Sporadic Groups - Prof. Richard Borcherds - The Archimedeans

i'd like to introduce our speaker for today we are very privileged to have with us professor richard barchards uh richard did his undergrad right here in trinity and he stayed here to do his phd supervised by john conway after his phd he introduced the idea of vertex algebra which he then used in 1992 to prove monstrous moonshine an important theorem relating monster group to modular functions and this was conjectured by conway and simon norton a decade earlier for this work professor borchards was awarded the fields medal in 1998 since then he has moved to uc berkeley where he is a professor currently and he has worked on applying these ideas of his earlier work to quantum field theory he's a fellow of the royal society of the american mathematical society and the national academy of sciences so today he'll be talking about uh some group theory we're very glad to have you with us today richard um okay that seems to come up with a with a reasonable first slide so this talk is um going to be about the sporadic groups um so i'll start just by recalling what a group is so a group is a a collection of symmetries of something and symmetries of something uh um well that's sort of what makes things look pretty so here are some examples of things with various symmetries so for example we have here a flower with some sort of eightfold symmetry and diamonds are considered nice possibly because they have various symmetries and here we see a potato and potatoes are not generally regarded as very artistic objects and as you can see a potato has very little symmetry um down at the bottom we've got a football which has quite a lot of symmetries either 60 or 120 depending on how you count them um so um we recall what a group is so it's just a collection of symmetries of something so for example if you take a tetrahedron um it's got 12 symmetries because it's got four faces and you can um so there are four ways to map some face to the bottom once you put a face on the bottom there are then three ways to to twist it so they're all together 12 symmetries um similarly a cube has 24 symmetries unless it's a unless it's some sort of rubik's cube in which case its symmetry group is very much bigger um because you can also twist the faces um so i'll just recall from the group theory of course a number of elements in a group which is the number of symmetries is called its order so for instance the the the symmetries of a tetrahedron has order 12. um and you also need the the order of an element of a group is is the number of times you have to repeat it to get back where you started so for example if we rotate a tetrahedron by fixing one of the vertices if we do that three times we get back to where we started that element is order three um so um a fundamental problem in groups here is to find all groups and obviously this is hopelessly difficult um you can at least start by finding all finite groups and you might try and find all finite groups of a given order and there can be more than one sort of group of a given order so for example if you take order four there are two different groups and i've sort of drawn pictures of them on this slide so at the top um there's a lot of objects with a symmetry group which is one of the groups of order four this is the climb four group um and you see if you take a rectangle you can um either flip it along a horizontal axis or you can flip it along a vertical axis or you can rotate it by 180 degrees or you can do nothing so it's got four symmetries and all these other objects also have four symmetries if you look at the bottom of the slide you have various other objects with four symmetries because these ones you can rotate by 90 or 180 or 270 degrees and you can see these two groups are different because they've got different numbers of elements of order two for instance that all the the groups on the top half order three elements of order two and the groups on the bottom half have only one element of order two so we have two different groups of order four um so here's a puzzle for you if you take a tennis ball it's actually got a symmetry group of order four and so the question is which of these two symmetry groups of order four does a tennis ball have so leave that as a something for people to think about um slightly unexpected answers what is the set of symmetries of an apple and you might think it's got an infinite number of symmetries because you can rotate it but if you cut it in half you find it's got um this this funny sort of five-fold symmetries it's actually got five seed pods arranged like this and if we allow reflections as well it's got ten symmetries because um we can either do nothing or there are there are two different sorts of rotations by um one-fifth of a revolution or we can have two rotations by two-fifths of a revolution and there are five symmetries where you just reflect in a line for instance you might reflect in this line here um and there are obviously five of these so it's got ten different sorts of symmetries and um these four classes of symmetries are called the conjugacy classes so roughly speaking a conjugacy class of a group is a set of collection of symmetries that sort of look the same in some sense um there's another group of order 10 which is just the rotations of a of a a 10-gon um and you can see these two groups are different because this this one whoops this one here what's it doing that has has four symmetries of order ten whereas um the symmetries of an apple has no symmetries of order tense so at least two groups of order ten and you can continue this and ask you know for each small order how many groups of that order are there and there's always at least one group of any given order because you can take an n sided figure and for instance for order seven you can just take a seven-sided polygon and just just look at its group of rotations um it's sort of not quite clear what the symmetries of a of a seven-sided polygon are because you could either just look at rotations or you could look at rotations and reflections so so you get a group of order seven and a group of order fourteen but there's at least one group of every order and you find well we've shown there are two groups of order four there are two groups of order six because we have rotations of a hexagon or rotations and reflections of a triangle um there are five groups of order eight which i'm not going to write down because um they're a little bit complicated um and you can go on to ask um how many groups are there of higher order and the answer is there are rather a lot so if we go to order 64 for example um there are 267 groups of order 64. um and um these were classified by hall several decades ago before computers um there are this ridiculously large number of groups of order two to the ten which were classified by computer um so you might think the number of groups of given order really increases really fast with the order of the group but it doesn't because if you look at groups of order a million there's only one group of that order um it turns out sorry a million and three never mind so um the number of groups of order n doesn't really depend very much on the size of n it depends very much more on the number of primes in the prime factorization of m and it depends um even more on on what the highest exponent of some prime is so here we've got a um if if the order is divisible by some prime to the power of 10 then that means there are going to be a absolutely incredibly large number of groups anyway um the conclusion is that there's no reasonable way to classify all groups of order n there are just too many of them um and they're just a confusing mess so for example if we look at one of the groups of order 64 we can ask what its subgroups look like and here is a sort of picture of its subgroups so the subgroups of this particular group of order 64. all of order one two four eight sixteen thirty two or six two four so so here we have the the group of order 64. here are some of the subgroups of order 32 and order 16 and order 8 and order 4 and order 2. and so you can see the structure of these groups can be quite incredibly complicated um this is worked out by hand in fact hall worked out the subgroup structure of every one of the 267 groups of order 64 by hand and drew these rather amazing pictures of them um so the question is how can you study groups and this was sort of um the the first idea for doing this was was done by galwa who um um he's rather famous for dying at the age of 21 when most of us are just doing our undergraduate exams so so um um you know you he he managed to do something that made him famous for the rest of time before most of us have done anything at all anyway galwa came up with this idea that you could divide groups into um basic groups that the the that are called the simple groups so roughly speaking some groups can be kind of split up into smaller groups and the ones that can't be split up into smaller groups are called simple groups and the name simple group is not very good the groups aren't really simple at four but we're stuck with it so an analog of simple groups is to compare them with um chemical elements so if you're trying to classify all chemicals and you obviously can't possibly classify all chemicals so for example here is a typical chemical it's a it's a piece of a dna molecule and as you see it's incredibly complicated and there is simply no way you can possibly understand all possible chemicals so so so what chemists do is as they discovered that you every chemical is built out of elements and you can at least um find all possible elements so here's a um a collection of all known elements which come to think of it needs somewhat dating because they haven't named a few of the ones at the bottom um so the the the connection between groups and simple groups is is just like the the connection between compounds and elements we can't classify the compounds but we can classify the elements they're made of and we can't classify finite groups but we can at least classify the the simple groups that they're made of um in fact um here's a picture of the o of the simple groups there's this guy called ivan andrus who has a sort of joke invents arranged all the known um all the finite simple groups into a table resembling the periodic table um it turns out that the number of the the the simple groups can be classified into some infinite families and some left over the number of infinite families just happens to be the number of columns of the usual periodic table and this is just a meaningless coincidence i mean the the groups are arranged like this just as a joke there's no real significance to it um so so this is a sort of picture of of all the finite simple groups um so here's uh here here's one example of one of the finite simple groups so um first of all there are some rather uninteresting finite simple groups which are just the cyclic groups of prime power order the first simple group that isn't of prime power order is is the group of rotations of an icosahedron um so here's a collection of objects um with with that symmetry group so um this one is actually a virus um i i was trying to find a picture of a chrono virus um to be on topic but uh as far as i can tell from pictures of chronoviruses chronoviruses don't seem to have icosahedral symmetry but but quite a few of them do this one's a molecule of buckminster fullerene um and um uh this one here is um a reflection group which actually is order 120 but if you take the subgroup of the rotations of this reflection group it's again the icosahedral group so the icosahedral group turns up quite a lot um so here's um here's an actual electron micrograph of a virus if you don't believe it has icosahedral symmetry so on the left is the actual virus um it's not entirely obvious that this is lycosahedron because it's a bit fuzzy so so what i've done on this picture on the right is i as i put a white spot on each vertex of the icosahedron so you can see this vertex this vertex and this vertex will form a little triangle and here's a picture of the virus drawn a bit more clearly so if you look very carefully um the these circular things you can see on the left aren't actually atoms they are they are um molecules that make up the envelope of the virus there's some rather clever molecule that assembles itself into like her saheedra um so um sorry this thing here is is a little animal called a radiolarian and here are some more pictures of them um there are some absolutely stunning pictures of these drawn by this guy called heckle um who who who drew these very famous pictures of various animals and plants so here are some of his pictures of radiolariums um they have this little skeleton made out of calcium carbonate i think and these skeletons are often um in the form of icosahedra or something like that and you know i've often thought that you know maybe you know there are some people who like to bash evolution saying how on earth could various things possibly evolve and um they they don't seem to sort of radial areas because it's a rather puzzling question how do you evolve an icosahedron i mean you know evolution normally works by making small changes in order to get something it's very hard to see how you can make small changes to something and end up with an icosahedron i mean what are you going to do have a have a skeleton that consists of 17 triangles i mean it sounds kind of ridiculous so i've no idea how these radio layerings manage to evolve these icos evil skeletons so the question is how do you classify the finite simple groups and here's a picture of the classification the classification is the longest and most complicated proof in mathematics at least the longest that has been actually written out in journals there are some much longer com proofs whose proofs consist of computer calculations but those don't really count so so this pile of books here is roughly most of a proof of the classification and as you can see it's several feet high [Music] so so here's the conclusion of the classification of simple groups first of all there are 18 infinite families of groups and 26 left over called the sporadic groups which are what i'm mainly going to talk about so typical examples of simple groups of the cyclic groups that i mentioned earlier in the icosahedral group of order 60. some examples of the spread it groups and the smallest spreading group is the mature group m11 of order 7920 and the largest of the spanish groups has been named the monster group and the reason for the name is pretty obvious if you look at its order which is somewhere around 10 to 54 it's it's more than number of atoms making up the planet earth if you want to get some idea of the size of this group um so here are some examples of the infinite families of simple groups first of all they're the cyclic groups which are the obvious ones the next family of simple groups that most people come across are the alternating groups um so the alternating groups are almost the same as the symmetric groups so the symmetric groups is just the group of all permutations of n objects so the symmetric group is order n factorial and it's not quite simple but it has a simple subgroup of order two called the alternating group um so this is simple provided n is at least five the alternating groups of order for any one or one two three or four elements are not simple another typical family of infinite simple groups is a general linear group um at least if you ignore the fact that this isn't quite simple um so so the general linear group consists of all matrices n by n matrices over f q which is the field of q elements where q is a prime power and it's not difficult to work at its order it's orders given like this and it's not quite simple because um in order to get a simple group we first have to take this subgroup of elements of determinant one and then quotient out by the center and if you do this you get a group called psl2 fq the p stands for projective which means quotient out by the center and the s stands for special which means take determinant equal to 1. so the groups psl n of fq are simple except they're sometimes not because there are two exceptions and this gives you some hint of why the classification of simple groups is so complicated because there are exceptions to almost everything so um the groups psl nfq are simple except there are these two exceptions and these two exceptions have to be dealt with somewhere in the proof and um the there are exceptions all over the place turning up from classification all which make the proof longer and longer because you have to separate arguments dealing with all the different exceptional cases um well as well as the general linear groups um you can form analogues of things like orthogonal or unitary or symplectic groups and so any any group you can define over the real numbers in other words a lead group has has a sort of analog over finite fields and that gives you some of the infinite families of simple groups so the question is how do you find all the simple groups so so here's a very rough outline of the proof so here i'm compressing about 10 000 pages of complicated argument into three or four lines so obviously i'm going to miss out a few of the details the absolutely fundamental idea of the classification is due to brower who suggests you classify symbol groups by finding the possible centralizer of an involution so an involution of a group is an element of order two and the centralizes the things that commute with it and the the the way the classification works is that um you first of all find all possible centralizers of involution and for each centralize of involution you find the simple groups of that centralizer well in order to start off the classification you've got to show that every non-cyclic finite simple group has at least one involution otherwise you can't get started um i missed that word none cyclic down here so it's it's not true that every finite simple group has an involution and the start of the classification is usually dated to this theorem by um walter fight and john thompson saying that every finite simple group has an involution and this theorem is the scariest theorem in mathematics i know about and let me show you why it's scary i'm going to show you um their proof let me see okay here we have the um classification of um simple groups of odd order by water fight and john thompson so here that their paper is called solvability of groups of odd order this paper takes up an entire issue of the germ also it's about 250 pages long and it is amazingly scary let me show you some of the um bits of it so let me try and zoom in a bit so here for example we have a definition somewhere in the middle and the definition starts up here and goes on down here and covers a fair amount of the next page as well so we have definitions that are more than a page long if i go to a later part of the proof um let's look at this page here so this is a bit of the proof of one of the following lemmas so you can see the proof consists of defining these weird numbers here and we have this um incredibly complicated calculation i have no idea what is going on in this calculation and just onto a few lines of this this sort of calculation goes on and on and on for several pages um i have just no idea how water fighting john thompson managed to think of this this proof um now the problem is that well so you know when i see a proof in mathematics and difficult proof i i normally think to myself well you know i could have found that proof if i'd worked a bit harder um but um the the the solvability of groups of odd order by fight and thompson i just can't fool myself there is no way i could possibly have found that proof um no matter how long i've worked um [Music] so let's see if i can get back to the my slide so so that that that is the proof by fighting thompson that every finite symbol group has an involution right that is you know maybe 200 pages the full proof of the classification is maybe 50 or 100 times longer than that paper by faison thompson and quite a lot of it is just as difficult and as complicated as the proof by fighting thompson so we have 10 or 20 000 pages of impenetrably complicated arguments to classify the finite simple groups um so let's give an example of a little piece of a proof so one particular possible centralized evolution might be a group of border two that's c modulo two z times the group psl2 of fq that's more or less two by two matrices over over a finite field of order q now if q is a power of 3 the groups with this as a centralized of an involution turn out to be one of the more obscure infinite families called the regroup so they're an infinite number of them if q is anything other than a power of three you usually get nothing with one exception if q is equal to four or five then you get a new simple group of order one seven five five six oh this was a very famous group discovered by yanko in the early 60s and it caused a sensation because um it was the first new sporadic simple group that had been discovered for almost a century i mean people have been discovering a lot of groups that are similar to league groups over the reels and we're hoping that was all of them but then yanko upset everybody by finding this group um so um the first sporadic groups to be discovered were actually discovered almost a century earlier by the sky mature um these groups are called m11 m12 m22 m23 m24 where the m stands for mature and the numbers 11 12 22 23 and 24 come because these groups are groups of permutations on 11 or 12 and so on points so for example m11 um acts as permutations on a set of 11 points and has order 7920 if you want to see an explicit description of m11 it's not terribly difficult um we can take these two permutations so so this permutation here is just a cyclic permutation that takes one to two two to three three to four and so on whereas this is a permutation that takes three to seven seven to eleven eleven to eight eight to three and then takes four to ten ten to five and so on and if you take all permutations you can obtain by multiplying these two together you get the group m11 so it's not difficult to define m11 although as you see this definition by two permutations doesn't really give a whole lot of insight into the group you have to do a lot of calculation to see what's going on um similarly m12 has order nine five zero four zero and it's actually an example of something called a five-fold transitive group um for any 12 points sorry for any five points there's a unique element of m12 taking those five points to any other five points you choose that's why it's order is 12 times 11 times 10 times nine times eight um you can actually describe m12 is a sort of sliding block puzzle fairly easily if you take the projected plane of order 3 it has 13 lines and 13 points and it's order 3 because each line has four points um so i guess somebody miscounted when they were figuring out what the order was um but what you can do is you can put 12 counters on 12 of these 13 points you have a sort of sliding block puzzle we're allowed to move any counter to an um you're allowed to slide it along one of the lines to another to an empty point but whenever you do that you have to switch the other two counters on that line and if you do that um you get a sort of sliding block puzzle and m12 consists of the permuta or or all permutations you can get by sliding things around that leave the empty point in the same place so now i want to talk about a rather famous simple group discovered by john conway john conway found this by looking at a certain sphere packing so you can ask the question what is the densest way you can pack spheres in some given number of dimensions well one-dimensional sphere packings are kind of trivial two dimensions you can pack them on a sort of square grid like this if you like and that's obviously not the best possible packing you can pack them more tightly by packing them hexagonally um so in three dimensions um this was actually a problem that the british navy worried about because they wanted to store cannonballs in the most efficient possible way so so they actually got some guy to figure out what the best way of storing cannonballs on the ship was i think this was possibly the earliest study of sphere packing um and um there are actually several ways of doing this um first of all we can take a cubic packing um so you might take something like a salt crystal and that they're packed in a cube and this this is obviously not the best sort of packaging um there are two obvious dense packings first is one called face centered cubic where the spheres are arranged like this there's also something called a hexagonal close packing so here's an example of hexagonal plus packing so what's the difference between face centered cubic and hexagonal close packing well suppose you take a hexagonal layer so so all these white um circles are going to be the lowest layer of spheres and then um you can put on a second layer of spheres which are these black ones and then it turns out there are two ways to put a third layer of spheres whoops because you can put the third layer there either above the first layer not like like this one here and if you do that you get hexagonal close packing and if you put them so they're different from the layer that's two below you get face centered cubic so you you can think of there are actually three ways to put the spheres in each layer you can put them in layer you can put them in positions a or positions b or position c and face centered cubic you sort of put the successive layers a abc abc and so on whereas um hexagonal close packing you put them in layers a b a b a b and so on um in fact there are an uncountable number of other equally dense ways because you can choose any collection of letters with no two consecutive letters and that gives you a dense packing princess you can patent like a b a b c a b a b c and so on so there are an uncountable number of equally dense packings in three dimensions so obviously you can ask what is the best sphere packing in various dimensions and in dimension one this is trivial in dimension two it's easy in dimension three face centered cubic is the densest possible packing although it's not unique um and this is incredibly difficult to prove as i mentioned in a moment in dimensions four to seven we haven't managed to prove what the answer is i mean we we sort of think we know the answer but we can't prove it um in eight dimensions we have managed to find the densest packing this is something called the e8 lattice that i'll describe a little bit later in dimensions 9-23 we're not sure what the answer is in dimension 24 we know the densest possible packing something called the leech lattice and in dimension greater than 24 we don't know so so this is very weird you would have thought the next most difficult case after dimension three was dimension four but we don't know how to solve it in four dimensions you have to jump to eight and twenty four dimensions and so something very bizarre is going on in dimensions eight and twenty four um so the proof the best sphere packing three dimensions was done by thomas hales in 1998 um this was the kepler conjecture and this is one of the longest proofs in mathematics it's far longer than the classification of finite simple groups but it cheats because it uses computers and so the proof consists of several terabytes of computed data so if you wrote it all out it would be um vastly bigger than the classification of finite simple groups um the the fact that the e8 is the densest packing in eight dimensions was proved by marina villaska in in 2016 who came up with this absolutely stunning proof of it um it was this was an extension of some work by by henry cohen and noam alkeys who who showed that e8 would be the densest eight-dimensional sphere packing provider you could find a certain magical function with with certain rather bizarre properties and quite a lot of people trying to find such a function i i spent a few hours on it and gone absolutely nowhere and marina villaskia managed to come up with this absolutely incredible way of constructing such a function um i must admit i tried to understand her arguments and can't really figure out what's going on um so how do we describe the sphere packing in in eight dimensions well the easy way to describe a sphere packing is to give the coordinates of the centers of its spheres so if i want to describe the usual cubicle packing like this one here i can just say the centers are all numbers a b c with a b c integers um and you can see there are six spheres touching a given sphere um for face centered cubic you that the sentence consists of all numbers abc whose sum is even and then you can see there are these 12 spheres or touching the sphere at the origin for e8 you do something similar you take um these numbers must either all be integers or they must all be introduced plus a half so for example um these vectors here are all in the lattice packing and again the sum must be even and now you can see there are 240 pieces touching the sphere at the origin because there are 128 vectors like this you might think there are 256 because there are eight choices of sign but in fact the sum has to be even so that cuts things down by a factor of two and then there are um eight times seven times two um vectors of this form so that gives you another 112 sphere so all together we have 128 plus 112 spheres touching the sphere at the origin which is 240. um so we've got an arrangement of 240 spheres in eight dimensions and um here's an attempt to draw a picture of it this is a this is the this is the 240 spheres and eight dimensions except they've been projected down into two dimensions so um and there's a line drawn between two of the spheres in uh if i think if they're touching each other in eight dimensions you get this rather complicated picture here um so um the automorphism group of this of this the e8 lattice is actually a reflection group so a reflection group is a is a group of symmetries of euclidean space or a sphere generated by reflection so here are some examples of reflection groups so so the first three are just euclidean reflection groups um you see that there are lots of reflections that preserve these patterns for instance if you reflect along this line here it preserves this triangular pattern only it exchanges black and white diagrams and similarly this is another reflection group these three spheres at the bottom are examples of spherical reflection groups you see you can reflect in these great circles for instance there's a great circle um here at the bottom and this gives you three spherical reflection groups and you can also get reflection groups hyperbolic space which i'm not going to talk about much but here are a couple of examples so on the top right here we have a sort of model of the hyperbolic plane and these are in fact triangles with straight edges and the reason these edges look curved rather than straight as they get a bit distorted um because um we're working with hyperbolic space rather than euclidean space anyway so so this example here and the example here on the bottom left give you examples of hyperbolic reflection groups um so here's so the e8 reflection group um the symmetries of the lattice is actually a reflection group and you can describe it explicitly um because it's just generated by 120 reflections which are reflections in the hyperplanes orthogonal to the 240 vectors giving you the centers of spheres um you can think of this is in in three dimensions the analog would be that the symmetries of a cube are generated by reflections um giving you a reflection group that looks like um this you you're reflecting in the in these great circles so so the reflection group of the e8 lattice is sort of similar but lives in in eight dimensions rather than three dimensions now we get to the symmetry group of the leech lattice in 24 dimensions that was investigated by john conway this is a picture of john conway demonstrating well one of the things he was most proud of was that was that his his tongue was more flexible than the tongue of anybody else he ever met and he he claimed he could do some very large numbers of weird patterns with his tongue and here he is demonstrating one of them anyway among mathematicians he's actually best known for the conway group which can be thought of as a symmetry of a sphere packing so here we have a sphere packing in two dimensions and it's got 12 symmetries because you can rotate it or reflect it in three dimensions it's not difficult to figure out the sphere pattern is 48 symmetries in eight dimensions we get this e8 sphere packing and the number of symmetries is is this large number here in 24 dimensions every sphere touches one nine six five six oh other spheres and the symmetry group was worked out by john conway and he got this rather impressive number um and this was particularly exciting because this symmetry group was more or less a new sporadic simple group um you have to sort of quote it out by standard order too but anyway um so john conway said you know this was sort of made in world famous he went around the world giving lectures on this new group and i actually found three different groups because there are three there were three new ways of getting spread groups out of the leech lattice so how do you construct each lattice well let's go back to this question ask how many cannonballs are there in a pile well the top layer has one cannonball the next layer is two squared the next layer is three squared and so on and so the total number of cannonballs is one squared plus two squared up to n squared and now you can ask the apparently silly question um can we find a pile of cannonballs like this such the total number of cannonballs is a square and apart from the trivial solution when there's only zero or one cannibals there's only one solution you have to have 24 layers and there are 70 squared cannonballs and we can use this to construct the leech lattice what we do is we start with 26 dimensional laurentian space what this means is it's like euclidean space except the distance is defined in a funny way you have a minus sign in front of one of the variables so if you've done special relativity you've come across this except in four dimensions it's just called laurentian space um and you can sort of define distances in the usual way except that sometimes distances are imaginary because this number here might be might be negative and that gives you time-like vectors and so on and special relativity anyway i'm going to take the following vector w which is 0 701 2 up to 24 and the key point is that w has length 0 because the sum of the first 24 squares is 70 squared so so now we're using this weird fact that the whoops that the if you've got a pile of cannonballs with 24 layers it has a square number of cannonballs and now we can use this norm zero vector w to form the leech lattice as follows we do is we take all vectors such that the coordinates are integers or half integers and the sum of the coordinates is even so this is exactly what we did for constructing e8 lattice and now we take all vectors orthogonal to w and we should actually divide out my w as well but i won't worry about that and that gives you the leech lattice so the leech lattice in 24 dimensions depends on this rather weird fact that the sum of the first 24 squares is a square and conway's group is the group of symmetries of the leech lattice um obvious question is can you do something like this in any other number of dimensions and the answer seems to be no this is something weird that only happens in 24 dimensions um the rest of this talk i want to explain john mccain's t-shirt so john mccain is a canadian mathematician and he had a special t-shirt custom made pointing out his great discovery that one nine six eight eight three plus one is equal to one nine six eight eight four and let me explain what the significance of this is um well um 196883 is the dimension of a representation of the monster simple group and 19684 is a coefficient of an elliptic modular function that i'll explain later and john mccain pointed out these two numbers only different by one and most people dismiss this as being a meaningless coincidence so if you get a large number of numbers you can find weird coincidences between them for example e to the pi minus pi is very very close to 20. and as far as we know there is no significance to this it's just this meaningless coincidence and um john mccain when he put out this was told by several people this is just another meaningless coincidence um so um um the explanation um involves um something called the elliptic modular function or sorry skiptor slide um another apparently weird coincidence is if you take e to the pi times root of n then these numbers here are often very close to being integers so here are three examples they aren't always for instance if we take e to the pi times the square root of 1. this is e to the power of pi and this is 23 which isn't nearly an integer it's very nearly 20 plus pi but as far as i know uh there's no good reason for that um anyway so what's going on here why are these numbers here very nearly integers um well the explanation involves klein's elliptic modular function so so here's a picture of felix here's a picture of his bottle which he's most famous for and by the way it wasn't originally called a bottle it was originally called klein's surface but the german words for surface and bottle look very much the same and i think some translator got a bit confused between them so um actually apparently although it was originally called a klein surface in german even the germans have now started calling it a klein model but anyway um so here's klein's elliptic modular function um it looks like a really bizarre definition for a function um i mean if you look at this definition here it looks like someone wrote it down as a sort of silly joke you can work at its power series expansion and it looks like this so here's this number one nine six eight eight four that appeared on john mccai's t-shirt um and what's the point of this function well it turns out to be the simplest non-constant function that has the following three properties so it's invariant under taking tau to tau plus one and it's also invent under taking tau to minus one over tau um functions in event under tau to tau plus one are very easy to define these are just periodic functions and they're just fourier series and e to the two pi i tau has period 1 so any function of q will satisfy the first the first equality it's the second equality that's very difficult to arrange and this really does turn out to be the simplest solution um so here's a picture of the graph of klein's elliptic modular function well it's actually a slightly different elliptic modular function um so what's going on here is that this is the upper half plane in the complex numbers so down here we have the real axis and you see along the real axis there's this whole chain of poles where the function becomes infinite and if you go away from the real axis the function becomes becomes very small so it's as you can see it's a really complicated function and by the way this picture has only drawn a few of the poles for simplicity that they're really an infinite number of smaller and smaller poles getting denser and dense along the real axis so um [Music] you should really think of them as just a wall of poles not not just a finite number of them um so now we can explain why these numbers like e to the pi root 163 are almost integers well klein's elliptic modular function is given um like this and if you take q to be the number um e to the minus pi root 163 then the elliptic modular function is exactly an integer and the reason for this comes from a mathematical theory called complex multiplication and the integer turns out to be 640320 cubed on the other hand the number 1 over q is approximately e to the pi root 163 which is this number up here this number is an integer and this term here one nine six eight eight four q turns out to be incredibly small i mean you you might think it's rather big because one nine six eight eight four is big but it turns out that q is so small that all these terms here are incredibly tiny so the result is that e to the pi root 163 is is very nearly this integer here um plus 744. um anyway um now we should talk about the monster group so the monster group is the largest of the sporadic simple groups um so the icosahedral group lives in just three dimensions and has 60 symmetries conway's group is in 24 dimensions and has this number of symmetries the monster group lives in one nine six eight eight three dimensions this is the other number appearing on john mccain's t-shirt and its number of elements is given by this number here that i'm not going to read out explicitly so it was a and discovered by um fisher and grice um in the early 70s um so uh well well or rather fisher and grice at first suggested the possibility of of a group like this existing but for a long time it was unclear whether there was such a group because it's actually incredibly difficult writing down such a group i mean what are you going to do write down matrices at 196 883 by nine six eight eight three they would have about a billion entries i mean it's it's you know you can't write them down and calculate with them um well you couldn't with the computers at that time um well it was its existence was originally proved by robert greist who actually managed to more or less write down explicitly matrices in this number of dimensions i remember i was at cambridge when this proof came out and the when robert grice managed announcing construction this group um people at cambridge basically could almost not believe it because you know they'd known it was possible in principle to construct the group by acting as matrices in one nine six eight eight three dimensions but assumed the calculations would be so phenomenally difficult that no one could possibly do them but robert grice spent several months working really really hard on this problem and managed to come up with this you know paper that was well over a hundred pages long where he showed how to construct um suitable matrices constructing this group um his construction since been simplified quite a lot but it was a really incredible construction when it came out anyway um so you can look at the monster character table now the character table of a group tells you what dimensions it can live in and here is a very tiny piece of the monster character table and so we see this entry here is a one and here's one nine six eight eight three which is the smallest dimension it lives in and it also lives in two one two nine six eight seven six dimensions and so on with all these numbers um the full table actually has about 194 rows and 194 columns so i've just shown you the the top nine or ten rows of it anyway if you remember if you remember the elliptic modular function had these coefficients and it turns out the coefficients of the elliptic modular function are very closely related to the entries in this column of the monster character table these are the dimensions the monster can live in so the first row we get one nine six eight eight four is one nine six eight eight three plus one um well if that was the only information you had you might write it off just as a meaningless coincidence however the next coefficient is equal to the sum of the first three dimensions of representations of the monster and the next coefficient is also equal to a simple linear combination of dimensions of the monster and at this point you can no longer say this is just some sort of meaningless coincidence um so how can you explain this well john thompson suggested there should be an infinite dimensional graded vector space acted on by the monster such that the dimensions of its various pieces were given by these coefficients and this was constructed by frank lepowski and merman in the early 1980s um so so we've got a vector space v which is a sum of finite dimensional pieces v n and the dimension of each space v n is one of the coefficients of j um i should say that you have to sort of throw away this number 744. um the elliptic modular function is only defined up to an arbitrary constant and this this number 744 um only occurs for historical reasons that there's no great significance to it so so you throw it away and all the other coefficients are coefficients of franklin powerskin merman's algebra um so this um graded vector space is actually related to string theory so you remember we construct the leech lattice by taking a 26 dimensional laurentian space and then fiddling around with it well um 26 dimensional laurentian space is all that much used for physics because we live in a four-dimensional anteater space not a 26-dimensional one however one of the more promising approaches to high-energy physics is string theory and the early versions of string theory only worked in 26 dimensions and this is a bit of an embarrassment for physicists working on string theory because um you know we don't live in 26 dimensions however it's um although physicists are not very happy um 26 dimensions is very nice for mathematicians because the 26 dimensions of string through turn out to be exactly what you need for dealing with the monster single group and the leech lattice so um frank le pasque in merman's construction of of this graded vector space in the monster action in fact uses um things called vertex operators coming from string theory and again um there's something called the no ghost theorem in string there you need in order to get things working properly which only works nicely in 26 dimensions um so um so that all works for the monster's simple group it's it's got a graded vector space um whose coefficients are dimensions of weird functions you can ask does something similar happen for the other um spreading simple groups the answer is it does for some of them for instance another of the groups fissured and discovered is called the baby monster group um in fact fischer discovered this before he found the monster group and he found the monster group by by suggesting the baby monster group or rather a double cover of it should be the centralized of the involution in the monster group anyway if you look at the baby monster group it lives in dimensions one four three seven one nine six three five five sorry that should be a two there that's a misprint 96255 and so on and there's a more complicated version of the elliptic modular function whose coefficients look like this and you can see this coefficient 4372 is very close to this coefficient 4371 and similarly all the other coefficients of this function are related to representations of the baby monster group and many of the other sporadic simple groups are also related to elliptic modular functions in a similar way there are however some variants of this that haven't been explained yet so if you want to um do some uh try some research that isn't understood yet we can take um a mock theta function so theta functions were introduced by remanager in one of his letters to hardy just before he died and one example of a mock theta function has fourier coefficients that look like this on the other hand if you take the mature group m24 that we discussed briefly earlier it lives in dimensions 23 45 231 and so on and now if you look at these coefficients 45 231 770 2277 they're the same as these numbers forty five two three one seven seventy two two seven seven and so on and nobody knows why um so there are quite a lot of other variations of this connecting m22 to various lattices in 24 dimensions called nema lattices so it's clear this is not just a coincidence but no one has really come up with a good explanation for it you can ask is there some sort of algebraic structure like a vertex algebra that's underlying it um well maybe but it's a bit difficult because this coefficient here is minus one which means you would need a minus one dimensional vector space which is a little bit hard to deal with so we have an open question why do these numbers turn up in two different places um i just finished with a sort of philosophical question why does sporadic groups exist at all um it's i find it's absolutely bizarre because if you look at the actions for a group they're completely trivial you can write them out here and they just they're just half a line long and they're very natural and simple and somehow the monster group and all the complexity of the classification of finite single groups is hidden inside these almost trivial axioms so we can ask is there some sort of uniform explanation of the spread it groups um you know why do the spread it groups exist at all um is the classification really that complicated or are we missing some fundamentally simple approach to classification which would explain what's going on so um i think i'll just finish there just by leaving um a few suggestions of further reading so if you want a popular book on the monster and spread it groups there's one by ronan if you're feeling really ambitious and want to find out what's really going on then two um good sources for this the book by conway and sloane on sphere packings lattices and groups that describes the leech lattice in detail and if you want a survey of the finite simple groups there's a very nice book by daniel gorenstein on finite simple groups um okay i think i'll stop there and answer any questions that the audience has okay yeah thanks a lot this is really interesting um yes yeah now we have some type of questions and i think i'll start off with one i'm kind of related to what you're talking about at the end um do you have any intuition at all as to why the number of uh spreader groups as finite no no one has this is a question people were discussing in in in the 1960s because most of the sporadic groups were discovered in the 1960s and 70s and people kept finding more of them and there was a lot of speculation about whether there were a finite or infinite number of them um for instance when the yanko group j-1 was discovered it looked a bit as if it was the first of an infinite family of new groups so people tried rather hard to find more groups like it but never managed to come up with any and no one has ever come up with a good reason why the number of spread it groups is only finite the only explanation for it is 20 000 pages of the classification of finite simple groups where you just show by brute force there aren't any others yeah thank you does anyone else have any questions yes you can put them in the chat or you can ask them out loud could i ask something sure um so it like you said at the start groups are the the study of symmetry in a way so do you find it unsatisfying that the classification is so unsymmetric and you know does it bother you in that way well it it it it it does sort of annoy me that it's too complicated for me to understand i mean i mean i sort of probably it's kind of obvious that even if i devoted my rest my life working eight hours a day trying to understand the classification i'd never be able to do it and it's just simply too difficult um my supervisor john conway on the other hand had the opposite attitude he was he was kind of disappointed that the simple groups had been classified and thought it'd be much nicer if there were an infinite number of spread groups and so we would never get to the end of them so uh is there a reason we like specifically tunneled in groups or is do you think there could be similar classifications of monoids or more complicated structures in groups um well it finite symbol group seems for the absolute limit of what we can reasonably classify so if you try weakening the axioms slightly like like going to semi groups or monoids you seem to get such an awful mess that there doesn't seem to be any sign of a useful classification um um on the other hand there that there are some related things like um lee algebras over finite fields and people have sort of classified these um the problem is that nothing terribly interesting turns up you you don't get anything any analogues of sporadic groups you you mostly just get known infinite families plus a few ones and ends that don't seem to do very much so i think this was a bit of a disappointment to people because a lot of techniques were developed for classifying finite groups and these techniques just don't seem to be useful for anything else in mathematics so they're going to be just forgotten because you know there's nothing else to do with them here's a question i say i got from someone um do you expect the proof of the classification to become simplified in the future or are people just going to leave it alone and say it's done you can move on to more interesting things people there some people are working on the classification and so um the the original proof can certainly be simplified and shortened quite a bit because you know when it was originally developed people didn't know what they were doing and you know you know prove things in ways that could later be simplified quite a lot so there is a revised calculation by um goernstein lyons and solomon i think gradually being published i think they published seven of the projected 12 volumes but it's still several thousand pages long and not easy reading um whether or not there's a fundamentally simpler classification that an ordinary person could understand their lifetime i i don't know people have sort of wondered if there is some fundamentally different way of classifying groups but no one has ever come up with any plausible way of doing this so maybe it really it really is that complicated can i ask if you have a particularly favorite simple group of your own well i i i guess the monster since that's the one i worked on most um or maybe just because it's the biggest one um the the the ones i'm the the ones that i'm most familiar with are either the monster or conway's group of symmetries of the leech lattice i guess those two would count as my favorites thank you i was going to ask uh is a particularly satisfying uh construction of a monster group because i've seen john conway's constriction simplification of the grice algebra constriction it seems rather piecemeal as in it controls it from below almost in a quite hodgepodged manner this is an open research question for anyone who's interested in find a natural construction of the monster every construction the monster we have is is a mess um most of the constructions are essentially simplifications of grice's original construction and the problem with all of them is you take two huge vector spaces which seem to have nothing to do with each other you take their direct sum and you put some sort of piecemeal algebra structure on this and by some amazing fluke it just happens with the monster as a group of automorphisms we we don't have a i mean you would never find this construction unless you already suspected the monster existed we're trying to construct it what we would really like is is as a sort of one-piece construction where you can structure just as in one piece for instance conway's group you take the leech lattice in 24 dimensions and that's a very natural single object in 24 dimensions and we don't think of anything like that for the monster yet i noticed that like you used the heger numbers i believe in when you're describing like how it like really showing how that also showed up in the elliptic curve so i wondered if like other higner numbers also showed up using the yeah yes or all the hegen numbers that are discriminants of imaginary quadratic fields show up that that e to the pi times the square root of something is very nearly an integer so those would also follow under the as in the monster group as well or would they be to like smaller groups sorry there was a bit of interference i didn't quite catch your question um so the question was like um so would those smaller numbers refer to like um smaller groups or were they also poor under this um monster group like relationship i i don't i don't think the heger numbers are directly connected with the monster group and they're just indirectly connected because both the monster and the hayden numbers are related to the electric modular function but i i don't i mean the the hegenera numbers don't really correspond to anything in the monster group by themselves there's a few more questions from the chat um yes first of all just a comment from gareth taylor saying that he remembers you lecturing modular forms back in 1997 he says it was his favorite free course um and since then we have a question uh asking so now that we have classified some fine and simple grapes does this mean that in some sense we have classified all the finite grapes or do you know what not not really because this is like the question if you've classified the elements have you classified all chemical compounds and obviously not because the ways you can see elements together are just so complicated that you can't really classify all compounds just knowing the elements and we have the same problem with finite groups that the ways you can join together finite simple groups to make a finite group are just too complicated to classify i mean even even if you just take copies of the finite symbol group of order two the simplest possible finite simple group and take a building of those then it seems to be just explain how you can join those together seems to be almost too complicated to describe um so no we we can't really classify all finite groups even knowing the simple ones um there is no simple uh and i can answer the other questions so someone has asked is is there a simple answer to how do we know there aren't any more spread of groups actually i think i i i just answered this a little bit earlier the answer is we there is no simple answer to why there are earlier why there are no more spanish groups the only answer is 20 000 pages of the classification can i quickly ask away um okay so um quick question that isn't so much about the talk with the people who you spoke about so in some of our lectures we always heard fun stories about conway doing weird and odd things do you have any fun stories of conway's oddities well there was the time he built a computer out of toilet components um i think he he once explained to us that if you look at the flush mechanism of a toilet it's really a sort of logical gate because you know you pour water in and once the water gets up to a certain level it gives an output and he claimed that he once built a a computer using these and it was he said it was exhibited in the guild hall in cambridge or something like that but they never invited him back because it had a some sort of overflow and caused a certain amount of damage to the flooring i guess it makes the concept of an overflowing computing rather more um dramatic than the usual meaning um i i have a sort of suspicion that that he he made up or exaggerated some of the stories he told about himself though so i take this with a grain of salt okay yeah thank you very much that's fun to hear yeah that was a great story thanks um does anyone else have any more questions um related to sporadic groups so i want to ask if tessellations in general are they related to classification of groups or well yeah yeah i mean tessellations are very similar to sphere packings i mean if you've got a sphere packing you can get a tessellation out of it and if you've got some tessellation you can quite often get a sphere packing so so you can regard the leech lattice as being a tessellation of 24 dimensional space for example do you mean for every tessellation we have a spare packing um well we have a spear packing and i'm not saying there's a one-to-one correspondence i'm just saying that that they're they're very closely related to each other so if you've got a destination you can quite often get us cool thank you very much um yes i have another question just kind of on a on the most basic level um how does any of this relate to beginnings of physics so uh yeah how does this relate to string theory or the quantum field theory why is any of this related um there's there's no short easy answer to that um well um the the the frank lepauskin merman used things called vertex algebras sort of vertex operators in their construction monster and the vertex operator in string theory is roughly something that tells you what happens when two strings are interacting and producing a third string so the word vertex actually comes from the vertex of a feynman diagram um so whether or not string theory is related to physics is a rather controversial question i mean physicists seem to be um have been arguing rather emotionally for several decades about whether string theory is really physics or not so i have to leave that question open um but certainly um i mean although even if it's not directly related from physics there are some ideas that go backwards and forwards um i mean the the vertex operators and string theory are quite similar to the um vertices of fun diagrams in quantum field theory yeah thank you um yeah and here's another question that i got from someone as so what's the relation to error correcting codes if they're saying relationships um yeah um well actually um i think one of the books i mentioned at the end was a book by thompson on error corrected codes and lattices and the relation to error correcting codes well there are several relations first of all you can use error correcting codes to construct lattices um so for instance the leech lattice was originally constructed using an error correcting code that the construction i gave um using 1 squared was 2 squared and so on equals 70 squared was only found later and there are other spradic groups that can be constructed directly from error correcting codes for example the mature group m24 is the group of symmetries of an error correcting code called the golay code which um is is a 12-dimensional subset of a 24-dimensional space over the field with two elements um for more on this see the book by conway and sloan which goes into much more detail okay thanks a lot um does anyone else have any more questions yes uh i was going to ask this is uh returning to the monster group uh towards the end of the conway paper there was a remark that simon norton had constructed a lattice inside the grice algebra by taking i think integer linear combinations of axes of transposition vectors but i couldn't find anywhere else in the literature anything about this particular lattice i was just wondering whether or not it's been investigated in greater detail anywhere yes um i think such lattice may have been constructed recently by carnahan if i've remembered correctly i think simon norton's lattice wasn't quite unimodular but i think the unimodular lattice has now been constructed um quite quite a lot of simon norton's things were never actually published so this is a common problem um yeah if if you look up papers by scott carnahan on the in on the archive you you may come up with something about this lattice thank you very much thanks any more questions right well if there aren't any more questions yeah let's all just say thank you to professor richard borchardt it was it was a really interesting talk yeah thanks a lot um yeah i learned a lot it was great to hear from you um yeah and so thank you to everyone for coming it's great to see so many people here um this is going to be as last talk of this term you

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Published: Mon Dec 21 2020