 Great. Okay. While we fix the last technological problems, welcome everybody for these new initiatives. And we are very, very grateful to Professor Don Zagir for accepting, in fact, even somehow proposing, but I mean, accepting our invitation to inaugurate a new kind of events for the math section at ICTP. It's a special series of lectures for all mathematicians. And the aim is just to have the best possible intellectual mathematical fun. So I'm sure I'm interpreting Don Spirit by telling everybody that we are free to ask questions and make comments and discuss at the end of the series of lectures. And it's also very nice, the idea to have a little bit of thrilling because we don't know how long the series will be because we will start from very classical and interdisciplinary stuff up to very recent, in fact, ongoing in the real sense, mathematics. So we will see during these 10 days if Don will be able to prove the last theorem. So thanks to Don a lot and enjoy everybody. And now? Now it's better. But you can also come closer than I can see you because the lecture room is very big and the number of people is not so big, so you don't have to sit where you can't hear. So as Claudia just told, this is maybe kind of new initiative, new kind of series for the ICTP that I don't know, but it's certainly new for me. I've been working at the ICTP for two and a half months a year for the last five years during which I was full time in the Max Planck Institute in Bonn and one of the directors. But I retired this year in July and so I'm going to be more connected with Trieste. I will have joined some sort of a double contract with the ICTP and CISA and actually I signed it one hour ago in Claudia's office the contract with the ICTP. And so I decided one should give something like an inaugural lecture. So it's not formally an inaugural lecture but that's the intention except that it's four lectures because it's such a pretty subject that I thought I'll give several lectures and we'll see how many people come. Also there's a theorem I would like to prove that's why I started this whole thing two months ago and the proof is not finished but I hope it will be by Friday in a week. So there might be something new next Friday and there might not and even if I don't succeed I still might give four lectures because there's plenty to tell even without that. So it's a huge subject or rather it's not a huge subject it's all subjects of pure mathematics interact in various ways and I don't think there's any major field that doesn't have something to do with what I'm telling. So there's an infinite number of choices. So the title of the series I think this got lost in the poster the Rogers Ramanujan Identities and the and I'll put this on the next board the Icosahedron. So to give you a feeling for how wide-ranging this subject is I found a very very nice article that I have here that I can also recommend by John Bayes just two years ago it's called from the Icosahedron to E8 and that will be also one of the themes I talk about possibly not today and it starts with the following words which form a perfect introduction so I'll just read. The first sentence you should all remember this is a beautiful statement about mathematics in mathematics every sufficiently beautiful object is connected to all others that's a very nice thought then it goes on many exciting adventures of various levels of difficulties can be had by following these connections take for example the Icosahedron that is the regular Icosahedron one of the five platonic solids starting from this it is just a hop skip and a jump to the E8 lattice a wonderful pattern of points in eight dimensions then he goes on to say that this story ties together many other remarkable entities and he lists maybe I'll list them from his list so we already had as everything connected everything the regular Icosahedron so one of the platonic solids the E8 lattice which as I said I will talk about but possibly not today but then he mentions the golden ratio I think everybody has seen this even in school and I might as well fix once and for all the notation I will always use phi will be the golden ratio one plus the square to five over two which is supposed to be the most aesthetically pleasing a rectangle that's just the right proportion and many famous buildings already from Greek times were made in these proportions because it's supposed to be the nicest and of course it has the property that if you move a square from a golden rectangle then you have a new golden rectangle and you can keep doing that okay so he lists the golden ratio then the quaternions but these are the ones listed by John Bayes and then I'll list a few more that I'll have the Quintic equation the solution of the Quintic equation so it's a famous theorem that you cannot solve the Quintic equation by radicals but you can solve it using other things that was done by Klein and he wrote a famous book called lectures on the icosahedron and the Quintic equation then highly symmetrical four-dimensional shape called the 600 cell I won't write all those words then he mentions a manifold called the Poincare homology three sphere I might talk about that I might not I love it I've done many things with it and then he mentions actually some more these are the ones he says he'll discuss in the paper and then he mentions other I'll just read them the McKay correspondence quiver representations octonians exceptionally groups maybe I'll mention octonians just and as I say many others these are the topics he lists that connect up with this but his interest basis in this it's quite a short article nine pages it's not about multiple forms I'm interested in much more so the topics we will have in no particular order are the representation theory of finite groups but in particular of the alternating group on five elements which is a simple group of order 60 automorphisms of the project of plane but not over C which I'll write P2 of C but here over the field of five elements then hypergeometric series which goes back there are usually called Gauss hypergeometric series I've never known why they were invented by Euler and Gauss begins his paper by saying he's studying Euler series the hypergeometric series then something that sounds very similar but is actually in practice very different called Q hypergeometric series I'm taking my list because I otherwise I won't possibly remember the whole thing singularity theory of surface singularities I might not talk about it well the road is from a neutral I don't have to say because they're in the title the Dedekind Eta function which is I mean I'll talk about it more of course but just for now if Q is less than one you take the infinite product one minus Q to the end you multiply by Q to the 124th for reasons that become obvious when you start studying it and that's called the Dedekind Eta function then the theory of modular forms the theory of elliptic curves and in particular if it will come at the end baby complex multiplication which goes with those things then mirror symmetry that's the thing I hope to get to well we'll definitely get to in the last part of the series but whether I can prove the actual statement that I want to prove then the famous sequence of up a re numbers that up a re used to prove the irrationality of zeta of 2 now up a re of 40 years ago by now prove the irrationality of zeta of 2 and zeta 3 with the same method zeta 2 didn't make him famous because it did note it been known for a hundred years but it's a completely different proof and the same proof word for zeta 3 and that did make him famous because it was a very old open question but the particular connection I'm talking about is actually for the up a re numbers for zeta 2 then multiple forms which I've already mentioned but there's a relatively new theory due in particular to a book yama who's here in the audience today professor a book yama that's the theory of multiple forms a fractional weight and even this is only a sample of the things that I hope to at least touch on during these four lectures so it's very wide ranging I'm going to start at a low level I don't want to assume any knowledge about any of these topics and as Claudia already said you should absolutely feel free to interrupt we specially chose as the time 2 30 till 4 so I started five minutes late since there were a couple of late comers the lectures meant to be an hour but I usually go at least five minutes over time just by myself but it's meant to allow time for questions afterwards not just at the end of the series but the end of each lecture but that doesn't mean you have to wait till the end please and it's really a serious request please ask especially there are a few younger people diploma students but also more senior people might well not know a particular thing in a particular field don't hesitate to ask I'll try to explain I mean it's meant to be a fun and it's meant to be informative so I'm going to start with the world is from nutrition identities without any real explanation yet of where the functions that that occur there come from sort of deep down because because I don't we'll come back to that when I talk about multiple forms so the rose revolution identities are called and I wrote this in the abstract by many people it's not just me but I kind of agree but it's that the most beautiful pair of it's two identities but they go together you can't really discuss one without the other in mathematics that doesn't mean it's the most beautiful statement most beautiful theorem because most mathematical theorems the statements are not formulas and determinants this theorem is very beautiful but it's not well it's kind of a formula I guess a square plus B squared equal C squared but without the drawing it doesn't tell you much as a formula but this is really identities so it's a non-trivial identity between two expressions and as I said there are two of them and I'll use the classical notation I defined two infinite series so Q will be a formal variable but it's it means that you can think that it's infinitesimal but actually everything I write is corrective Q is a complex number of absolute value less than once it's just in the unit disk closed unit disk so g of Q is defined as an infinite sum 1 minus Q in the denominator 1 minus Q squared up to 1 minus Q to the n of course I wrote at least three but as usually in mathematics if n is 2 these aren't that you just stop here and if n is 1 you stop here and if n is 0 you stop in before it's just the product 1 but in the numerator you put Q to the n squared and the other one is almost the same the denominator is exactly the same I won't write the 1 minus Q squared again but here it's Q to the power n squared plus n so maybe I'll give up on the idea of keeping that board for the icos-eat when I prefer to have long equations so let's first look how this looks so it starts with 1 if n is 0 then it's Q over 1 minus Q squared then it's Q to the fourth it's the squares over 1 minus Q times 1 minus Q squared etc but since Q is small I can develop each of these in a geometric series so 1 plus sorry this Q over 1 minus Q Q over 1 minus Q is the infinite geometric series Q plus Q squared plus Q cubed plus Q to the fourth plus Q to the fifth plus dot dot dot but then I have to correct because then I'll have Q to the fourth plus Q to the fifth so these two coefficients will become two's whereas here if I multiply this one out I love Q to this Q squared and here Q to the sixth so here if I multiply it out there's there's a gap there's no Q but there's a Q squared Q cubed Q to the fourth Q to the fifth but then when I get to Q to the sixth there's an extra one because this would have again been an infinite geometric series 1111 when I get to the Q to the sixth there's a two so the coefficients of course will grow these are well they are in fact modular functions but I'll talk about that a bit later so now if you multiply this out as a product any power series that starts with one you can write it's a product of one plus Q to the n to some power as n varies or one minus Q to the n you have a choice here we want to take one minus Q to the n so one minus Q to the n starts one minus n Q but this thing starts one plus Q so the first n should be minus one so it's in the denominator now I see that that already matches one over one minus Q already matches this up to here so it's that multiplied by news series that starts one plus Q to the fourth but that's the same as one minus Q to the fourth plus higher terms so if you keep going and that's this was done by the way Rogers is I think 1897 and Brahman Nugent forgotten maybe 1915 he didn't know that Rogers had discovered it he discovered it independently published it then he was informed he met Rogers and they wrote another paper I think in 1918 or so a bit later in which they gave they had each given completely different proofs they gave a third proof together so it's really a joint work and what they discovered what you can discover if you just have the idea of looking at this and just multiply it out then if you begin you find that very surprisingly when you multiply this out as product one minus as I said one minus Q to the n to some power depending on n then that power is always either zero it doesn't occur like two and three and five and seven and eight or it's minus one it occurs exactly once so a general series would have one minus Q to some power one minus Q squared some power and similarly when you do this when you find exactly one minus Q squared one minus Q cubed so you see the one minus Q squared that would give Q to the fourth so then I need one minus Q cubed and that would already give together a I can't do it in my head but anyway believe me that it works sorry this is a cubed and the next one is one minus Q to the seventh one minus Q to the eighth and so on so if you just multiply this out this is what you find and now you cannot possibly fail to notice that this is very specialist I said first of all the fact that there are only terms of the denominator and always put the exponent one or zero it either occurs or it doesn't so it means that we have only terms one for one minus Q to the n and similarly here we have only terms one for one minus Q to the n and it's supposed to be here n is congruent well it's okay it's congruent to plus or minus one modulo five and this is the thing that makes it so startling that you don't see any five at the beginning but the final thing involves five and if you know the Legendre symbol that remind you the n over five if n is an integer is defined as plus one minus one or zero this is if n is congruent to a square but different from zero modulo five it's for any prime p so i can put p if it's not a square modulo p you put minus one if it's zero modulo p you put zero and then in the case of n over five one can work out exactly what this is I mean it's very easy it's one if n is congruent to plus or minus one mod five minus one if n is congruent to plus or minus two mod five and zero if n is congruent to zero mod five and that's five residues that's every residue modulo five so i can write this more intelligently as the product n over five is one one over one minus q to the n and i can write this more intelligently as n equals five one over one minus q to the n you can ask where's the identity because i now have if i start here one two three four five six and six more of 12 equalities on the board plus all combinations making hundreds so the actual world's revolution identity i hope there's colored chalk i was hoping for red i don't see red but purple will do that it's called g is just the name but because i have to refer to it later the identity is of course that that thing defined by the hypergeometrics here is i just multiplied it out to sort of show you you know how you find it experimentally the identity says that this one has this product expansion and that the other one has this product expansion okay so those are the two world's revolution identities and one can just enjoy them i hope at that level you do something rather simple looking one plus q over one minus q q to the two squared over one minus q one minus q squared and then by miracle it's an infinite product in which the prime five suddenly occurs so as a consequence of that there's a formula ah now i don't know i didn't write it down myself in my notes and i've probably lost it uh the formula that well now so first i want to talk about a formula that is you should write this i can now remove this list of topics and also the name john baez so there's a further thing connected with this just at the level of formal identities which is absolutely beautiful and so this is i forgot to put it on my list continued fractions but you know that for instance the golden ratio which whose definition i just erased remember i told you that it has the property that if you take a rectangle whose sides are in the ratio of the golden ratio it has the property that if you move a square from it that what is left is a new golden rectangle but that means when you move a square from that i can't draw it because then this isn't very square because i'm drawing on the board this is a new square and there's a new golden rectangle i can make a new square and a new square and a new square and a new square and so what this tells you is that this number is one plus so here's one here's phi but this is one this is phi minus one but phi minus one to one is the same as one over phi so it's the same number again so you get an infinite continued fraction and of course this has been known for a very long time maybe since the year 1000 or so in in india but uh but ramanujan found an absolutely amazing a generalization actually it's a little better if i take the reciprocal so i if i took the reciprocal of this one over that of course i'd get one over five which is well square to five minus one over two and so the what he does is let's replace the one by q but let's replace the next one by q squared the next by q cubed and so on and so what he discovered is that this thing is simply h of q over g of q it's the ratio and so if i'll put that in parenthesis that gives the identity very very surprising identity that this infinite continued fraction as i said any power series starting with one so this of course starts i i'm not going to try to work it out of my head it starts with one so you can expand this one minus two to the end of the something but here the exponents are simply the le genre symbol so maybe if one had to choose one the most beautiful single identity rather than pair of identities then rather continued fraction is not a very bad choice and you see that if you take well you don't exactly see what it has to do with the other one because here i'm not really allowed to let q equal to one in fact right if i try to make you see that there's a five and here you see that this is a five and this is a five and you see that this specializes to that if q is one but the problem is that the infinite series i write would completely diverge and this product doesn't make any kind of sense every term is zero so you're multiplying zero divided by zero divided by zero times zero it's complete nonsense but in the limit it's true and the fact that both things are true we started when q was very small here for all of these expansions as expanding in power series and also in this formula i'm expanding in power series so q is either infinitesimal or or simply small it doesn't have to be very small but it's less than one so again i'm in the open unit disk but this connection already tells you that in q equals one where the series doesn't work something is happening and the fact so i'll just say the fact that this happens that you get some limiting behavior has to do with the basic fact about these functions is that g of q and h of q are up to up to some tweaking so you have to multiply by power of q and change variables from q to tl but they are basically what are called modder functions which is not quite modder forms but a special case of modder forms modder forms of weight zero or quotients of mod forms all of this i'll define today later in today's lecture i'll remind the beginning of the theory of modder forms so but there's much more than this g of h and h g and h as a pair are in fact the two characters associated now i'll use words that you some of you know very well better than me some of you don't know it doesn't matter i i don't think i'll get to this in the lectures at all but it's the two five minimal model so in conformal field theory you study in particular things called vertex operator algebras some of them are particularly nice those are the rational there's some other conditions rational vertex operator algebra some of those are called minimal models and they're indexed by two integers and the first really interesting one in some sense is two five and every one of these vertex operator algebras of these nice ones has finitely many modules whatever that is over them each one has some associated power series here there are exactly two and those two power series are g of q and h of q and these formulas come from what's called the fuck representation in quantum field theory so there's a kind of a clear and the product comes from another interpretation in quantum field theory so one can actually give proofs of these identities using conformal field theory and therefore quantum field theory implicit i won't go into that but i want to have mentioned it so that's a very beautiful thing also g and h so the vector space spanned by this is the is the exact solutions i mean it's the full solution space of a certain second order linear differential operator and that linear differential operator is what's called the modular linear differential operator there's a whole theory of them now so i think i'm also working on with carthers and they're working with many others it's a popular subject and this particular linear differential operator although i didn't know that till two years ago is a special case of a differential equation so they solve differential equation it's the one called the kaneko sakeer differential equation which kaneko and i another japanese colleague had found 25 years ago in a completely different context but there's a parameter in our context the parameter at certain values and this is the same equation so just as a preview of things to come although i won't talk about probably the modular linear differential equation i surely will not talk about vertex operator algebras but just that these things are not isolated curiosities so when you see this you first just think let's prove this combinatorially so let's just count here if you think what this means you have you're counting a square of some size and then something like a young a young diagram well it's it's not quite a young diagram but it's similar you have a certain number of uh one a certain number of times a certain number of twos a certain number of threes up to a certain number of n's and when you're counting diagrams there's more intelligent way of writing of where it is in fact shaped like a young diagram uh you're counting young diagrams of a certain shape there's a square and a trapeze and then the other is something very similar but the square stops one earlier and then there's a purely combinatorial proof of these but it's very complicated actually there are several by now this this identity probably has 50 different proofs some by famous people like sure so as i said you can start just combinatorially forget that q is a complex variable just think of this as an identity and try to understand why these two series are those two products but if you do that you'll never really understand what's going on and to actually understand you have to at least learn the basics what is a multireform and in particular multireform relates different values of the parameter and that's why in this case we can relate the value q equals one where this identity in fact becomes meaningless but nevertheless we can talk about it and uh so the the specialization i want to say is this uh if you take so this is this is from another survey paper that i'll also quote several things from by Oliver Nash and he reminds the following if you have there's the following form of Ramanujan i hope you all know the story of Ramanujan or Ramanujan is how he became known in western mathematics he wrote a letter uh to Hardy but actually as we now know he wrote to three or four famous mathematicians in England and the other people at the very least didn't answer and probably didn't read it just threw it away the letter had and of course it was hand written so many times uh Hardy also he describes it or C.P. Snow in his book describes and Hardy again in his mathematicianology describes his reactions when he first got this letter it was a very long letter 20 pages with something like a hundred formulas and a few of the formulas Hardy could recognize could guess where they come from he was a top mathematician but even they weren't obvious and many others he said maybe if I spend you know a month I could prove that if it's true and there were many that he just had no ideas he'd never seen anything like it so he said after now he so he called his friend Littlewood they had a famous collaboration Littlewood came over to his apartment they spent the evening or half the night looking and at the end they decided this had to be the question is this this guy just a crazy you get lots of letters as a mathematician by crazy peoples with lots of formulas that make no sense is it crazier is it a genius it could only be one of the other and at the end they decided it had to be a genius because nobody could be original enough to invent these formulas if they weren't true and in fact they were all true except for one or two little mistakes but one of the ones that as Nash reminds especially called Hardy's eyes and I remember reading that in the mathematicians apology I don't know if you've all read it it's a book every mathematician should know because it first was beautifully written and it tells something of the kind of emotional side although Hardy was anything but emotional of being a mathematician and he apologized for being a mathematician because he felt it wasn't a very useful thing for the world and he was very proud that nothing he had ever invented would ever have any use but of course that isn't true because nobody's that lucky but anyway so he wrote the famous book where he tells the story and there he tells in particular that he was something that called his eye as completely crazy was this continued fraction you see it's a special case of that except for this factor where q is e to the minus two pi and the value of this that's the one I meant to have written in my notes and forgot to transcribe the value of this so this is the formula that was given so I'll put double exclamation mark this is the formula that uh Ramanujan wrote in his letter to attract Hardy's to get his attention and it certainly worked hard he was completely blown over so if I use the notation I've been using phi is the golden ratio this is phi and this is the square root of phi times the square root of five okay so that's an amazing formula and that's a direct consequence of the theory of multiple forms plus the theory of complex multiplication that I mentioned before so this is not meant to be an introduction yet to the theory of multiple forms that will come at the end of today if I get there and otherwise it will be the beginning of the Friday lecture but I'm just giving you formulas this continued fraction uh is equal to that product that's an amazing form as I said maybe the most beautiful single identity you can imagine and this is a special case if you know something of the theory of multiple forms okay I haven't really said anything yet but are there any questions I mean I'm just it's so far just been a survey so now I want to sorry yes oh no no no no no no when I say this is no of course not when I say this is the most beautiful form in mathematics I mean that it's somehow unique so the the thing we're talking about it's e8 it's the icosahedron and these are all unique objects e8 is the biggest dignity diagram the icosahedron is the is the last there are only five platonic solids well only three up to duality all of these are completely unique formulas nothing generalizes so it you can if you just ask is this thing always modular then in fact the answer is yes if you take any prime instead of five not seven not 11 the two examples you chose wouldn't work but if you took 13 it would work so I mean that's so that I can certainly say if p is the prime which is congruent to one mod four remember that I had two ways of writing that with the lejeonism but also plus or minus and plus or minus so if you put plus or minus it should be the same that means that you want the lejeonism to be an even function that's only when p is one mod four then I could take n over five is congruent to n over p is congruent to one and it will still be a modular form so the fact that I mentioned as the reason that these things have good properties that's because there are multiple forms and that property is always true but that's not the broadest ramanujan identity there will never be another identity like this so my friend Werner Naum was a physicist and and was studied this in the connection of vertex operator algebras invented a generalization of the that are now called num sums of this which are higher dimensional but in the one dimensional case so this would be the one dimensional num sums but it's really interesting for the higher dimensional ones I've spoken about them in this auditorium several times in the past but today I'll just mention them if you take a n squared plus b n plus c somebody's called me well they shouldn't I forgot to turn off my phone well it doesn't matter it's not so loud so if you take any form like that so as I said the right hands your question is is there an identity of this form in general of course not then it wouldn't be a special thing like e8 or the e8 lattice or the icosahedron all of these unique objects what is general is that this is always a moderate form what is not general is that something of this shape so Naum looked at this generalization and he asked when is it multinar and he conjectured and a student of his proved part of it a few years later proved the rest so it's proved this is if and only if abc belongs to an explicit list and there are exactly seven seven triples and only two of them have a product expansion so this kind of identity will never work again so at least this kind of maybe you can think of something new but nobody ever has and it's been a hundred years so no the whole point of such identities is that they're very very special but what is true and actually I don't really need that n is prime that's only if I take the genre symbol if I take this definition then if I take the product n plus or minus a modulo p actually p doesn't even have to be prime this function well if I'm very correct I should put a power of q this is always a moderate form but I don't want to continue on this subject now because I haven't get told and maybe not everyone knows what moderate forms are but whatever they are there their functions beautiful a beautiful class of functions with an infinite group of symmetries which is responsible for this kind of identity so beautiful identities come and these products for any number p it doesn't need not be prime in any a the product one minus q to the n that's q to jacobi it's part of the jacobi triple product identity so that thing is always modular but these q hyper geometric things it's absolutely incredibly rare so here there are infinitely many these are not even integers these are rational numbers a b and c are in q I didn't tell you about the c you could say it doesn't matter it's just a constant but if I'm very honest so maybe I should be very honest uh you have to actually multiply by fractional power of q so later I'll tell in detail what moderate forms are and you'll see why but for people who do know the definition it's actually q to the minus a 60th g of q and so that would mean if I put q to the minus 60th here then the quadratic form here would be n squared minus 1 over 60 and here it'd be n squared plus 11 over 60 and so there's really an a and a b and a c and you need all three changing c will destroy modularity even with three parameters and not even integers well this was already rational in the whole world q cube which is rather rather large infinity there are only seven cases so it's an utterly rare and unique thing okay so that is I wanted to introduce the two themes of my series today so one was the Roger Ramanuchan identities I've introduced I've told you that the reason it's interesting is because it connects up to the theory of moderate forms I've prepared an easy introduction to the theory of moderate forms but at the speed that things take I see that I won't get that that will be for Friday and so for today I want to tell you about the icosahedron so at this point I can okay now I have a problem which is that the icosahedron is a poly hydrant you should draw it I can't draw it all so normal people lecturing in this room use the screen and then of course they either make a picture themselves or there are 5 000 on the internet and I could draw but I don't like first of all I don't know how to do lectures with with screens I always get lost and when I listen to one I usually can't follow because things disappear before you've read them and so I essentially never give them so I'm going to be in the difficult situation of having to describe some geometry and draw some pictures and I'm an absolutely lousy artist so you just pretend that you're watching a three-year-old child trying to draw you know a horse crossing the street and you have to be very very favorable towards the child to agree that it could be thought of as a horse crossing the street so let's think of the icosahedron I mean certainly everybody has seen it so that certainly simplifies my life's time to start it's like all the regular solids you can embedded in a sphere I might draw the whole sphere well maybe just to orient ourselves so there's a north pole and a south pole and like all of the regular solids it can be inscribed in the unit sphere in such a way that all of the vertices inscribed so the vertices are points the icosahedron has a I'll let V well I'll let maybe script V because I want to use the normal fonts a bit later this is going to be the set of vertices they're going to be this is the set of edges and F will be the set of faces and the classification oh I didn't even tell you I do two lines of history the platonic solids like everything in mathematics pel's equation is not due to pel the Riemann zeta function is not due to Riemann the Gauss hypergeometric function is not due to Gauss and of course the platonic solids are not due to to Plato either they're due to his friend Theotatus I don't know quite how to pronounce it where's my write-up gone that I was using with all of my notes so this is actually mentioned there's a comment by Euclid in his elements he he tells that that Plato got them from his friend so since he's an important author I'll give his name Theotatus which is circa 415 to 369 and obviously these dates are BC because first of all he was a Greek mathematician before Euclid and secondly if they were after Christ then the dates should be increasing and not decreasing so if if people use negative numbers when we just put minus 415 to minus 369 but that's not what they do so they're all given in the book by Plato called the Timaeus so that's why they're called the platonic solids but they're not they weren't found by him so going back I'll try to so we're going to do it in the following way these two points all of no not all of these platonic solids but all of the ones with an even number of vertices here the number of vertices is going to be 20 the number of edges will be 30 and the number of faces will be 12 and as always there's a duality that's true for any of the regular polyhedrals in higher dimensions if you take the midpoints of the edges or first forget that take the midpoint of the faces they will be of in a smaller circle of smaller radius but still they will all have the same radius because of the symmetry there's a complete symmetry and they will be the vertices of the dual thing which is the icosahedron and that one the dodecahedron and dodeca means that one has 12 faces sorry 12 vertices v has sorry it's exactly I'm speaking and writing at the same time it's called the icosahedron because it has 20 faces because it is 20 faces it's what you see when you draw a picture is 20 face and they're all pentagons whereas the dual one the icosahedron has 12 faces and it's therefore called dodeca that's the number of vertices here so since 12 22 is an even number it there will be a symmetry of so you will always have the antipodal point so I begin with north himself and then at a height which shouldn't be too high and shouldn't be too lower the picture won't work at all but I have to distribute five points and I'm sure I won't succeed I think on another page I drew it more neatly so that I'll have a slightly better chance of copying it here well I'm not sure which I like more okay I'll try to do it so I'll put a point here a point here a point here somewhere a point here that doesn't seem quite right and a point here and then let me remove the equator because that's really confusing and then it has to be symmetric so I have a symmetric thing here and here a little point here and a point here somewhere and two points here and here and a point here somewhere so now if I use a dark chalk to connect then what I'll see is lines this one I won't see it's in back and similarly here the self-pull in this picture I'll only see two of the things it connects to but they're connected to each other just as these are connected to each other and then this one is connected it's not actually quite as bad as I expected it came out more or less icosahedral so there is the icosahedral and of course in back you would have the back faces which of course are in no way inferior to the front faces except that they're in back and so you can't see them in this picture and I don't know I might have left out one but maybe even I didn't so oh yeah here there's obviously a connection here and I guess here there's a connection here so there it is so that's the icosahedron now I don't need any of this so let me erase and I'm going to tell you first at a very naive level kind of high school geometry how we can you know write it down write down the coordinates and then a little bit how it looks but the important thing is that there's a group that acts on it and I'll call that group I for the icosahedral group and it's uh so it's the icosahedral group it's the group of orientation preserving of the morphisms of the icosahedron I'll just put icos because I'm not writing that so often so this is a polygon you can rotate it if it were a square you'd have four rotations and you could flip it over this in three space I don't want to change the orientation at least for now so there are 60 elements of this because it's a it's completely symmetric so i acts on this on the whole thing and therefore in particular it acts on the set of vertices and on the set of edges and on the set of faces so we can either move every vertex so each v goes to some v zero then since I have 12 vertices you can rotate any vertex by an element of the group to this vertex and then I have the things that fix one vertex and since it has five neighbors but at equal distance they form a cycle and I want to preserve the orientation I can just rotate cyclically around the vertex so this is rotation around a vertex around one vertex you pick one it doesn't matter which one and therefore that's where this comes from because you have 12 vertices you can pick and then five rotations around that and every element of the group uh sorry okay but I can also write it as 20 times 3 so you pick this is the number of vertices that's the and this is the number of faces uh yeah now I'm doing it correctly and this is here once you've picked a face every face is a triangle then once you've picked a face you can rotate but again only reserve the orientation by 120 degrees so rotation rotate a triangle by 120 degrees so here it's a rotation by 108 degrees and here it's a rotation by 120 degrees is that correct no it's 72 degrees I can't divide 360 by 5 correctly okay so and that means we also have a presentation of the group and I'll come back to that later but not today if I pick one vertex like the north pole and I let v be the rotation around that vertex then I pick one edge that contains that vertex like this one and let e I'll write in a second be the flip around that so the you take the midpoint of the vertex and take the symmetry so you join that point to its antipodal point and rotate by 180 degrees so you flip so that'll flip to this to this so that I can call e and the last one would be f you take a face that contains the vertex you started with the edge you started with and both so it's for instance this vertex this edge and this face and then you see that since rotating around the vertex is order five rotating around the edge flipping the edges order two so all of these will be one rotating around a face since it's an equilateral triangle has order three but there's one more relation if I look at where this vertex is under under e it gets if e was this edge it gets flipped to there but now under translation under e maybe in the other direction so I'm not sure which is the direction uh well I could have chosen this edge if I chose this edge then v this vertex will go here now when I rotate no it's it was this edge if I rotate clockwise then it'll come back to this and then when I apply v it's fixed but in fact normally this vertex is fixed in some order you have to choose the order this sorry equals one so that's the description of the group okay so it's given by the presentation where you have three generators of orders two three and five whose product is one now because the product is one I can eliminate for instance f so I could also write this as generated by v and e with only two relations so v is equal to by the way the binary acosetron group that I'll talk about a little later is when you drop one then the last of this common element is the central element of order two but for the moment this is the group so now v5 equals e squared equals and now f up to inversion which doesn't matter is ve so you can and you can also write it as generated by well you can pick any two v and f or v and e and then each time you'll have three relations like that so now you might think that's surprising wow this group is generated by only two elements I started with three it's not so surprising the general rule as far as I understand I'm not a group there is is that if you take just any old group it's always generated by just two elements so the monster group which as the name says is monstrously big and monstrously complicated the theory of it took 30 years to develop the size of the group is 10 to 71 you listed its elements it would I guess you know it would fit into the into the universe if you used empty space but not the atoms of you I'm not sure it's a very very big group but it also is only two generators so you know that's not as as exciting as all that that there are only two but for what it's worth that's the group and now as I said this group is acting and of course it acts transitively on the vertices because it brings every vertex to that one I mean it acts transitively so I should have added that word so every vertex is sent to every other every edge to every other in every face to every other and since the group is order 60 and we have 12 30 and 20 the stabilizer for vertex which is obvious has order five of an edge is order two and if a face is order three that's the rotations for instance around the face okay so that's uh the beginning and now I have two different ways of describing this in coordinates so I want to actually introduce I'm going to go over a little because it's uh I don't want to interrupt this story quite in the middle so the first way that I could do it is if I just think of this as the unit sphere so I think of x y and z in the unit sphere which means it's the set of points in r3 where x squared plus y squared plus z squared is one and that's z then the north pole would be the point zero zero one and the south pole would be the zero zero minus one which is the negative of that in three spaces the antipode so it's kind of obvious from the symmetry since I'm rotating uh you can think of you just have to pick one point here then you just rotate by 180 by the 100 by 72 degrees you'll get the others and then you take their antipodes and minus those points and the only thing you have it doesn't matter where you start on this circle once you've picked this ellipse remember when I drew the picture I picked the ellipse I started somewhere it doesn't matter where you start you just rotate so what I did is I picked a circle it looks like ellipse because I'm looking from the side I picked five points equally space it doesn't matter which one and then the only thing I have to do is make sure that the distance between two of them is the same as the distance of two adjacent ones is the same as the distance to the north pole and so that gives a very simple quadratic equation and when you solve it it turns out it's even ridiculously easy I'll do it in a second so let's write the points with the north pole which I said is zero zero one but actually let me not use these coordinates r3 I could write as r as c times r and that's much more convenient so I want to think of this direction the horizontal direction this plane as c and this direction is r okay so then I won't you and I won't call it z anymore I'll call it t so I want to think of w and t in c cross r so t is a real number w is a complex number and now the absolute values the norm the absolute value squared of w plus t squared should be one so then the rotation is let me write use the notation zeta 5 and later I'll use it for any z to n e to the 2 pi i over n so there's a rotation in the complex plane by 72 degrees by a rotation of order 5 then my top point in this notation wt the top point is simply n and then I'll have some points pj where j is an integer multiple of 5 and then I'll have some other points qj which will be simply pj and then I'll lift the south pole which is simply minus n so here j is again multiple of 5 that's going to be my 12 points and so in the wt notation but I'll have another rotation shortly so if you're taking notes you might want to leave some space for later uh well actually no maybe you maybe you don't want I'll come to that when I do the second is I'm going to give you two descriptions so now wt in this description here it'll be zero and one and of course this is zero and minus one here as I said you only have to pick a certain number and then multiply the complex number by fifth roots of unity so the one I'm taking is one over the square root of five times well two over the square root of five and here one over the square root of five so you see the sum of the absolute value squares well they're already positives you get four fifths plus one fifth is one and here I multiply by the jth power of this thing and then these are the antipodes so it's minus two of the square root of five zeta five zeta five to the j and minus one okay so and as I say it's very easy if you just work out assume that this happens at height t then the radius is squared of one minus t squared so if you take one of the points to be real it's the t is t and the w is squared of one minus t squared and then you rotate by fifth roots of unity and you just compute what it means for this distance to be equal to that the distance squared is very easy and you find that t has to be exactly plus or minus two over the square root of five so that's that picture okay so that's one picture but there's a completely different picture that I also want to give and which is almost easier so I mean I mentioned two survey articles by John Bayes and by Oliver Nash and they're both they both talk about more sophisticated topics one about the connection we solved in the quintic equation and one about the connection with the e8 lattice but each of them explain the icoshedron and completely differently and they give a different description coordinates so the one here is actually the one that you would get from from Nash's description from Bayes' description it's completely different so a different description is you take the point zero one it doesn't matter in what order so I'll take some order you take the point zero one and five remember five is the golden ratio and you take all possible signs so here's the second description you take this thing with all possible signs and then cyclic permutations so if you have a triple there are three cyclic permutations x y z y z x and the last one z x y permutations so there are three times four that's 12 vertices so v now looks completely different it still is closed under the antipodal map because with everything I've designed but you no longer have the point one zero zero no point on this has two coordinates equal to zero it's a different description but this has a nice property because if I take the original four points then they just lie in well maybe I should make it a little nicer it doesn't matter you have to do it the right way let's put it in the z here maybe it's still not the one I want but then I can talk about x y coordinates which is what we all learn in school and so here you see that x is one and y is phi and here x is minus one and y is still phi and here this is minus one and minus phi and here this is one and minus phi and so you see that these four vertices that I started with form a golden rectangle but then I have the three-circuit permutations so there are three and this I really cannot draw there's a nice picture in the paper if I was using the beamer I could draw it so the next when you take this is length two phi this is length two or I could rescale and make it one and five doesn't matter now I take a slit here of the same length and I put another golden ratio like that which goes through that way and then there's a third way that's this way and then there's one the other I won't even try to draw it there's a third one where so this is where z is zero so it's in the x y plane there's one where x is there and the y z plane one where y is there and the x z plane so you get three of these rectangles that as I say I won't even attempt to draw but roughly uh they look like that and the third one looks like which way does it go I can't do it but anyway if I look at the vertices here I'd four vertices which form a golden rectangle but here the others also so therefore v in this description has a completely different symmetry v is the union of four of the vertices of the vertices of three golden rectangles but it's not unique if you look at the across even if you look at this picture and try to figure out which two are the golden I should have prepared this to be able to tell you I think it's this one and those two but I'm not sure that they form a golden rectangle but anyway there are lots and lots of them and even if you take this triple so the triple is three golden rectangles whose vertices are disjoint because the 12 of them have to be all 12 but there are five different ways to do this this can be done so there are five there exist exactly five such triples so what I mean is the triple of embedded golden ratios you take four of the vertices which form a golden ratio you have three that are disjoint they have nothing in common there are exactly five ways to do that but then of course I permutes them the group I my acosahedral group permutes this and therefore and that's an injection of the group so I is actually a subgroup of the symmetric group on s5 and since it is what this is order 120 which is five factorial this is order 60 so it's index two and it's very well known that in abstract group theory there's only one such thing and so it is a five but here I've given an explicit isomorphism because I've tell I mean if you just write these things down and write down the 60 automorphins you can do it more intelligently you see that you're getting exactly the even permutations of these five things so in this description you have that but in this well actually in any description you have something else namely there I have to look at my picture so I don't get lost drawing it so that's one way of doing it but there's another way to do it again I'll use another color of chalk here is a triangle and here's its center then I want another triangle that's disjoint from a completely disjoint not even touched in the corner so this vertex isn't allowed but here the first one I can take here's a good one okay and it is its center here that's a third one now I need another one and there's a perfectly good one here and its center and then there's a fourth one which is the one in back I'll again draw it dotted and then I have to make dotted shading and its center is in back so it's the antipodal point of this point here in front and those are four disjoint triangles so this is a completely different thing we can also divide divide the vertices well no sorry the phases remember there were 12 phases we choose three phases four phases the 12 phases but we choose four phases four phases in the set of phases of my dodecahedron so I've chosen these four and then if you look at it the symmetry is perfect and as a result these four myth points of these phases they form the vertices how could it be anything else of a regular tetrahedron so you have among the vertices you can pick four vertices which together give you a regular tetrahedron so this gives me a regular tetrahedron in v and v is then the union from one to five of the vertices of a regular tetrahedron sorry something is wrong because now it again doesn't come out what I want to say if I rotate around the north pole then this triangle so these vertices these these four triangles that the properties you can see in the picture that they include all the vertices every vertex this vertex belongs to that this vertex belongs to the back one but they don't touch so every vertex belongs to a unique one there are four and each one is a triangle four times three is 12 that's the 12 vertices but it's the midpoints which form the tetrahedron sorry so the it's the midpoints of phases and there were not 12 there were 12 vertices but there are so there are 12 vertices but there are 20 phases it's nycosahedron and so indeed it's the phases sorry which are the the phase midpoints if I think of f is the set of its midpoints then it's the vertices of a tetrahedron of a regular tetrahedron so the 20 phase middle points become but again when you can do this here if I rotate this whole picture around this vertex I get another one and those have to be different so there are five different ones and those five are completely disjoint no phase can belong to two of them because each vertex belong to a unique one so these have these are disjoint phases disjoint interiors these disjoint phases at least except on the edges so we've divided up the 20 phases into I can't get the numbers right into five you have to do it right so there are five ways to do this we've divided there are five there are five disjoint patterns like this actually that thing is not unique either but I don't care there are five and they're permuted by the group two and that gives a completely different map from A into S5 and they're actually not even conjugate to each other this is some external automorphism so all of the beautiful properties of of this special group S5 are coming in so I've done about half as usual of what I prepared for today and have run over time but only a little because there was a long introduction we started late on Friday I'll finish this I want to tell a little more about the connection with the symmetric group on five letters the alternating group and also the group of automorphs of the projective plane over a field of five elements but then also start with multiple forms and you'll see how they connect and then through the rod's ramanution identities these things will come together in a completely coherent way so for the moment all you can really see is the rod's ramanution we're full of the number five and the icosahedron is full of the number five and that's not a coincidence it's the same five but I haven't yet so it's only the first of four lectures so I'll stop for today and say I'll be very happy if anyone has questions no no no well well actually there were two formulas I mean the continued fraction so the continued fraction identity I wrote it twice the continued fraction which was one over one plus q over etc I wrote first that it's equal to g over h and then I wrote that this is the product one minus q to the end so I don't think anyone can go directly from the continued fraction to this so this was the rod's ramanution and then I'd already done and what ramanution also showed is that g and h and it's because they're multiple forms multiple forms but are related to each other by multiple transformations g and h individually are multiple forms again for people who know the definition but I'll explain it in detail on friday on the group gamma five which is a subgroup of index also 120 in in sl2z that's where multiple forms come in the rod's ramanution functions individually are multiple functions for this group but the pair of them is a multiple function on all of sl2z that's why the ratio this this divided by this the quotient is exactly psl2 of f5 and that is our icosahedral group and that that's why where the connection but that means since g and h are together a multiple form on sl2z that when you apply multiple transformations of sl2z you send g to a linear combination of g and h and h but it's very well known and you know it extremely well that under iteration the continued fraction is the same as matrix two by two matrices that you successively keep applying a matrix to the quotient and so if you apply that that's essentially what's behind it so the multidarity is what's used well he didn't use the words multidarity but ramanution he didn't really know it yet when he wrote that letter but that's why g over h the continued fraction and that g over h is this is two separate identities the g is what it is in h and if you remove that and ask to go immediately for the continued fraction as far as I know nobody would know how to do that so both deep down are used in the multidarity and basically a general statement is this any non-trivial identity in mathematics comes from a multiple form I mean all actual identities I don't mean formulas when you compute something and it's you know long calculation but if if two different expressions very different expressions are equal to each other that's usually a piece of number theory it essentially always comes multiple forms obviously it's a slight exaggeration but it's not very far so when you see such amazing identities even if you didn't know you would say where are the multiple forms but the answer is there right there so I can't hear can you speak louder the coordinates apple's right it's not correct the coordinates are for qj and it's not the opposite of for pj oh sorry because I was talking and writing at the same time in my notes I took out the square root of five and the point was this point and then you don't have to write it twice and then you you rotate so excuse me all four of them were wrong this is square root of five this is also square root of five and this is also square root of five sorry thank you very much that wasn't a question that was a correction I tend to write lots of notes on the board and then I see it after people leave and I'm very unhappy that nobody told me so you will get you will lose points if you don't point out mistakes and if you do point out you get you gain some brownie points so thank you don't usually it's a question conformal field theory very simple I mean yes or not usually character and conformal field theory are serious because you are just counting the space that's right the definition is well the original definition is a trace so it's a trace but I know I know that there are many characters not only the the one of the model two five you mentioned that admit the infinite product representation that's true and I don't understand quite why so what is true so I'll I'm not an expert at all but I have many friends who are experts in particular van der Nam said gay guk of others so roughly we're talking about conformal field theories we're talking specifically about what are called rational or simple rational vertex operator algebras c2 co fine that they're various technical words and as I said such a thing is an algebra not a usual algebra with the multiplication it's a vertex operator algebra infinitely many but just as for usual uh so let's call it a for algebra instead of v since I've been using v I have modules over it and for these nice ones there are only fine at the many irreducible modules up to isomorphism and each of these spaces is a graded vector space so as Giuseppe as professor Masata said the the character is defined as the sum n from zero to infinity q to the n times the dimension of the nth piece so it is integer coefficients which are positive numbers from its very nature but actually that's not entirely true the n is shifted by an amount called the conformal weight so it's typically I remember I had that q to the 11 60th there's some funny shift and that's the character and there's a general theorem that's always true dong or maybe there are different people well I don't know the literature well and I've forgotten but it's known for for this class of functions that all of the characters of all of the modules are modular functions I haven't yet defined but many of you know modular function is a modular form of weight zero or a quotient of two modular forms of the same weight so g and h were examples so that's always true for every algebra it's it's always a modular function and what's more it's even better remember I told you briefly and answering a question on in the main lecture yet it'll be next time that g and h are modular functions on this subgroup gamma 5 which is index 120 in sl2z actually 60 you have to divide by plus minus it's really 60 the thing I want that it's 60 anyway come to think of it but g and h together are a modular form on sl2z if you think of it as a vector valued form as a pair then you can transform by any element of the full modular group and you go into g and h that is again a man's from which are special and which are general that second property is also general that's part of dong's theorem for any of these nice vertex operator algebras not even just minimal models there are many other types it is always the case that the individual characters that are fun at the many that's also a theorem those finally many characters are all individually modular functions on some subgroup of sl2z but jointly the vector space they span is invariant under all of sl2z and that doesn't mean sl2z permutes them it acts on the vector space so sl2z sends each of these functions into a linear combination next time I'll do it explicitly for the world's ramanujan so those two things are general now you as so often what you like is the specific form and not the general theorem you asked about the product expansion already in the theory of modular forms that's super rare it doesn't mean it's not interesting there's at least one entire book by kubernetes on such modular forms that have a product or in another one by german professor that I can't remember on a different there are these two big textbooks research textbooks monographs on modular forms that have product expansions they have a name they're called modular units and that's the title of the book by kubernetes where is the book by the german's name I block on especially about products of the dedicated data function that I'll speak of also later so that's a very very special thing and then you're right but you know more than I do there are some other cases where these characters have product expansions but it's definitely a rarity they usually don't so I think that's not at all something you always expect they always non-negative integer coefficients by their very definition the coefficients are dimensions they always are mounted on some group and they're always jointly bought there's a collection under sl2z but they don't usually have product expansions but I on the other hand but they do always have I think it's like q-hypergeometric expansion by some fox-based decomposition but I think in general there'd be like a finite sum depending on you know conformal blocks or something I don't want to even say the wrong words but I'm pretty sure that it's not a general phenomenon and another beautiful question thank you no young person has asked a question it's all senior people either you don't dare or you know more okay there's no pressure if you come back on Friday you can ask questions even at the beginning about today or you can interrupt at any point and there'll be plenty of time I'm going by my standards incredibly slowly and I hope sufficiently slowly in presenting this material okay so thank you