So I want to welcome you to the Mathematical Institute for Not a Public Lecture.
So this one is very special.
Martin Bryson started as head of department this October and we thought it would be a good idea for him to, to, to be to give a public lecture.
I see is the public face of the department and to present him to the whole public as well as the staff.
It's a very special one because it's an inaugural lecture, but is the first inaugural lecture for head of department in this department.
So it's the inaugural inaugural lecture of the department.
So before I introduce Martin, let me remind you that in case of emergency, we have exits here and there.
If you go through this door, you'll go down and then you find the exit.
But make sure not to disturb the ghost of all the head of departments, previous heads of department still roaming around into the never.
They never quite leave. Okay.
I actually wanted somehow we sung the previous head of the department, Professor Harrison,
to give the introductory remark so they could do all the usual jokes about being head of department.
But he was very wise. He left the country right after he was done, but he left a couple of notes for Martin.
So I hear the first envelope. So let's see.
It's quite short. Dear Martin, there is no money left.
Best wishes and good luck. Okay, so that settles it.
So now we know why he's not here anymore.
So the second one with letter number two and number two says to be open only in the case of a massive, massive invasion of Belgium mathematicians.
So I don't think that would happen. Okay.
So let me tell you a little bit about about about Martin.
Martin did his undergraduate here out for college in the Mathematical Institute.
After that, he left. And it is a Ph.D. in Cornell in New York State and worked for a while.
Professor at Princeton University, before he joined Oxford in the nineties,
is currently the white head professor of pure mathematics at the University and a fellow of modern college.
Its main interest are in geometry, topology and mostly in geometry group theory.
So that's a lot of work and I hope that today's lecture will go into detail and explain that to us.
So thank you very much, Martin, for accepting to give this lecture.
And I thank you all for that very nice invitation. So since there's no money left for myself, just concentrate on the mathematics.
So we're giving public lectures.
I much prefer to do it in a village hall or a school where you're sure the audience,
the trouble of giving it in the maths department is that the mathematicians show up.
And so I have to say, I'm not going to talk to you, my colleagues.
I'm going to concentrate on the the genuine public, as it were, and the genuine public, particularly the young people, I would say.
I'm going to flash a lot of things at you. And don't worry if the details pass you by, the bound to pass you by.
There's far too much information on these slides to absorb in a sort of symbol by symbol way or even a line by line way.
Just let it wash over you and and feel the beauty of the thing.
I just. Just try and hang on to the big ideas and let the pictures tell you something that you don't have to translate into words or symbols.
So I'm going to talk about the words in the title. Very efficient of me.
I'm going to talk about symmetry.
I'm going to talk about spaces, that is to say, geometry and different sorts of environments in which geometry takes place.
And in particular, as I said, I'm interested in geometric group theory or geometric group theory.
A lot of it's about is how you can extract geometry from algebraic descriptions of symmetry.
So I'll try and give you some sense of that, the way in which geometry emerges from algebraic structures that are designed to describe symmetry.
And then there's this word undesired ability that my spellcheck keeps telling me doesn't exist but is commonly used and in mathematics,
and that I want to try and convey to you that many natural problems in mathematics turn out to be too hard to expect as a global algorithmic solution.
Okay. And so we'll see examples of that.
And I'll try and convince you that this isn't some fringe phenomenon way beyond the boundaries of mathematics,
mathematics fading into the sort of logic that only interests old men in wheelchairs with long beards,
but it's really close at hand in natural problems.
So that's why I want to talk about symmetry, the way that geometry emerges from thinking about symmetry,
and then we'll try and touch on on the side ability and how it really comes, comes into play with with core mathematical problems.
Right? So let's start with something we think we know. So we think we live on the surface of a sphere, that particular sphere.
Okay, we might question that a little bit later, but let's think that's home, first of all.
Now we might zoom in a bit.
On those European islands there. If you don't know me, you might be puzzling as to my accent and exactly which one of those islands I'm from.
We won't take a vote. I'm from that one in the middle.
It's the Isle of Man. If you don't know me, you probably didn't that.
And now the flag of the Isle of Man is what's called the Tri Scullion.
That Viking symbol up there. And the motto that goes with it is Coca-Cola to stop it.
Whichever way you throw me, I stand. Now, that's rather a nice symbol, and we can talk about symmetry.
So let's start with that. So what symmetries does it have and what is a symmetry?
Well, a symmetry is where you take something and then you say to the person observing you, close your eyes, you do something to that thing.
And they say, Open your eyes, and they can't tell whether you did anything or not. That's what a symmetry is.
You transform the object, whatever the object may be, in a way that doesn't destroy any of its essence.
That really leaves everything a central preserved. So here I took this picture.
I left alone, and I took this picture and I rotated it.
And I took this picture and I rotated it twice. If you know how weak my computer skills are, you won't believe me.
You'll know. I just cut and paste but just pretends. But that's what I did.
It would have the same effect. So not so called rotation.
Rotate by one click. And I've just noticed here that if you do it twice, that's the same as if you did the inverse operation of the first click.
That is, instead of clicking to the to the right, I might have to anticlockwise I might have clicked one clockwise.
Okay. Okay that this little symmetric nice little bit of symmetry and when I what let's think so that has three fold symmetry right.
Has three symmetries. Do nothing, rotate once or rotate twice now.
So that motto come quite a stab at that.
That means whichever way you throw me, I stand in English.
Now, in pigeon mathematics, we might say what we've just observed.
It is, I have three symmetries in more sophisticated, fluent mathematics.
It is something about my symmetry group statement, not just about how many symmetries I've got, but what those symmetries are.
And so I might sort of be being a studious person.
Write down the symmetries I found. I found I could do nothing, which I write as one for doing nothing.
I could have rotated at one, so I could have rotates it twice.
And that's all okay, now in my notebook that I'm going to keep because I'm going to try and make an extensive study of symmetry.
I'm going to write more brief notes. I'm to say, Well, there's only one serious thing you can do to this.
You can rotate it, and I'll make a note for myself that I don't forget in case I lose my flag,
or at least to my notes, that if I rotate three times, I get back to where I started from.
Rotating three times is the same as doing nothing. Okay, everyone happy?
So just make some notes. Right now, mathematicians should be feeling a warm glow at this point.
We're doing mathematics. We're observing what's in the world around us.
We're making notes and we're starting to abstract, right?
So this is what mathematicians do. They observe, they look for patterns.
They say, Well, I'll make some notes, but I might come back that later.
There seems to be some structure that might be interesting here. But the first thing I'd like to ask is, well, is the pitch in maths,
in the fluent maths the same does having the same amount of symmetry does the fact I only had three symmetries,
does that mean that if I find anything else with three symmetries and make notes, I'll have the same notes.
Just the same amount of symmetry always mean you have the same type of symmetry.
So I've got the question. So another way of saying that is, is number the correct counting, the correct way to try and talk about symmetry,
is it enough to say, well, when we're discussing symmetry, we'll just count like we do with many things.
What you need is there in more natural language that you should develop in order to properly discuss symmetries.
Okay, so if everything only had three symmetries, it seems that well, it's probably they're probably all the same.
But let's think of some other objects with three symmetries. Exactly.
Three symmetries. Well, this is well, it's called a strange attractor. It arises in dynamical systems.
It's a rather beautiful thing that, you know, that that that fluid will flow to if it's agitated in the right way.
And that thing, again, has three symmetries, and it's obviously got more than a superficial resemblance to that flag up there.
Okay. And that that's not because I went out looking for a representation of Manx ness in fluid dynamics.
It's just because if you take something with three symmetries, that's what you'll find.
You'll have the same sort of sense of affair, of rotation, of a three fold symmetry, just being some sort of rotation.
That's another object with three symmetries. This time some Celtic not work.
Again, it's not exactly three symmetries, but more than that it's got the same sort of feel that you just can rotate it right.
And if you think of anything with three symmetry, that's what you're going to find, right?
It's not just that has the same amount of symmetry, reason has the same type of symmetry.
Well, I may or may not be interesting. Maybe we should look at a bigger example.
How about six is a number? Okay. Well, there's lots of natural objects that have six symmetries.
Fascinating fact is that if you take snowflakes, they tend to have this six fold symmetry.
And so I call that a photo of a slop over that for reasons the experts know.
Imagine that's a photograph of a snowflake. What can I do so that it's a real symmetry?
So if you close your eyes, then open again. You won't notice I've done anything. Well, again, I can rotate it.
Okay. I can rotate it once or twice or three clicks to four clicks.
So five clicks, the six click will get me back to where I started from.
Okay. And I might abstract that in all sorts of nice ways with the patterns again that
come out of studying dynamical systems as a sort of a snowflake like object.
And it's got the same sort of symmetry, right? Different object.
But I can say exactly the same thing about the symmetries. Remember, I'm right in my notebook.
So again, I notice I'm studying objects where you have a rotation that that somehow accounts for all the symmetry.
That is to say, if I just keep doing this rotation, I'll get all the symmetries.
And if I do it six times, I'll I'll get back to doing nothing if I'm happy.
We just made similar notes as when we had three, but he's a different object now.
That doesn't look the same. Okay, this is so we're not looking at nature.
We're looking at snowflakes or or abstracted versions of snowflakes.
Here's another object that goes nature, an ethane molecule.
Okay, so now how many symmetries does that have?
Well, you just stared. I don't will count again. Well, so these these big ones are carbon atoms.
These these little ones are hydrogen atoms. So. So how can I turn it so that after I've turned it, it'll look the same?
Well, I might if I decided I'll leave that carbon atom where it is, then I can obviously just rotate by a third of a rotation.
Right. So I can rotate fixed that one, I can rotate those three around.
So that would give me three possible symmetries, do nothing, rotate one third of a circle or take two thirds of a circle.
And I call that t for twiddle or turn right and I could flip it right.
I could just take hold of it and flip it around.
So I interchange the two carbon atoms on that and that's essentially all I could do, except of course I could twiddle it and then flip it.
And so you count how many things can go on here. Well, I can do nothing.
I can always do nothing. Ha. Stay in bed in the morning or twiddle or twiddle twice.
I can flip it like a toilet and then flip it or I could twiddle it twice and then flip it and you can convince yourself.
But anything else you might try to do to that is on that list. So again, it has exactly six symmetries.
Now, what am I going to write to my notebook this time? I'm going to say, well, I found two basic things you can do.
You can twiddle or flip these two little three times that scratch back to do nothing.
If you flip twice, that's obviously the same as doing nothing. That's what these notes mean.
Then this time I'm going to note if I flip and then twiddle and then flip, it actually has the effect of twist twiddling in the opposite direction.
There. Don't see that. Think about it. I should say.
But I'm going to put these notes on on the Web so you can come back and look at these slides at your leisure.
So. So there's another object six symmetries. But that doesn't look the same as the symmetries here.
In particular, try and convince yourself that here you see this one basic operation, and I have to do it six times before I get back to doing nothing.
Here. You convince yourself that anything you do, you either do it too, twice or three times before you get back to doing nothing.
So that's something that's really different.
Okay, so here we can see that these two objects have the same amount of symmetry, but they don't have the same type of symmetry.
They don't have, in any sense the same symmetries.
So so now we know that number counting is not enough of a language to help us describe the natural things we find in nature.
Wanting to describe symmetry. We need some other structure. We need some better language.
I'll just give you one example, which I'll do quickly,
because I want to do this because let's think about the nature of symmetry, what I'm saying symmetry.
And I hope you're not flinching at that. You know what I mean?
I'm twiddling things around without breaking them so I can put them back before the museum curator comes.
But according to the type of object you're interested in, what you mean by a symmetry will change, right?
So if you've got a glass object, then you better not try to stretch it because it will break.
So if symmetry is a rigid motion, right? If you have a rubber object, then you allow a symmetry to be.
I might put my finger on it and stretch it a bit. Right, and then stretch it back.
Right. That would be a perfectly good symmetry. I haven't broken anything.
I could put it back. I didn't. I preserved all the nature of the thing.
So what you think of as a symmetry depends entirely on what sort of thing you're doing.
What sort of objects are you interested in? Okay, so in this case, I'll be very naive.
I'll just have three balls. And now I'm interested in how can you rearrange three balls?
Okay, so I don't. There's no rigidity now I can just move them around as I like.
And so I claim once again, the answer is the six ways of doing this.
You can do nothing or what I've called Alpha.
You just take the first two and you interchange them or betas take the second two pair and you interchange them.
And I claim that there are exactly six ways doing this. And if you work it out, if you alpha, you could do beta.
If you do alpha then beta that has the effect of kind of cyclically permitting them or I claim the alpha beta alpha is the same as beta alpha beta.
Okay. So once again, just check. It's lots of ways to convince yourself is exactly six ways of rearranging those balls.
And, uh, and I'll make some notes on my notebook.
I'm trying to develop a theory of symmetry in my notebook. And so this time I write down I've got two basic things I can do.
They were flipped, but all I'm going to write down is the bare essentials because I don't have much space.
My notebook, there's two basic operations, and I gave them Greek names and I observed again,
I'll say, if you do eight alpha twice, beta twice, that's the same as doing nothing.
And I observe this other little rule I noticed. Okay, because that seems sort of interesting.
You write notes and things seem interesting. So now we've got three things that have six symmetry.
So we've got that for which we had this description here.
We got that's a more primitive one where we just rotate and we've got that object.
Okay. Let's skip over that. Right.
Okay. Now we reflect, right? We go and sit in the pub with our notebook and we think, what have we just been doing?
We've been making notes about symmetries of things. But in that last one, we changed our mind about what we were, what we had symmetries of.
So now we're going to do mathematics. We already are doing mathematics.
Okay. Now, what's object? What? What's mathematics about?
It's about studying objects called X. The only question you have to decide when you start doing mathematics is what is X?
Right. You have to decide what interests you. And that that really is the start of a lot of mathematics.
Right. So you observe patterns, you reflect on truth and beauty,
and you think I care about these sort of problems and the objects that I want to think about have these properties.
So I'm going to make a careful definition of those properties, and then I'm going to study all the objects to satisfy that.
Okay. So maybe I'm interested in the geometry of rigid objects.
Or maybe I'm interested in the geometry of rubber objects, how you can deform them.
Maybe I'm just interested in counting things and moving them around.
It's up to you. It depends entirely on what sort of problems you're trying to solve.
The first thing you've got to do when you start being serious about mathematics is decide in order to do this type of mathematics,
I'm going to study objects of this sort. You make a careful definition and that's your axes.
And then no matter what you're doing, you sooner or later get interest in the symmetry.
So that object. Okay. I'm interested in the properties of this object, the property that I use to define it.
Now, what are all the things I can do to it that preserve all the properties I care about?
Okay, so. So in the objects, we started with this on most primitive ones.
You're interested in the ones you really can't see any difference. You preserve all the geometry when you change them.
But no matter what sort of mathematics or science you're doing, you're really interested in the symmetries of your system.
Okay. And so those symmetries. So, in other words, automotive isms.
So I'm not going to use that word systematically, but it should be where it exists.
All right. So you think about them. You got an object you care about now.
You care about all the ways of preserving the essence of that object.
So if you're just interested in sex buckets of balls or something with no extra structure, you're just interested in the ways of rearranging them.
But if you're interested in glass objects, then you want to move them.
So you preserve all distances. You don't break them. If you're interested in rubber objects, you might stretch them.
And if you're interested in monolithic objects, you might require a finer structure, whatever.
But just just just think I have objects, and there's ways of deforming them.
Matt. This is the first point I really want to get to.
So let all this wash over you, but then absorb this no matter what sort of mathematics you're interested in.
The symmetries of your object. The old emotions. Your object.
Always. Always. Always form something called a group.
Okay. So this is a source of propaganda or.
Or persuasion that groups are the appropriate mathematical language to study the symmetries of any object in any context.
Now, for most, if a lot of primitive mathematics, the right concept is number you count,
and then you develop number systems and you manipulate those numbers.
Systems get tremendous mathematical power to solve questions that ask things like how much or how many?
Or Is this greater than that?
But when you're studying symmetry, automotive isms, symmetries of objects, the right question isn't it's a different nature.
And numbers are not the right thing to talk about. You want to talk about these algebraic structures called groups?
Now, this doesn't come out of thin air. It comes out of your experience.
It comes out of your experience that we sort of were pretending with which we were running through examples of here you study objects,
you make notes, and you start developing the sense of what the appropriate mathematical structure should be.
So what is it, group? Well, you just try to make the best possible requirements that capture the essence that are common to all situations.
But you care about symmetry. So what do you have to do?
Well, as I said, you can always stay in bed. Every object of every sort has at least one symmetry.
Just leave the damn thing alone. Just don't touch it. You certainly preserved all of its essence.
So every in every setting, you should always have an operation, which is one for historical reasons, of just do nothing identity.
Right? Just leave it alone. Stay in bed. Don't worry about it.
And then the other thing you can do with what's a natural thing.
If you if you to twiddle this in different ways, if you don't break it by doing that,
you don't break it by doing that, then you can do that and that and you still haven't broken it.
Okay. So you can compose when you've got two symmetries of an object, then you can do one followed by the other.
And now you have a new symmetry of the object.
So you have this notion of composing things, do A, then B, and if you didn't break it, then you can undo it.
Right. Just run the movie backwards of you being caught on CCTV picking up that Ming vase in the museum.
Right. As long as you can slowly and break it. As long as you've preserved all the essence.
You can always undo what you did. So those are obvious inherent features of any system of symmetry, of any sort of object whatsoever.
So we'll just make those our requirements, those and only those.
Okay, that's a slight, right? But not really. And then what does it mean to say two things at the same?
Well, if we were dealing in the realm of number, the appropriate notion of the same is having the same number of whatever as they are.
Right. So we make the abstraction of two because we get fed up of saying that's the same that pair of apples and that pair of volunteers.
That's the same number of apples as oranges. That pair of chestnut.
Pair of apples. I can match them up so they have the same that there's the same amount of apples as there is of chazz.
So we make the abstraction of a number. And then instead of having to compare it to something, we can then say there's two of them.
This is basic mathematics. But you look at the common essence of a class of objects you're interested in, and you abstract it and you say,
Now I have my ideal gold standard, one that I can leave in the Bank of England, and so that I can use this to to count pairs of things.
And the same with groups instead of having to say, well, that's got the same type of symmetry as that,
I want to make a list of groups and say, Well, whenever I find a new object, it ought to have symmetries from my list.
Somehow I want to describe all possible symmetries of all possible objects.
And when do you say two things are the same? Well, now it's not just a matter of matching them up.
You should match them up in a way. So if I match A's with B's, A's being symmetries of one object to B's being symmetries of another,
I should be able to match them up so that when I compose them,
if I do a one of the A's and then another one on this side, it should be the same as that twin's on the other side.
They won't have these. You have to match them up. Not just that they're counting the same, but they they respect the structure.
We care about composing or inverting. Okay, that is the most technical bits.
If you survive, that's okay. Just let it wash over you.
If you didn't survive that, try to wake up now or go to sleep.
As you wish. Right. But let's go back to our examples.
So now we have a language we have the language of group theory.
So now I'm going to just talk freely about groups. Okay.
So member groups, a group is a system of symmetry that describes the symmetries of some object.
But I've abstracted so just like I abstract from counting to having a notion of number.
And I can then talk about numbers. Now, instead of having to talk about the symmetries of a specific object, I can just talk about groups.
Okay. Oops. I didn't wanna do that.
Let's go back to our three basic objects. We had this bucket of three balls with those symmetries.
We had our cyclic thing. Except I traded in four different cyclic things.
And when I made those notes and I had my ethane molecule where I had those notes back groups.
Now I have three different descriptions of groups. Remember here I said there's two basic things you can do.
Here are some observations about how these things behave. Here.
I said, Well, there's one basic operation that gives you the whole shebang.
And the only thing I know is if you do it six times, it's the same as doing nothing.
And here I made some different notes. Now all those groups are the same.
Well, we sort of argued earlier that that one was different to that one.
Okay. But in fact, those are the two. That one and that one are the same.
Okay. So that's just a little exercise. We've never seen it. Think about how to match up the symmetries of that object with the symmetries of
rearranging three balls in such a way that you expect all the compositions and things.
Okay, that's nice.
On one hand, we sort of getting somewhere and be able to describe groups, but we're probably starting to be a little bit worried because.
It's a moment at which you should not feel alone.
If somebody just walked into the room and said, I found this lovely object and here, here are the notes I made about cemeteries.
And somebody else came in and said, Well, oh, I found this other object.
I don't think it's like yours. But here's the notes I made about its symmetries.
And he said, Well, it's your system of symmetry, sort of the same as mine, although we talk about the same group in a different language.
Well, the answer is yes, but it's not obviously yes right now that's going to worry us because we got so used to number two.
So when I say two and you say two, we have no doubt that we talk about the same thing.
But now when we talk about groups, we've described the same group,
quite an easy group, just something with six operations in it in two different ways.
And that's making us nervous to think that actually recognising what you've got might be a bit tricky.
Okay. Hold that thought. Okay, so we've got a language for describing symmetry, but we're a bit worried that when we have different descriptions,
it's not very easy to see if we've really got the same thing in disguise.
Okay. Well, we've counted. We've done some finite things.
Let's do some infinite things. We feeling strong enough for infinite things. Some of you look strong enough for infinite things.
Right? So imagine this, the paving going on forever.
Okay, here's a nice, simple, infinite system of symmetry.
An infinite group. You look at that paving, stare at it for a while.
You realise there's two nice things you can do.
You can shift this, this paving stone I'm pointing out along to that lawn and the whole system will repeat.
Okay, so that's a paving. That will be a symmetry of that paving. Imagine this paving goes on forever.
It's a mathematics institute, not an engineering one. So imagine this paving goes on forever.
That's obviously a repeating pattern shifted along. Or you can shift up.
You can shift that one up to that. And it's doubly periodic.
It's periodic in that direction and that direction. Everybody happy?
Those two basic operations, I'll call them A and B, and the only note I'm going to make is, well,
look, if I shift along and then up, that has the same effect as if I shift up and then across.
So that's the only note I'll make as I've described myself.
Another group of note to this two basic operations that seem to give me all the symmetries if I repeat them and I only notice one rule,
it's a rather nicer pattern. This is a Charles Rennie Mackintosh wallpaper.
I think now it's different.
The previous object, right. But clearly got the same symmetries.
All the symmetries of this again, what can I do? Well, I can either shove it along one column to the next, I can shove it up by two rows.
And again, I'll call those symmetries and B and the only thing to really note is that doesn't matter what order I do them in.
Now, that's clearly the same group. Okay. And it doesn't come as any surprise to you say that that has the same symmetries,
is that they're different objects, of course, but it's intuitively clear they have the same symmetries.
Everybody happy, right? How about that?
Okay. Well, that's got a few extra symmetries, right?
So so that is rather like the previous one. It's just a tiling of some plainer surface.
Again, you can again shove it along by one column.
That's a perfectly good symmetry. Observe all the shapes and all the colouring.
So you have to show off that little. You have to shove that.
So you have to shove that point over to that, get a symmetry and you shove it along does that, or you can shove it up.
It has those symmetries again, but now it has a bit more symmetry.
So how are we going to describe it this time? Well, I want you to focus on some particular point.
Let's take that one. Let's take that little purple triangle there.
Now, I have to get this order right here. If I if I focus at that corner of the triangle, and then if I turn it through 180 degrees.
Right. If I stick my finger in the pot and then turn it by 108 degrees, that will be a nice symmetry of the pattern.
I'll call that a and if I do it twice, I'll get back to I started from I just rotate by 180 degrees.
If I go to this corner, then I have to rotate by a third of a rotation.
I'll call that B, right. So that operation stick your finger there and rotate by a third of a circle.
So if you do that three times, you obviously get back to I started from.
And if you go to the final corner of the triangle, that one there, then you have to rotate by you can rotate by a sixth of a circle.
And if you rotate six times, you can actually start from there.
So I've spotted three extra symmetries that aren't to do with pushing it around, but now it's doing rotating.
And I'll make some little notes myself.
Look, I found these three cool things, like B and C, if I do A twice or B three times, A 66 times the same as doing nothing.
And I make one other observation. This is one for you to check at home.
If you do A, then B, then C you'll get back to I started from everything will be moved back to where it started from.
Now I've made some observations about the symmetries of this,
and I've made notes in my famous notebook that I have to take to the pub and reflect on later.
Now, have I captured all of the cemeteries of this in particular?
What happened to that? Shoving it along a bit and shoving it up a bit?
When I say I've made these notes and I've got it right.
Well, I mean, two things I'm asserting, first of all,
that every possible symmetry of that picture is obtained by doing these basic operations A, B and C repeatedly.
Maybe I'll do A once and then B twice, then C twice, then I'll undo A, then I'll do B twice, then I'll do C three times.
So you just make combinations, these basic operations exercise for all those bright eyed young students over there and prove
that every way of moving this pattern to itself can be obtained by repeating those operations.
That's the first assertion.
Second assertion, any rule that you tell me, connecting some combination of B and C, so you say, look, man, I've I've been working on this all night.
I can see that this massively long combination of AP and CS.
Well, in an unexpected way. Get me back to where I started from.
I claim. Oh, I knew that. I would say that, wouldn't I?
So I claim that anything you can tell me about combinations of AP and C follows from these simple rules here.
Okay. Now, I'm not going to delve into what I mean, but I honestly mean that literally anything you can prove,
any combination of moves that you can prove are related to each other,
any equation relating A, B and C can be deduced from these four simple equations here.
So I have nice notes explaining exactly the symmetries of that object.
Well, that's good, right? So what I'm getting to is my notes I've been making all along, my famous notebook.
I've been jotting down the symmetries of every object I come across.
And I have developed a sort of calculus, a sort of notation that can let me talk about all the symmetry groups I ever come across.
And what I'm going to do is I'm going to make notes. I have some basic transformations, some basic operations.
I do my thing. When I have my fingers, I'll make some notes, some equations about some rules that are satisfied.
Like here, I've noticed that these little rules are satisfied.
Excuse me? Are satisfied. And I'll start making notes when I'm convinced that everything,
every possible equation relating my basic operations can be deduced in a purely formal way from the ones I've written down.
Okay. So for example, let me give a simple example. If you tell me, look, I'm a clever boy.
I've noticed that if you do a four times, it's the same as doing it, doing nothing.
Well, I'll say of course it is, because if you do a twice, it's already nothing.
And if you do nothing twice, it's the same as doing nothing. And so, I mean, that is that sort of obvious deduction from the rules.
Anything you can say about the linking A, B and C can be deduced from these simple rules.
And so I'm interested in making accurate notes in that sense.
You have to have enough basic operations to generate everything,
and you have to have made enough notes to have information that allowed you to use any equation that holds between these generators.
Now. So that means we can start just writing down systems of symmetry.
So knowing the theory is a very powerful thing, right?
Once you've decided I know what I mean by numbers,
you start manipulating them and you get you find fine structure that will tell you about real world objects.
I now have a we now have a way of just describing systems of symmetry.
Right. So you might before you ever meet something with 1,000,000,517 objects in it, you sort of have a conception of what that number is.
Right. And you believe there are things that that have exactly that many members.
So now how about if I just write down a system of symmetry and say, well,
that must be the symmetries of something because it's a perfectly good system of symmetry.
So, for example, his four things I might write down, I'll say I'll,
I'll look for objects that have two basic symmetries, A and B, and it doesn't matter what order you do them in.
Be followed by A is the same as a fall by B. Well, we saw objects like that, right?
Our wallpaper and our paving were objects like that.
But now I'll write down some more interesting rules, and then I'll go and seek objects that have those symmetries.
And what I'm hoping is, well, I love geometry,
so if I write down some system of symmetry and go hunting in some intelligent way, there's something that has those symmetries.
Maybe I'll discover new geometries, objects that really have interest, partly because they have these interesting systems of symmetries.
So how about this one? So instead of saying it doesn't matter what order do, and B, and suppose I have two basic objects.
And I'll say if you do B that A, it's not the same as doing A than B, it's actually the same as doing A twice than B.
I had to write this down so I didn't get it wrong in the wrong order. So.
So can you think of anything like that? Okay. Well, here's his here's a couple of simple things you can do.
Take a line. Okay. Which one's which yet? And say A is the one that translates this line to the right.
So it sends X to X plus one. Okay. So any point on this line, it moves it along to X plus one is move the line rigidly.
Okay. Now, let's mark something called zero on this line.
And let's say B is the thing which contracts the line by a factor of two towards the origin.
Okay, so imagine this line is made of rubber. You can shrink it by a factor of two.
I can move it along by one. I leave it up to you to check that if you do B, then a a has the same effect as doing eight twice.
And then b. So found an object that has those symmetries.
He has two symmetries of a rubber line that satisfies the one we want.
We've got that group that was just given to us by some some some abstract dry algebra.
And now we've conjured up an object that has the right symmetries.
Another thing you can do, by the way, for those of you who know what matrices are,
if you take B to be the matrix like that, you take A to B, the matrix like that.
Okay. So these are ways of moving the the the to the plane to itself, that all lines go to lines and you can find those matrices satisfy these rules.
So that's a different object that has the same symmetries, right?
You can take a rope a line or you can take a plane and transform it in a way that always sends lines.
Two lines is two ways in which you can realise this abstract system of symmetry.
Right. This is a moment to pay close attention.
And I hope you're not colour-blind. The red group. Sorry, the red group.
But don't worry. It's called something different. G three is red. G four is blue.
Now they're modelled on this one, right? So now got three basic operations and they sort of follow the same rules.
Here. I've got four operations. They follow the same rules. Let's hunt for objects that have these symmetries.
Okay, so maybe it's like this. Maybe we'll be able to do something just for the line.
Well, maybe we'll be able to write down some matrices and some dimension, maybe some bigger matrices because the bigger groups.
And we'll look for ones to satisfy those rules. Hands up if you think this will be successful.
My. Might be right.
It might be. Might be. It won't. So instead.
Maybe I'll just look for some finite object and look for some symmetries of this finite object,
like maybe some nice big a molecule and ethane, and see if it has three symmetry satisfying these rules.
Again, you'll fail. Now. Why will you fail?
How do I know you will fail?
Because it's I know the following non-obvious fact, but neither of these systems of symmetry arises as the symmetries of any finite object.
Nor can it be realised as a group of matrices.
You can't get it to act on any linear space and any way that preserves lines in any geometry.
Now, in one case, there's a very good reason for that. One of those groups is the trivial group, which is to say from the rules I've written down.
So maybe it's that one. Maybe it's that one. Let me just pick on this one arbitrarily.
I if it if this group is trivial, I would say you've written down a system that's just stupid.
It cannot be the systems, the symmetries of any object.
How do I know that?
Because by a long and complicated calculation, I would figure out that these rules imply that alpha, beta, gamma and Delta are all the identity.
They all do nothing. Now.
Maybe I'm lying to you. Maybe that's not true of that one. Maybe that's true of this one.
Well, actually, only one of those groups is the trivial system of symmetry.
The other one is an infinite group. That is to say, it's the symmetries of an infinite object.
But it's not the symmetries of any finite object.
And has no finite dimensional manifestations of its manifestations on infinite objects and infinite dimensions.
Okay, let's have a vote. Who thinks G3 is trivial but G4 is infinite.
You'll have to vote one way or the other. That's always interesting. Okay. Thank you.
I forgot what I said. I said that was that was trivial.
I said, okay, who thinks G4 is trivial but G3 is in front.
It's always the same, it's always more or less 5050 it's actually G three that's trivial.
And G four is an infinite group that has no finite manifestations.
If you've got it wrong, don't worry. This is not obvious.
Right now this is a manifestation. So that's not a very those aren't very big examples and it's already extremely hard to tell which is which.
Now, what we're looking at here is a problem, a decision problem.
Now, it's not undecidable that I knew the answer, for example.
So can't be that undecidable, but it's a manifestation of the fact that problems about even fairly small group
presentations can be intensely difficult and in fact generally impossible.
And I want to explain what I mean by that impossibility, that underside ability.
This is a picture of Max Day, the beginning of the 20th century, and he was the first one in this famous paper who realised there's a problem here.
And what he saw, what he said was, was all very well writing down systems of symmetry and then saying,
now I know everything about it because I've made clever notes before I went to the pulp and I have a complete description of this group,
but it's very hard to see what information is written down.
It's very hard to extract real information from those notes, even if in theory you've got complete information.
So, for example, so what he asked is, is it actually impossible and do that actually exist?
Algorithms, processes that you could always be confident would work,
always be sure would work that would allow you to answer basic questions, such as is the group I gave you trivial?
Is there really nothing it could possibly be the symmetries of or whether certain
equations I give you can really be deduced from the relations that you think it can.
Okay. So those are those illustrate the fact that it's intensely difficult to take the
information out of out of systems of symmetry that are written down like that.
But I want to pick up on why Dane was interested in this.
Dane wasn't just interested in the way I described as just thinking of symmetries of things and coming to it from a algebraic point of view.
What Dane was trying to do was understand all models of possible three dimensional space, so called three dimensional manifolds.
And he was completely particularly interested in knots. So look at this comment.
One is led to such problems by necessity, and it really is by necessity.
When working in geometry, in topology that knotted space curves in order to be completely understood.
The man solution of these of these algebraic problems.
So let me try. So I'm going to explain why that is, but I'm going to try and illustrate it so that let it wash over you.
The following slide is the right impression. Oh, not that slide. There are some knots.
There is some knot. Now that that is not table meaning.
All of those knots are different. Now, those are the simplest knots.
How many there are that is overall is six, eight, 15, is it?
Those are the 15 simplest knots in some sense. Okay.
Now. How do you know that? Different. That is to say, if I take two of those.
How do I know I can't take a piece of string depicted in one of them and manipulate it to be the other?
How do you know that's an honest list with no redundancy on it?
That's exactly the sort of thing Dane was interested in. Okay.
How can you tell that not that appear to be different or space appear to be different, really are different.
Now, people are very good at this. In the 19th century, though, he here's a 19th century not table.
Actually, I think this is exactly the one that was on the murderer's wall in the last episode of Lewis.
Yeah. So he's not.
So how are you going to tell those the different. Well, you think maybe that's not so hard, but let me show you some examples.
You have to stare at that picture for quite a while to see that if you give it a good tug, you cannot, not it.
That really is the are not disguised. This one is the example of Wolfgang Harken that is also the UN, not if you pull out around enough,
you'll find that you can deform it without passing it through itself to a circle.
Now, that's not obvious, right? So Dane, being a clever guy, I thought, I know what I'll do.
I'll translate this into an algebraic problem. So we associated a presentation of a group.
He associated a group with the sort of description I've been given to each not.
And he then thought he proved you actually made a mistake, that somebody fixed it later.
He thought it proved that if you can say something sensible about the knot about the group, you'll know exactly which knot you've got.
And this is so you see lots of theorems in mathematics to say I have reduced this difficult problem to this much easier one.
But one should always beware of such statements. There's some sort of the some sort of principle of preservation, of hardness.
And this is a good example.
After thinking about it for a while and realise that the problem it translates into is harder than the one he started with.
So it's hard to tell that that is the UN, not that it's really a circle and it's extremely hard to see that its group is the group of the unknown.
Oh dear. This is all getting a bit bleak. So here's the underside of ability.
So. So these are these aren't utterly esoteric problems, right?
You want to know what a nut is? You want to know what a group is. And it was unknown from the time of day until the very end of the 20th century,
whether it's actually possible in an algorithmic way to declassify or not.
So you have these not tables that go up to about 14 crossings, but it is actually unknown whether it's an algorithmically solvable problem.
If somebody gives you two pictures of or not, can you decide to the same knot or not?
Not the UN not, not. Not yet. How do you decide if to not two different.
Okay. It turns out and I'll try and explain in the next few minutes that that problem is solved.
But it uses a huge amount of mathematics throughout the 20th century to answer that simple question.
Can you distinguish between knots in a systematic way?
What came earlier, what came in mid-century was the realisation that the algebraic version of that problem,
these basic problems that Dave ran into of Can you tell what group you've got when you write down
a system like can you tell if that's the trivial group that is algorithmically undecidable?
Okay, now what? That me and I'm not not that we haven't yet invented an algorithm or the techniques we have so far insufficient.
But one can prove that the can never exist a systematic way of deciding if a group is the trivial group or not.
Okay. You really, really. You might at this point think, well, I'm smarter than some dumb machine.
I could decide. No, no, you can't. Right. So anything that you could reasonably do could be done by a computer.
And so there's a real theorem here that says there is no algorithm that,
given an arbitrary, finite group presentation, will certainly give the correct answer.
Yes, it's trivial. Oh, no, it's not. Now, that doesn't mean if you're given a particular instance of a problem, you can't solve it.
What it means is, before knowing which which group I'm going to give you, you can't be sure that if you give me that, you can't think.
If he gives me that, I'm going to do this, this and this, and I'm sure it'll work. You can never be sure.
Okay, so this is undesired ability. Now this isn't way out.
So I want to convince you this isn't this way out. An idea as you might think. So you might think.
Well, decide if an arbitrary thing is trivial. That's undecidable.
Okay. It took until 2050.
Well, it took a few years ago, but something just appeared in the literature.
So Welter and I proved if I if you like those groups, G three and G four, I had if you give me a group and ask,
can this be realised of the symmetries of some finite object, we now know that's an undecidable problem.
We cannot tell, given a system of symmetry, whether as a finite object with symmetries of the prescribed type.
And here's an even perhaps more concrete term anyway.
So Andrew Wiles up there, of course, is famous for telling us that if you pick an integer that's bigger than two and you try to solve the equation,
let's pick L1, all the integers X, Y and z non-zero integers.
So that's fine. That equation. And famously the answer's no.
But one might hope that with advances in technology, eventually, if I give you any polynomial, it's called diophantine equations,
a polynomial equation with integer coefficients and ask you Does that have any nontrivial solutions or not?
You might hope that eventually the theory would develop far enough that no matter what equation I give you, you'll be able to tell me yes or no.
Right? That's never going to happen. But not you say, which proved in 1970 that that sort of diophantine problem,
that very concrete problem of given a polynomial and several variables with integer coefficients, does it have a solution in integers?
That's an undecidable problem. So as Andrew proved, for certain equations you can decide, right?
But if you don't know what equation I'm going to give you, there's nothing that you can hope.
You can be confident will work in every case. Okay.
So we've talked about symmetry and we've talked about underside ability.
Okay. We haven't talked so much about geometry.
And I realised that there are halfway through my slides and I have 5 minutes to go and I'm not going to rush, but I do want to talk.
So we've had a lot. So. We need action.
It's no use just sitting there. We need action.
It's no use. Just sitting there staring at these algebraic descriptions of systems of symmetry.
You've got. You've got to get it to act somewhere or you're never going to understand it.
So we need mechanisms. If you give me some algebraic description of a group,
I need to have some mechanism to conjure up some action on something so I can use geometry to try and understand it.
And hopefully in doing this, I'm going to discover some nice new geometries. So let me skip ahead to this construction.
So here's a group we do understand. We keep seeing this got two basic operations and they commute.
Doesn't matter what order you do them in.
Here is a purely geometric description without understanding that group where I can conjure up the sort of things that acts are so in
I so often in mathematics you've got some data and it's come from some situation and now you want to extract more power out of that.
So you sit back. I seem to be saying there's a lot of sitting back in mathematics, but there is.
And you think, well, how else could I interpret that data?
What else is naturally associated? That data that would change context.
And maybe from that change perspective, I could actually get more information.
So you look at this, at these symbols. Look at your notebook with the eyes of a topologies to a geometry instead of an algebraic.
Okay. And I'll treat that as some meccano assembly instructions.
And the meccano instructions are take one point a attach one wire for each of my generating objects,
take a little rubber sheet and attach glue it and attach it to the wires according to spell out the word a B than a inverse and b inverse.
If you do that, you'll get this bagel like object over here.
It's called the Taurus. Okay. And then the magic of algebraic topology, which I'm not going to explain,
is that if you're wandering around on here, imagine yourself a little ant wandering around on here.
And imagine you're a member of the Flat Earth Society. You haven't yet learned to think globally.
You're just really a little, aren't you wandering around on here? You don't know whether you're on here or you're on here.
Right, because all you see is this little dot around you. If you're there, you've got two arrows.
You've got to be arrows and all looks the same. So with a limited horizon, you don't know whether you live here or here.
And so this unwrapping procedure gives me something where this group acts as symmetries.
Now, I rather like that, right? Because I just took some systematic way of taking the symbols of the system of symmetry and conjuring up an action.
I've got this group now acting on that thing up there. And so I'll take some less familiar a skip over this.
I take some less familiar. So, hey, here's another. I do that with this group we had earlier.
I do that little construction downstairs. Now I unwrap it, and upstairs in that picture, I'll find this nice tessellation.
Now I'm about to get new geometry. Ready for new geometry. Observe.
Here's something. Whose symmetries? Look my notes at B and C, you do them twice, you get nothing.
And these numbers here, tell me what happens when you add a couple times two, three and six.
See those numbers? Two, three and six. Let's make the least possible change.
Let's turn two, three and six into two, four and six.
And repeat that operation. Okay. Take that group presentation.
With the minimal change that was previously given in a copy of the usual tiling at the plane,
we're all used to go through that system of generating objects on which the group would be symmetries.
And you got this wonderful object. Okay. Make another slight change.
And you got this very similar looking object. And what's happened here is we've discovered something called hyperbolic geometry.
Excuse me. So Euclidean geometry is the one we.
We sort of naively used to have squares going off forever. Hyperbolic geometry is what's pictured here.
And if you think of this as an infinite space that you think of, this is a road system.
You've got to pay a pound to go along each edge.
And so the distance between two points, how much you've got to pay to get from one point to the other.
So even though these look like they're getting tiny to go from there to there, you've got to pay that.
Not not not that. So is that the infinite? Because you got across infinitely many edges to get to the end of the disk.
There's an infinite geometry, different to Euclidean geometry that we've discovered by first,
starting with the system of symmetry and then asking, what is it? Symmetries of.
Now this is a rather beautiful thing that has all sorts of properties that are different to Euclidean space.
It's very well-studied and particularly has thin triangles.
And if you're wandering around in this space, you wouldn't think you on a flat plane, you'd think you at the point of a saddle all the time.
So locally, if you're anywhere in the space and you're trying to, you think you're in a saddle.
But this really happens in nature, right? So what happens in hyperbolic geometry is things try to grow exponentially quickly.
If you count the number of triangles within that, then PN steps of that point, you'll find it's growing exponentially.
It's multiplying roughly by two at each step. Now for a while you can do that in Euclidean space.
Vegetables try to do it and they end up looking curly like that.
A coral tries to do it, ends up being negatively curved, hyperbolic like that.
Even bizarrely, some crochet enthusiasts do it.
Well, that's a lot to think about. And I just want to finish the couple of remarks.
Oh, I should have stopped. So this hyperbolic geometry happens in every dimension.
Okay. More than that, there's all sorts of other geometries I haven't told you about that come out of fact,
that come out of thinking about arbitrary groups.
So you go to think, you go away and you think about this and you realise that as I was standing there in the Isle of Man, I should have zoomed out.
And when I assumed out. At the next step.
I really can't be confident what I get. Right.
There's lots of geometry, so maybe I have flat geometry or total geometry or spherical geometry or some hyperbolic geometry.
Okay. So think about this.
All the young people in the audience think about can you decide what sort of surface you live on by just making experiments?
Can you do some ordnance survey just on the earth and figure out whether you live on a sphere or doughnut or flat
plane or high agena surfaces that call something like that or something with three holes or four holes or more holes.
Think of experiments. Can you do that? And it might not always be easy to recognise things.
So that surface there is the same as that surface there.
By which I mean if you cut all the tiles up there and reassemble them so they have the same edges in common, you'll get that previous surface.
So again, it's hard to recognise which surface the same, but it's possible when you go into three dimensional space it's even harder.
It takes most of 20th century mathematics. But you can make a complete list of all possible models of reasonable three dimensional spaces.
The key people have Poincaré who really developed that picture of hyperbolic geometry.
I've explained to Bill first and who first thought it might be possible to classify space in three dimensions.
And Gregory Fischer Perelman was the one who actually proved it.
Okay, so having gone from 2 to 3 to 4 to 3 dimensions, should we go to four dimensions?
No. Because just as it's impossible to tell what a system of symmetry is.
Well, it's impossible to tell if an arbitrary diophantine equation has a solution or not.
When you get to dimension four explicitly because all groups come into play in Dimension three, there's some restrictions I mentioned for all groups,
all systems of symmetry you can never describe arise as symmetries of models of space time, so-called four dimensional manifold.
And because you can't recognise what group you've got, you can't recognise which space time you've got.
So once you can classify all models of two dimensional space and all modes of three dimensional space,
it's proved impossible to make a proper, exhaustive, ever done list of all models of full space explicitly because of group theory.
Okay, that's a good place to stop. Thank you.