modular form by integrating the symmetry of this the modularity behavior you find out exactly the failure of modularity here and since this is actually modular it has no failure that's the same as the failure of modularity here so what this implies is exactly what Freeman Dyson said that the original F with the formula complete F itself is of course not modular because then it would be an ordinary modular form but it's holomorphic so the difference the failure of modularity is holomorphic and this is explicit and of course it's determined completely by G so it's a completely different form not the one i wrote not non holomorphic a different thing in some integral or some sum but there's a completely explicit function that you write down once you know the shadow G and that tells you the failure of modularity and if G is zero well then it was modular and there's no failure so it's Freeman Dyson with a fantastic person still by the way going strong i met him for the first time a few months ago he was 90 gave a fantastic lecture he's full of beans and full of ideas and very impressive and his wife's who's presumably much the same age still runs marathons which i can't even think about anyway so he predicted or he said it would be wonderful if someday this could enter the machinery of string theory and to do that you would need to first understand the mock symmetries of mock modular forms and that's what has happened so let me just say basically i can say nothing i think about the physics background i wrote down a few notes in case i said a few words i mean i also took the opening page of the paper with me so that i could look at formulas lovely i told you that there's a modular space and that for any point in that modular space you get a power series which counts something what they call the quantum degeneracies which we would call the dimensions of the eigen spaces of certain operators and so that modular space well first i should say what string theory is it it's a type two string theory based string theory as i mentioned earlier that you have a three-dimensional Calabi all times sir minkowski force base so this is the string theory where x three in this product exits itself a product of the k3 surface which i've also mentioned and the two torus and there's a famous duality and you could also do this would be a type two string theory you can also do a so-called heterotic string theory on something else which turns out to be t4 times t2 or simply t6 and then some wonderful duality that's 25 years old by now and also very mysterious certainly to me tells you that you get the same numbers and everything works nicely it's again some kind of i think some kind of mirror story or i don't know what kind of a story some kind of a brain story probably so this is where we're living and so t2 well t2 as i said already in earlier lectures it's c modular lattice that corresponds to h modular sl2z so you have a modular space but if you go to the universal covering the discrete group that's acting as sl2z but the other one k3 k3 is there a lot of k3s and it's a 20-dimensional space but actually here it's even a bit bigger it's a so-called 22 6 so it's an orthogonal group of size 28 so 28 by 28 matrices with that signature over z and roughly of this group or not i don't know if it is in fact the product acting and so if you think you don't have to know any of the words that i'm saying what this acts on well sl2z acts on two by one vectors of length two of integers and 28 by 28 integral matrices act on vectors of length 28 so you've all together 56 quantum numbers and the first row of this those 56 corresponds somehow to the in one version in one frame to the electric quantum numbers the other to the magnetic so this is the i think the s numbers and the t numbers there's no point in trying to say all the right words since i don't know what they mean you won't know what they mean even if i get them right unless you do know what they mean and then you'll know i get them wrong so it's called the s models and the t models then we have a whole bunch of quantum numbers so now we're down to something very simple we just have to study some integers that depend on twin on 56 numbers plus maybe some more stuff so that looks pretty hopeless but then it turns out that because of this group duality the thing that depends on this makes this independent is invariant under both sl2z and the other and that comes down to saying that if you take this matrix y which is 28 by 2 and you take y by y transpose that's now just 2 by 2 and what's more it's symmetric so suddenly you have only three coefficients to worry about so now you have a function that depends on three parameters i'm being terribly rough and that would remind you if you're a number theorist of zeal modular forms and it had been found by various physicists there i don't want to mention because i'll leave out names then in this case the counting function that you need is actually a zeal modular form so it depends on three variables at tau a tau prime and a z and a z but it's the it's a famous zeal modular form of weight 10 i haven't defined any of this one of the famous zeal forms that may be the most famous it's the exact analog in one of the one variable function delta of tau that igusa discovered i don't know 40 years ago very famous function but it's in the denominator and because it's in the denominator it has zeros and then when they're in the denominator the new function is poles but usual poles you can go around them but in higher dimension the pole gives you a wall and you you cross it and and and also if you do some contour integral to computer Fourier coefficient then when you cross this pole well you pick up a residue and so things jump so somehow it's because this function is the reciprocal of a holomorphic multiform here is zeal form that you get the wall crossing and the final physical interpretation of the whole thing again i'm going to skip all of all details so these three numbers a b and c are some quantum numbers of the states if you imagine two if you take the simpler one one over delta of tau that would correspond to much simpler multiforms which would just be a one by one matrix whose coefficients would be the coefficients of one over delta that one has been much studied and that would correspond to some simple kind of black holes in some model but if you had a diagonal matrix a zero zero c but that would correspond let's say to two of these simple things but very far apart so there's somehow in the theory too but the ones we care about are somehow bigger there are twice as big but they only have a single center so they're they're sort of single centered black holes don't ask me again too much to explain but artisans right there and he's happy to answer all questions so we put two little dots in a circle meaning they're sitting there together and these things move around but they're parameterized by point in multiple space when that thing in multiple space crosses a wall these are called the walls of marginal stability then it can sort of degenerate it can throw off little single centered black holes let's say it turns into a different one and so it's kind of messed up but among the single centered one there are also single centered ones which are twice as big but they're those are what they call wonderful terminology immortal immortal black holes don't die because they're immortal when they cross a wall they simply cross the wall but there are also these horrible ones which are kind of a coalescence of two single ones and when they cross these walls they can fall apart and split up and so they mess up your counting so the question for the physicists was really can we take this whole counting theory that they had already computed which were the coefficients of this but which jump around and then split off split it up into two pieces one of which doesn't jump and will count the immortal things which are immortal and so they're well defined numbers and then the other part is something explicit of the jumps were completely well known we knew how much it jumped by just since there are jumps it can't be a nice multiform so if you're going to count for the jumps and it turned out it works on the nose the original thing this maramorphic zeagle form when you take its Fourier expansion in tau prime gives you maramorphic Jacobi forms when you take their Fourier coefficients according to Sander's vacas you split in its two pieces one of which is a mock multiform and the other is a completely explicit thing called an upper large sum but you write it down it's known and so you split the whole thing into two pieces this each coefficient that you want to know is one coefficient plus another and the second coefficient has the jumps and it's completely explicit it's an elementary a little complicated but elementary sum and the other doesn't jump that's counting immortal black holes and so we went through the whole theory it took as I said four years to work out all of the wrinkles because you need a very interesting family of mock multiforms it turned out that this immortal part was exactly a mock multiform and therefore the abstracts I read to you we show that the generating function of the quantum degeneracies of single sentence immortal black holes in type 2 you know n equals 4 super swing theory is a mock multiform with way to half now just and I've still six more minutes with a few uh well I won't go beyond six even though we started late because of the announcements let me just say a few words about a related development that I would have mentioned anyway but somebody asked about the tea two days ago it's been a little bit in the papers recently and you may have even heard so I would have probably wanted to mention a few words anyway so very very briefly many years ago John McKay noticed so John McKay was is both a number theorist and a group theorist and so like every number theorist he knew the j function and knew had seen and recognized this coefficient which only number theorists kind of would have immediately spotted but he also was a group theorist and so he knew that uh I'll just put m o the monster group which is a group of order roughly 10 to the 70 or something like that or 10 to 59 I've forgotten some huge monstrous number this group has approximately 170 representations so as many as it comes class it's of that order less than 200 but it's a huge group the order is 10 to the 60 or something and the first one of course the dimension is one that's true for every group of the smallest here to use but that's the trivial representation but the second one is one nine six eight eight three and so everybody in group theory many people like Conway many others knew that number by heart many number theorists knew this one by heart and who came in both camps he recognized he said that can't be a coincidence he showed it to lots of people in particular Conway and a whole theory blew up and this was originally called moonshine meaning just blah blah blah because nobody could prove anything it was all guesswork and looked a little crazy and since it was connected with the monster it was called monstrous moonshine and then in the course of the succeeding years there were many refinements I could tell the story but not in this time and it doesn't I decided not to what it fit this series beautifully but it's very old this is from the late 70s I think I don't remember exactly then there were various developments first Atkin, Fung and Smith that is sort of the numerical check showing that the predictions were at least compatible with modular behavior and then Frankel, Leposky and Marmon constructed some infinite dimensional module of something which had the sort of the right properties and finally Borcherts completely wrote down the thing which showed in an intrinsic way why this connection and it goes way way further this was just the beginning that started the story why the whole story is true and he got the Fields Medal for that so I'll just put Borcherts as the sort of final solver of that and that's a very very famous chapter in the theory of in mathematics of recent years which involved modular forms and groups very thoroughly so then four years ago but I guess five now because it's 2015 but in 2010 there was another absolutely beautiful discovery by three Japanese mathematicians and they found that if you take so again a very so the monster group is the biggest of the 26 sporadic groups and it I think it even contains all the others as subgroups but there's another one a few of those I think six or seven of these sporadic groups were in the 19th century or very early 20th century and the biggest one of those is called the Mathieu group well actually there are three of them 10 no 11 12 and 24 it's 23 and 24 maybe there are four of them it was the biggest one if you took this then if you looked at its representations you found some numbers which I don't remember exactly I think those were the first three and they notice that these are equally it was just an observation first complete mystery more moonshine coefficients again of something multiderm but it was a big surprise because here this is a multiderm function so it's an ordinary multiderm form rational but of I mean with not allomorphic because it's way but of weight zero but these things were the form a very specific one which had already been studied by various people actually even I hadn't as everybody had studied this particular one it again it weighed just like all of Ramanujan's examples it weighed a half it wasn't I think it was one of his last notebook anyway it was set in example but the point was that this was neither weight zero but had weight half nor was it multiderm but multiderm and this time it was even better you weren't even off by one and here the next dimension you have to take a combination the first I think three coefficients of this multiderm form were all on the list of your useful representations so it was clearly not a coincidence and so then we had a new word in mathematics or rather in physics to my great amazement I mentioned in many lectures and to many friends nobody in the pure math world heard of it everybody in physics knew and I'd gone to several conferences where it was talked about but they were always physics conferences by people like you know Gabbard deal and and the mathematics somehow didn't make a splash at the beginning now it has so this was called Mathieu Munchain it was this wonderful discovery and then three mathematicians maybe I'll skip their first name so I mean I know them all personally Miranda Cheng John Duncan and Jeff Harvey all mathematical physicists but all very good mathematicians as indeed these three are too they discovered that actually there's a generalization of that Mathieu thing namely and I'll just say it very quickly if you think about lattices in particular even unipodular lattices then it's very well known it's been known for a long time and uses multiple forms to prove that it the dimension the rank of such a lattice must be multiple of eight for eight there's exactly one for rank 16 they're exactly two but well okay there are two rank 24 they're exactly 24 and rank 32 they're at least 80 million so these 24 of course are very famous they were completely classified by Niemeyer one of them is the famous Leeds lattice the other 23 are not the famous Leeds lattice so these are called the Niemeyer lattices and what they discovered was that there's a moonshine story attached to each one of these 24 and one of them I or maybe of 23 I'm not sure the exact how it works and each one was very similar to what the three Japanese names I apologize to them I just erased and so this was called because it was everything was mock and I had the shadow and shadow in Latin of course is umbra so they call this umbral moonshine so now we had and they formulated very very precise conjectures for all 23 or 24 cases which in the case in one of the cases the group was the mature group and in the other case they were smaller subgroups of m24 and in each case but also in the usual months case there were many subgroups there was a similar story so they did this and they've done a great deal of work since two huge papers of I don't know 200 pages and then as I understand it but I'm not an expert I haven't seen any of the papers a few months ago Terry Ganon proved a big part of the umbral moonshine conjecture but only the numerical part the analog of what I mentioned here that Atkin and his two students Fong and Smith did which is to check the compatibility with all the modularity maybe a little more so he did that for the original case of the three Japanese of the mature moonshine and then about a month ago I think uh John Duncan who was one of the three I won't put initials anymore one of the three authors of the conjecture the same John Duncan a graduate student uh and uh and his supervisor Olner who's a mathematician who's an expert on mock monitor forms uh they well they've announced in a big way to the press I mean Olner in particular that they've solved the full moon umbral moonshine conjecture but as far as I can understand they've done something in between they've done the analog of what Atkin and Fong and Smith did in the old case which is sort of this numerical part and it generalizes what Ganon did so it includes that case but he had done the that case first we just in some sense the the central case on the other hand their proof is uniform and apparently very elegant I haven't seen it yet I'm sure it's a very good proof but I don't know to what extent it's true that the full umbral moonshine conjects are true or not true but I want to at least mention this conjecture and I'll just say that since there are roughly 23 cases you get roughly 23 very special mock monitor forms all with more or less the same shadow I mean up to a constant so they all differ or linear cognizant differ by ordinary multiple forms but they're all different and so you get a bunch of very specific forms of low level and in the work with the Atisha Dabholkar and Samir Murti we also had very specific multiple forms in several certain cases that we had in the last chapter of the book where things worked out very well in terms of this physics model and then the two lists are almost identical our list of special multiple forms and the one coming up in the umbral moonshine conjectures which are now at least at that level theorems but those two lists are almost the same the divergencies were very mysterious two years I think has now completely understood them so there is seriously a connection between this these models in string theory and the models on the group theoretical and multiple form side so that's the story I wanted to tell you today so thank you very much. Questions for Don? Atisha is sitting there giggling he can hardly hold his sides for for laughing at the way I described his work but you know it's that's how it is you know it's what you get by working with people who don't know what they're doing but he's very forgiving and he's being nice about it but as I said if you want to know what it's actually about don't read the paper for heaven's sake read the book later it's going to be beautifully written he and Samira writing an introduction that will make all of quantum field theory all of string theory and all of the classical and string theory black holes crystal clear to any mathematical reader and I'm looking forward very much to that final version but the current version of the archive if you can read it you're a better man than any of the authors so or woman if as the case may be I don't think any of us can read the whole paper and actually know what each set and says but presumably at least one of us knows what each set and says so it's it's not an easy read but but if you want to know the truth just ask a dish above all don't ask me any questions any further questions that was sort of my question but come on somebody students can ask afterwards but if you've got a phd this is your only chance out of the order that's very nice well there's a rough answer and a precise answer the question is what after all the dust cleared what was this order of a multi-multi-form when Ramanujan said three of order three ten of order five and three of order seven there were various partial answers vaker said it's more or less this so more or less is easy I told you that for instance for the ones of level five he had ten Ramanujan I mean but he himself gave identities showing that it was actually a five by two matrix they were five groups of two and each vector different from a multiple of each other vector by usual multi-form so it was essentially just one but it was vector valued and each of those forms separately I told you that the whole vector transforms under all of sl2z but if you take the subgroup which preserves up to constant at least each term that roughly is level five it roughly has gamma five but actually it's got a little bit more so very very roughly the order is the level except that is fakers noticed uh five and seven were correctly named three was actually misnamed two of them have level three but one is level two and you've applied the heck you operate it there's some technical thing it's actually not quite level three but roughly it's the level which I've told you is just the subgroup but I mentioned today the word jugobi forms I didn't talk but it's not that hard I could have but I didn't I just mentioned that you go from a zeal multi-form like this one of igosa a zeal multi-form is a function of three variables tau z and tau prime put together a systematic two by two matrix and then if you Fourier develop in tau prime you get a function of two variables one in h and one in c and that's called the jugobi form and then if you Fourier develop in in z then you get a function just of tau those are multi-forms except that if this is a meromorphic instead of holomorphic then that was some just big discovery you get a mock multi-form so somehow we have a jugobi form and it turns out actually in the theory of the three authors have been mentioning we developed a theory of something called mock jugobi forms which sort of their coefficients are mock multi-forms and a mock jugobi form a jugobi form doesn't have a level or rather it does but ours all with level one so they were on the full multi-group except that because there are two variables as well as the multi-group and tau you have the translations of the elliptic curve in z and so you actually have the jugobi group so this is gamma gamma j the jugobi group which is an extension by c2 but it's still got level one so all of these things have level one but a jugobi form has two indices one is called the weight and the other is called the index and it's not the subgroup it's the other morphi factor there's a form as well as the c tau plus d to the k there's a second function in the transformation law which is e to the 2 pi i m times something and that m is called the index and so the actual answer turned out to be that the order in all of the case except one of the weight three cases is one sixth of the index so in other words I told you we got these special families two of the most beautiful special families had indexed they all had weighed one half but one of them at index 30 and the other at index 42 and that was these were on the nose ramanujan's chi one and chi two and the ones for 42 the cofists were on the nose it's a triple it's a vector his three functions of order seven so turn out the precise answer was completely unexpected the order is a sixth of the index of the associated mok jukobi form something he could not have guessed but anyway thanks for the question it's kind of a fun answer that you know there's six of course all of the forms of sixes and 12s and 24s everywhere I think we all need a drink