Symmetry and the Monster

SYMMETRY AND THE MONSTER

Professor Mark Ronan

We will start off slowly with a little early history.  Everybody learns the formula to solve quadratic equations when they are at school, even if you everyone tends to forget it after you've left school or university.  I remember I forgot it completely, and had to recapture it by working it out for myself!  But the Babylonians solved quadratic equations using essentially that same formula - don't believe anybody who tells you otherwise!  I have read the original tablet - they really did solve it by this method.

The Babylonians solved a few of the equations of degree three, but Omar Khayyám solved cubic equations.  Of course, he is often better known for his poetry, but he was actually a brilliant mathematician and astronomer.  So it was that he managed to solve equations of degree three by a graphical method, drawing graphs and measuring line segments between certain intersection points.  But he could not get a numerical formula, and that came later, in Italy in the first half of the 1500s. 

Let me just talk about that for a minute.  What happened was that a man called Scipione del Ferro found a method for solving cubic equations, and when he died he left this method to one of his students.  This student then dined out on the secret by challenging other people to mathematical contests.  Then he made the mistake of challenging a man called Tartaglia. 

He was an unfortunate chap - indeed, 'Tartaglia', which wasn't his real name, means 'the stammerer'.  As a boy, he had been slashed across the face by some visiting soldier and was disfigured, and perhaps that contributed to his stammer.  But at any rate, he decided he would work on cubic equations, because he realised that you needed the solution in order to be able to deal with the questions that this student was putting forward, and so he solved cubic equations. 

It then became known to a man called Cardano that Tartaglia had solved cubic equations.  Cardano then tried to get him to explain the method, and Tartaglia refused to do it.  This went on and on - it is a long story, and you can read about this in many books on the history of mathematics at the time.  Cardano had a lot of friends among influential people and so was a powerful man and so Tartaglia was being persuaded to give him the formula, but Cardano said, 'I will promise never to reveal the formula'.  He didn't... until he did!  What happened, you see, was that he had a terribly bright student called Ferrari with whom he worked on these things, and Ferrari solved equations of degree four.  Then they found out that del Ferro had originally solved cubic equations, so Cardano felt that let him off the hook, so they went ahead and published.  So now, in mathematics books you will see that the formula is called Cardan's Formula, after Cardano, which is jolly unfortunate for Tartaglia.  At any rate, these four were brilliant people, but let me move on to the quintic equation. 

By the first half of the 1500s they had solved degree three equations and degree four, but equations of degree five seemed to pose a particular problem as nobody could quite see how to solve them.  This went on for quite a while, until one of the greatest mathematicians of all time, Carl Friedrich Gauss, thought that perhaps, since so many people have tried, the formula was impossible and contradictory - and Gauss was absolutely right.  About the time he said that, Paolo Ruffini, an Italian physician and mathematician, published a proof that you could not solve equations of degree five by means of a formula.  When I say by means of a formula, I mean using square roots, cube roots, fifth roots, and anything else, in some complicated method.  But people found it terribly difficult to go through Ruffini's writings, and nobody really went through it; nobody actually nailed an error in it anywhere.  Ruffini was very frustrated, and he tried writing other versions of it, and he either couldn't get them published, or he couldn't get them read.  I think it has been agreed by people who have looked into this in more recent years that he did in fact have a serious gap in the proof, but nevertheless, his idea was the right one. 

In 1824, 25 years later, a famous young Norwegian mathematician, Abel, solved the problem and showed you could not get a formula for quintic equations.

But to return to Galois, the problem is this: you have got an equation, say of degree five or degree six, you know that there is no general formula, but on the other hand, some equations can be solved by a formula.  For instance, if I say to you x5 = 2, you have no problem solving it - it is just the fifth root of two.  So then you want to sort out which equations can be solved using square roots, cube roots, and so on, and which ones can't be solved in that way.  This is the problem that Galois dealt with, and let me just explain in outline how he solved it.

An equation is irreducible if you cannot break it up into two equations.  In an irreducible equation, you can permute the solutions among one another because they are all, so to speak, equivalent, but you cannot do all possible permutations.  What Galois did was to consider not the solutions themselves, but the possibilities of permuting them among themselves, and he considered a group of permutations.  This is now called the Galois Group of the equation.  Galois was the first person who used the word 'group' in this context, and that is why Group Theory in mathematics has its name.

Here is an example:

x4 - 10x2 + 1 = 0

You can see this equation has only even powers of x.  So if you have a solution, whatever it is, the negative of that is also a solution (because when you square a negative, you get the same result as when you square the positive).  So there are four solutions - a, b, c, d.  The number of permutations are rather restricted because if b is the negative of a, and cis the negative of d, then there are these restrictions on the possible permutations.  The Galois Group is a certain set of permutations that might not be all of them.

So, what did he do exactly, apart from have this group?  The key idea that Galois had was the idea of deconstructing a group of permutations into simpler groups.  This means that you deconstruct it as a tower of groups.  It is a little bit like deconstructing a molecule into atoms because you get the same set of atoms however you do it.  That is the case with these groups: however you deconstruct them, you get the same set of sort of atoms of symmetry (which is what I am calling them here).  Of course, the idea of deconstructing things is terribly important.  The alchemists wanted to turn iron into gold, but once we have got the idea that iron is composed of iron atoms and gold is composed of gold atoms, and you can't turn an iron atom into a gold atom, no matter how clever the chemical process, we see the importance of these atomic bases.  So the idea of coming down to the basic building blocks is very important in mathematics, in science, and, actually, just in general. 

A group that cannot be deconstructed is called a simple group.  It is quite a bad term in mathematics, because these simple groups tend to be pretty complicated!  The reason for the word 'simple' is that you cannot simplify them, but they can still be quite complicated.  If you take a pentagon (a five-sided figure) you just take the rotations of that little group, called a cyclic group of order five, by doing a turn five times and getting back to where you started.  But it is the non-cyclic simple groups that can be very complex.  The first family of these things was actually discovered by Galois.  Other families came later in the 19th Century, and they stemmed from work of a man called Sophus Lie.

Let me actually mention something about Galois and about Lie, because these are two very interesting characters. 

I almost hesitate to give the story of Galois, because everybody has heard it before, but it is still a very good story.  Galois was a young man and he was never anything more than a young man because he died when he was twenty.  He died in a duel, or, rather, he was mortally wounded in the duel and died the next day.  Why did he fight the duel?  Well, I don't quite honestly know.  People who are better on the history of mathematics than I am have looked into this, and it is actually quite confusing. 

There was a story of a woman that he had been in love with, but there were also other stories revolving around his being in prison.  It is something of a tragic story because the chap was born in 1811, and everything was going swimmingly.  His father became Mayor of their town, they sent him to a wonderful boarding school in Paris called the Lycée Louis-le-Grand, named after Louis XIV, and he was doing just brilliantly well and was a very clever boy, until he got told to retake classes that he had already passed.  He then got so annoyed with the whole thing that he decided he would concentrate all his efforts on doing mathematics, which he did, and he became brilliantly good at it.  A year later a wonderful teacher came to the school and encouraged and helped him to submit some research papers, but things just went wrong.  He would submit research papers, and somebody would say, 'This is a wonderful paper - I must give a talk on this at the French Institute,' and so he would get ready to give the talk, but then when the date came, he would not give the talk because other things crept up.  He would re-write the paper and he would submit it for a prize competition.  One of the examiners was Fourier, after whom Fourier Analysis and Fourier Series are named, who was a very old, respected mathematician, so it was wonderful that they gave it to him.  The trouble is, he died, and then the paper was never found.  So Galois had a lot of terribly bad luck.  He applied to the École Polytechnique, which was the place in France at that time to go to.  He applied earlier than he should have done and he did not tell people that he was doing it - you understand, we are dealing with a teenager here - and he failed the entrance examination.  In normal times he could just have applied the next year, but the next year occurrences surrounding the revolution put pay to that plan.

The story is that Galois' father was very much on the Republican side and he had been a great supporter of Napoleon, so he came to lose his job as Mayor.  There was political intrigue, and he was made to look an idiot, because the Jesuit priest in the town wrote some scurrilous rhymes under the name of 'the Mayor'.  Normally, this would not work, but the Mayor did actually write funny rhymes, so it did work in this instance because it did look as if he had written them.  This brought the Galois' father to commit suicide.  This happened about two weeks before his son, the young genius Galois, was due to take the entrance exam for the École Polytechnique for the second time.  He failed.

What actually happened in the exam was that there was oral exam and one of the people who was asking the questions liked to ask very simple, perhaps rather provocative, questions - and Galois was provoked.  There he was, up at the board, and in response to this man's question, he threw a board duster at the man - or so the story goes.  I don't know if that is true.  I don't think it is one of those hardboard dusters, I think it's probably just a rag, but at any rate, he failed the exam.

But then things went from bad to worse, and he was  eventually put into prison.  And it was only a month after he came out of prison that he died.  So, was it a police plot to kill him or was it about this woman or was it a mixture of the two?  Was it also because he wanted to foment a riot and thought he was ready to give up his life and didn't care what happened to him and they would have a funeral for him and there would be a great riot at the funeral, and it would change the course of history in France?  I don't know.   At any rate, poor Galois died at only twenty.

Sophus Lie is quite another character entirely.  Whereas Galois started as a teenager doing serious research, Sophus Lie had never done any research.  He went to university and he did his degree, and he had no idea what to do.  He wrote to a friend, a couple of months later, and said, 'When I said goodbye to you before Christmas, I thought it would be for the last time, because my intention was to become a suicide, but I didn't have the strength for it.'  Poor Sophus Lie - he suffered from bouts of depression, but he could also be a tremendously energetic man. 

He was also physically enormously powerful.  Sophus Lie, when I read about this a few years ago, I was absolutely flabbergasted, because I had no idea!  Apparently he was a great walker - he would walk fifty miles in a day, if he went out hiking, and one day, he walked home to get a book he needed and walked back to the university - that was 36 miles each way!  He would go out for great hikes in the country, and he would show the country boys how he was stronger than they were and could do pull-ups to a beam more than they could and so on and so forth.  He was an amazing character!

One summer he went to his sister's after he had left university, and he decided that he would teach his nephew to swim.  I don't know if he was correct or not, but Lie thought that the way to teach kids to swim is just to throw them in.  They went out in a boat in the fjord, and he strapped a life jacket on this young chap and tossed him over!  Well of course, there was a good breeze coming down the fjord, and his nephew was carried off on the wave tops.  I actually do cold water swimming, so I know what this is like - it must have been very dangerous because if you don't know what you are doing, you could die very easily.  So, another boat took off from shore, because they knew all that Sophus Lie was a dangerous sort of chap and another boat took off from shore and they went and rescued the boy.  Then there was an argument because Lie wanted him back in his boat, and they wanted his clothes back in the their boat.  This argument went backwards and forwards until the clothes were handed over and they took the boy ashore.  Actually, mothers in the local town used to warn their children, 'If you don't behave, Sophus Lie will come and get you!'

Anyway, Sophus Lie wanted to do for differential equations what Galois did for algebraic equations.  The point is that an algebraic equation, if it is an equation of degree three, there are generally three solutions, but there are at least a finite number.  With a differential equation, you can have a continuous range of solutions.  For example, there is a differential equation that describes a vibrating string, but the way it vibrates will depend on where you plucked it.  So there are a whole lot of solutions. 

Lie wanted to consider merging one solution into another, and so he looked at continuous groups rather than the finite groups that Galois was looking at.  His groups could also be split into basic building blocks, and these were classified by a German called Wilhelm Killing.

All these characters are interesting.  Killing was obviously a very clever man, and also a terribly nice man. Apparently, his father was Mayor of some small town and the school was about to close, for lack of resources, and so he stepped up and taught all the classes, which meant teaching for 36 hours a week!  At any rate, he then worked for a place teaching Catholic clergymen.  During this time he did his great work on classifying the basic building blocks for Lie Groups.

Killing wrote to a man called Felix Klein in Germany.  Felix Klein knew Lie very well because they had gone to Paris together as young men.  That is an interesting story too, because that was 1870, and for those of you who know history, that is the year of the Franco-Prussian War.  So, as a Norwegian and a German in Paris, with the Franco-Prussian War was about to start, they thought that they'd better get out.  It was the correct decision because Paris was surrounded, and they even auctioned off animals from the zoo for food, and there were recipes for marinated giraffe's neck and things like that. 

After leaving Paris, Lie decided to go to Italy before going back Norway, and of course how do you get from Paris to Italy?  Well, if you're Lie, you walk.  So he got as far as Fontainebleau, but unfortunately he would sing Norwegian songs as he was hiking, and so he was taken as a spy.  He could have sounded German to most people, and all his mathematical notes, with points and lines, would have been considered to be artillery and cavalry.

Killing didn't have such exciting stories.  He was just an awfully nice chap, but the point I want to make about Killing is that not only did he do fantastic work, but having done this fantastic work, he then got a Chair of Mathematics at the University of Munster, where he had originally been an undergraduate.  Therefore he found that he had all the time in the world to do his research, and so he could now work on what he really wanted to work on, which was the foundations of geometry.  This is a topic which is now totally forgotten, but he wrote a great book on it.  Friedrich Engel, who collaborated with Lie, recommended this book for a prize, but he actually said privately it was a load of rubbish.  But, you know, he knew what Killing had done in this work on Lie Groups and he really deserved a prize for that.

It is an interesting thing though, because people like Galois and Killing were under great stress.  Killing had no time for research, and yet, he did the greatest research of his life - and Galois was under tremendous stress in prison and what have you.  But you give them a comfortable position and they just don't do anything good anymore.  I am sure you can come up with lots of examples of this yourselves.

Groups of Lie type are finite versions of Lie groups.  These were created by a man called Claude Chevalley in 1955.  Actually, a lot of them had been done by Leonard Eugene Dickson in 1901.  Dickson was the very first successful PhD student from the University of Chicago, and he visited Leipzig and Paris, and then he came back and wrote this terrific book, in which he created finite versions of these groups.  So these give you finite atoms of symmetry, if you like, the finer building blocks for all finite groups.  Chevalley found a uniform method of doing them all, and then other people made variations on Chevalley's theme, and we got, by 1961, absolutely everything.  We could portray this as being in a periodic table, by analogy with chemistry, so that we have a periodic table of all the finite building blocks of symmetry - except for the ones that don't fit in!  There were in fact five that didn't fit in, but we will come to those later.

In the meantime - this is early 1960s now - Walter Feit and John Thompson proved a fantastic theorem.  What they proved is that these finite simple groups, these finite symmetry atoms, must have cross-sections, and if you could locate a cross-section and you work with it, you could in most cases nail down the whole of the symmetry atom.  So this was very important work because it gave a method for trying to complete the search for anything that did not fit in the original periodic table.

By 1965, it looked as if you either had to be in the periodic table, what I call a group of Lie type, or you had to be one of five exceptions.  They were discovered by a chap called Mathieu in the 1860s, and these are really very remarkable groups of permutations.  There is nothing quite like them and it looked as if that was perhaps going to be the end of the story.

By the way, before I go on to the rest of the story, let me just mention Walter Feit and John Thompson.  I can now give some personal stores, because I actually knew Walter Feit reasonably well.  I was once at a conference with him in Paris and we arranged to go out to dinner.  We met at his hotel room, and he we decided on a restaurant before he picked up the phone to ring them to reserve a place.  I said, 'You speak French, right?'  He said, 'No, I'll just talk to them in English.'  - Walter Feit was an American mathematician, awfully nice guy, but he has never used any languages apart from English.  He has lived in America all his life, and he has probably only travelled abroad later on. 

Imagine my absolute astonishment when I later saw Walter Feit giving a special conference at Oxford in his honour, and he got up to speak and said, 'Well, thank you very much for organising this conference for me.  It's a great pleasure, particularly since Oxford is the place I went to school.'  People were astonished - we no idea he had been to school in Oxford!  And it wasn't that he had been sent over to boarding school - Walter Feit was on the last kinder transport from Vienna on the first of September 1939.  His parents were going to follow two weeks later, but on the third of September 1939, War broke out and he never saw his parents again.  So that was how he came to be in Oxford.

Feit and Thompson did really great work.  I mentioned the five exceptions.  What Thompson then did was to work on a particular family of these symmetry atoms and show that the cross-sections that we see in this family can only occur in this family and they cannot appear anywhere else.  It was one of the steps that needed doing in this big project.

What he did was send round copies of this work, and a man named Janko in Australia wrote to him and said, 'I'm having a bit of difficulty with the smallest case in your theorem.'  I have this story from a person who was there at tea on the day Thompson received the letter, and he said how, 'Thompson showed me the letter at tea, and he was laughing about it.  The next day, he wasn't laughing.'  Because there really was a problem with the smallest case, and Janko actually was pretty close to finding another exception, using that cross-section.  I do not mean to put John Thompson down by that story - he was an absolutely brilliant guy.  It is just that one can miss things, and he just missed a small thing there.  He is a really brilliant man.  In fact, this colleague who told me the story at tea said, 'Thompson's unbelievable - the only person I can think of who would just do mathematics all the time - lunch, dinner, sleep, all the time!'  The only person I can think of like that is Bobby Fisher, the chess player.  Thompson is a most remarkable man.

But Janko and others then tried to find these other exceptions.  There were various methods of finding them.  One was this cross-section method: you would take a potential cross-section and you would see if you could get something from it.  Others were done by groups of permutations, which is how Mathieu had discovered his, and some were found using geometry.  This particular one, the Hall-Janko group, is called J2, because J is for Janko, and the two is for the second one he found.  Actually, it was not the second one, it was the third one because the second and third were discovered at the same time, but that is one story too many for now.  So, he found it using the cross-section method.  A chap called Marshall Hall, in California, found it using permutation groups, and Jacques Tits, who worked in Germany at that time, constructed it using geometry.

There is an interesting story about this discovery by Marshall Hall.  He came to Oxford to give a talk on this at a conference in 1967.  Hall gave a talk on a group of permutations that permuting a certain set of symbols around among themselves, and there are exactly 100 symbols.  When you analysed the technical details, this number 100 had to split in 1 + 36 + 63.  Two of the people who were watching this talk said, 'this 1 + 36 + 63, that reminds me of 1 + 22 + 77 - doesn't it you?'  They agreed, so they worked together on this, and they produced another exception, within two days, within 48 hours.  So it looked as if these things were popping up all over the place.

An example of one that was discovered using geometry is the Leech lattice.  John Leech constructed a remarkable lattice in 24 dimensions.  I can give you reasons why lattices in 24 dimensions are very worthwhile to look for.  Basically, it is to do with looking for good lattices anyway, packing points together but keeping them a reasonable distance from one another, and the more dimensions you have, the more points you can pack in.  On the other hand, the lattice has to be fairly efficient, so it is a trade-off.  This lattice was absolutely brilliant.  He constructed it using Mathieu's largest group of permutations, and then he hawked this lattice around to a number of mathematicians, trying to get them to work on the symmetry group of the lattice. 

John Conway took this up.  Now, Conway is a big name in mathematics, but at that time he wasn't well-known.  He had a wife and four children, so he was a fairly busy man.  He said to his wife, 'Look, this is really exciting - I really think this lattice is worth looking into.  I have to have some time on my own to do it.'  So they had an agreement that he would have from Wednesday 6pm to midnight, and Saturday midday to midnight.  So on the first Saturday, he got himself all set to work on this.  He took an enormous sheet of paper, a great long roll of paper, and started to write down everything he knew about the Leech Lattice.   He worked and worked, and after about six hours of this, he finally decided he was getting somewhere.  He was quite excited, and he picked up the phone and talked to John Thompson, because Conway and Thompson were both at Cambridge University.

He said, 'Look John, I think that the size of the group is either this number or it's half of that number.'

Thompson said, 'I will think about it and call you back.'  Twenty minutes later, Thompson called back and said, 'It's half that number!'

He said, 'But have you really got it?' 

He said, 'No, but I need to find one new symmetry that we don't already know about.'

So he got terribly excited and he worked, and then by about ten o'clock he phoned Thompson again, and he said, 'I've got it!'

Thompson said, 'Well, that's great.'

And he said, 'I'm going to bed now - I'm really tired.'

Then he thought, well, it is pretty stupid to go to bed, because I haven't actually got it; I have almost got it, but not yet!  So he stayed up until after midnight, and then he rang Thompson one last time and said, 'I've got it,' and the next day, they worked together on studying this fascinating group of symmetries.  At any rate, it was a wonderful group of symmetries 'very important, and it contains most of the other ones that were known at that time. 

You understand that these symmetry atoms could contain other things - it is a little bit like the Earth.  From outer space, you look at the Earth and it looks like an atom, but when you actually come down and look at it more closely, you see things like trees, and so those are there as well - you have to take those into consideration - and then there are animals, and then, if you look even more closely, you find insects.  So there is a lot of interesting stuff on the Earth, and in the same way there is a lot of interesting stuff in some of these symmetry atoms.

Fischer, a German mathematician, was approaching things in a completely different way.  He was using something like permutation groups - they were not exactly the same but they were similar to permutation groups.  He used the big Mathieu groups to discover some pretty monstrous things, and then, to cut a long story short, he found a fourth one that was even bigger, and this was by far the biggest thing that had been discovered so far.  Fischer was the sort of person who, as soon as he made a discovery, he just let everybody know about it - he just broadcasted it to the mathematical world!  When he discovered this huge thing, which I will here call the Baby Monster, he immediately looked to see if it couldn't perhaps be a cross-section in something bigger, and discovered that it could be, so it was called the Baby Monster, and the bigger thing was called the Monster.

When I say these groups were discovered, I don't necessarily mean that we had it.  I mean that the existence of these groups was predicted.  What we then had to do was to nail down all of the technical data so that we could really establish whether they existed or not.  This technical data is encoded in something called a character table, which is like a giant Sudoku, and they decided to look for the character table of the Baby Monster.  In order to do that, they decided to look for the character table of the Monster, and then they decided to look for the character table of the Baby Monster, and then they handed this information over to somebody who constructed it on a computer, but the Monster was too big for this.  The Monster is an absolutely enormous group, too big for computer techniques - the character table was like a 194-by-194 Sudoku puzzle. 

Actually, I should say, Fischer could not do this on his own, and so he went up to Birmingham to consult Dr Livingstone.  They worked together on this character table, which they produced, and it showed that the Monster needed at least 196,883 dimensions.  That is a lot of dimensions, and even for mathematicians, it is a lot of dimensions - 196,883.

Now, a very interesting thing happened with this.  There is a man called John McKay, who lives in Montreal.  He is the sort of person who attracts information from all over the world about all sorts of things in mathematics, and he likes to put things together and say 'This is related to that'.  He noticed 196,883 was one less than 196,884, which he found from something in number theory called the j-function.  He got very excited about this so he wrote a letter to John Thompson.  McKay is a professional mathematician, so it's not just anybody writing to you, but Thompson could have waved it away and said 'I've no idea about that,' but in fact, he checked with other data on the Monster, and he checked that there did actually seem to be a connection between the numbers you got out of the Monster, namely the number of dimensions it could operate in, and the numbers that appeared in the j-function. 

So, when he got back to Cambridge from Princeton, he mentioned it to John Conway, the chap who had done the symmetry group for Leech Lattice, and also the man who gave the Monster the name 'Monster'.  He was immediately very interested, and he shot down to the library to look at old books on the j-function, number theory books from the 19thCentury, and started to do calculations with the Monster. 

I remember Conway telling me about these once.  There was some number like 2011, or some number that had nothing to do with anything, which popped out, and then he found it in this number theory paper, and he thought 'that's it, that's it - there really is something here!'  Simon Norton joined him in this quest, and they produced a paper called Monstrous Moonshine.

Now, before I tell you more about that let me just go on to Ogg's observation.  Andy Ogg was a number theorist in California.  This had happened very shortly after the Monster had first been predicted, and I am not sure we knew the 196,883 at the time, but we knew the size of the Monster, and Jacques Tits in Paris, who was giving his inaugural address at the College de France, wrote down at the end of his lecture the size of the Monster on the board.  He wrote it, as people do in mathematics, as a product of prime numbers - 2 46 x 320 x etc.  All the prime numbers that appeared there were exactly the same prime numbers that Ogg had noticed in connection with the j-function.  They were connected in a way that would take me too long to explain in this particular talk, but they were connected with sub-groups of the modular group, which is connected with the j-function.  He thought it was absolutely extraordinary - it was exactly the same set of prime numbers!  It is not that one contains the other; they are actually the same set, and the numbers that are missing are exactly the same - extraordinary!

So he mentioned it to someone at the time, who said, 'Who knows?', and nobody had any idea what that had to do with anything.  It was only when the Moonshine paper appeared, with John Conway and Simon Norton, that in fact it was clear that Ogg's observation was connected with McKay's observation, and the whole thing fitted together into a big pattern, which was called Moonshine.

The point is this: the coefficients that appear in the j-function, 196,884, and then the next one, which is something like 21 million, these can all be viewed as dimensions of spaces on which the Monster acts, but somehow one had to understand it that way.  So what Frenkel, Leopwski and Meurman did was they put all of these spaces together.  And by the way, the j-function has infinitely many coefficients, because it is a power series in mathematical terms.  So you would have an infinite dimensional space, and they studied this thing and found it had the structure of a vertex algebra.  Now, vertex algebras are important in mathematical physics in string theory, where the interaction of elementary particles, each of which is a string, are described by these vertex algebras.  So, apart from a connection between symmetry and number theory, there now was a connection with string theory.

There were certain conjectures made by John Conway and Simon Norton, and Richard Borcherds proved these conjectures.  He produced the missing pieces that you needed to actually prove that they were true and that connected with this big infinite dimension or vertex algebra.

Borcherds is a man who is in Berkeley and stays there; he doesn't go anywhere - he just stays in Berkeley.  He is originally from England, but he is very happy and so just stays in Berkeley.  He doesn't like travelling.  So I was out in San Francisco once and I went to talk to him.  I said, 'So Richard, it's clear that the Monster is in fact connected to physics.'

He said, 'No.  It's clear that the Monster is connected to string theory.  But whether string theory is connected to physics?!'

But there are still mysteries here, and I wanted to just give you one little mystery.  When John Conway and Simon Norton were working on this Monstrous Moonshine, what they had was this huge 194-by-194 Sudoku puzzle, and each column in this thing would give a different, what I may call, j-function.  They found that these functions were not all independent of one another, and they started to wonder what dimension the space would be that these functions would span, and so you had to start crossing things off - 'well, this one is the same as that one, so that doesn't add anything,' and you start to cross things off.  They were crossing off and they were getting down into the 160s - from 194 straight down to 171, and then they started counting down to the 160s.  John Conway said, 'Let's guess what number we reach!  I guess 163.'  And that is exactly what happened!  What is special about 163?  Well, e to the power - times the square root of 163 is almost a whole number!  There is a mathematical reason for this and if I had more time, I would give some other examples.  It is whether this is connected, whether it is something we don't understand.  So there are still mysteries to find out here.

When the Monster was first discovered, and this would be back in the very early '80s, Freeman Dyson, a famous physicist and mathematician at Princeton, said he had a sneaking suspicion, unsupported by any evidence, that some time in the 21st Century we would find the Monster, somehow embedded in the very structure of the universe.  This seems like a nice thought to leave my lecture on.

Thank you.

©Professor Mark Ronan, Gresham College, 20 February 2008