In Our Time - The Poincaré Conjecture

Episode Date: November 2, 2006

Melvyn Bragg and guests discuss the Poincaré Conjecture. The great French mathematician Henri Poincaré declared: “The scientist does not study mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing and life would not be worth living. And it is because simplicity, because grandeur, is beautiful that we preferably seek simple facts, sublime facts, and that we delight now to follow the majestic course of the stars.” Poincaré’s ground-breaking work in the 19th and early 20th century has indeed led us to the stars and the consideration of the shape of the universe itself. He is known as the father of topology – the study of the properties of shapes and how they can be deformed. His famous Conjecture in this field has been causing mathematicians sleepless nights ever since. He is also credited as the Father of Chaos Theory.So how did this great polymath change the way we understand the world and indeed the universe? Why did his conjecture remain unproved for almost a century? And has it finally been cracked?With June Barrow-Green, Lecturer in the History of Mathematics at the Open University; Ian Stewart, Professor of Mathematics at the University of Warwick; Marcus du Sautoy, Professor of Mathematics at the University of Oxford.

This BBC podcast is supported by ads outside the UK. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio four. I hope you enjoy the programme. Hello, the great French mathematician Henri Poincaray declared, The scientist doesn't study mathematics because it's useful. He studies it because he delights in it,

and he delights in it because it's beautiful. If nature were not beautiful, it would not be worth knowing in life, would not be worth living. And it's because simplicity, because grandeur is beautiful, that we preferably seek simple facts, sublime facts, and that we delight now to follow the majestic course of the stars. And Poincaray's groundbreaking work in the 19th and early 20th century has indeed led us to the stars on the consideration of the shape of the universe itself.

He's known as the father of topology, that is, the study of the properties of shapes and how they can be deformed. His famous conjecture in this field has been causing mathematicians sleepless nights ever since. He is also credited as the father of chaos theory. So how did this great polymath change the way we understand the world and indeed the universe? Why did his conjecture remain unproved for almost a century and has it finally been cracked? Join me to illuminate these questions. I'm Marcus Jusotoy, Professor of Mathematics at Oxford University.

Ian Stewart, Professor of Mathematics at Warwick University, and June Barrow Green, lecturer in the history of mathematics at the Open University. June Barrow Green, let's start with Pwain Carey himself. Can you tell us something about his early? early career? Well, Poincerey promised as a mathematician really from about the age of 15. He was recognised at school. In fact, his teachers at the Lise called him a monster of mathematics. And he, from the Lise, he entered some national competitions and came first out of the whole of France in mathematics. And from there, he went to the Acole Polytechnique, which was one of the

premier schools in Paris, which prepared students for careers in civil service, engineering. It was for state careers, really. And Poincere did very well at the Accoled Polytechnique, with one exception, really. He didn't do very well at drawing. And there's a nice story about the fact that he got only one, I think, out of 20 for one of his drawing exams. And had he actually got zero, he would have failed his entire course at the called Pellot Technique, but in fact he came out first out of his entire group with 7% more than anybody else, despite doing so badly at drawing. From there, he went to the School of Mining

and looked set to have a career, in fact, as a mining engineer. And this was, of course, a very respectable career in France at that time, and still is. And he went, at that time, he was mostly in Paris studying. And he also... also were studying at the University of Paris doing a PhD in mathematics. It's interesting that going into mining, isn't it? It links up with what we're doing about the encyclopedia last week about the appeal to the artisans and describing their trades and crafts and how steady that's remained in France and how very respected it is. But can you just give us one or two notes on his particular talents?

Because as I understand it, from reading notes of you three people, He had an extraordinary capacity for geometric visualisations. Can you just open that up a little? Well, one of the things he was well known for was that in his lectures, he hardly took any notes at all. He had an extraordinary power of concentration. Apparently he could just sit and really exclude the world around him and take things in.

And the other thing was, as you've mentioned, this power of visualization. He could clearly see things unfolding in front of him in his mind. and that enabled him to make great leaps, actually, in mathematics, even though he couldn't actually draw these things often that he could see. So I think he was really remarkable in this kind of dual aspect of having this concentration combined with this visualization. So we have this brilliant young man comes from a brilliant family. His father's professor of mathematics.

His cousin was... Actually, his father was a professor of medicine. Sorry, Professor of Medicine. and his cousin ended up as president of the republic. So we're talking about a brilliant family. But when did he, when did Henri Poiréry come first come to public recognition? Well, he got his doctorate in 1879. And the following year he sent in a paper for a prize at the French Academy.

The competition was for doing a further development in the theory of differential equations. Differential equations are equations which study motion of things. like water going through a pipe or orbits of the planets or something like that. And although he didn't win the prize, in his competition entry, he actually came up with an entirely new piece of mathematics, a new theory. Probably the reason he didn't win the prize was because the theory wasn't absolutely developed, but it was a really monumental piece of work for somebody so young. And I think as a result of that, he then got called to a lectureship in Paris.

At that time, he was at Korn. and then he went to Paris and the career was begun. He's spoken of as the last of the universalists. What does that mean? Mathematics was getting bigger and bigger and bigger, broader and new areas of mathematics being invented all the time. And round about Poin-Carré's period,

which is since the end of the 19th century, it really became impossible for any one person to roam over the whole field and understand the whole field. he just about managed to do that. He worked in all of the important areas of mathematics, perhaps with the exception of mathematical logic. What would those areas be?

Analysis, algebra, topology, which he pretty much invented, various areas of applied mathematics. He worked on electromagnetism. He worked on celestial mechanics, the movement of heavenly bodies. So he pretty much covered most of what was around. After Poin-Couré, partly because of the stimulus he and others like him gave to the the subject. It got so big that people really had to specialise.

And then you get a period when specialisms become narrower and narrower, maybe deeper and deeper, but on the whole, mathematicians become experts in a tiny bit of the field. But his generalist view, as I understand it, helped him in the development of his key theatres. Can we just talk

a little bit more about what it gave him? And June referred to his experience that once is going to be a mining engineer, so have a practical society as well. It's important to realise that mathematics is just a little collection of walled-off specialities. They all interconnect with each other. And one of the ways that big problems get solved is people find new connections between different areas, connections that nobody realized were there, suddenly get an insight

into how to think about the problem from another angle and to use the mathematical techniques from another area. Now, if you've got a broad overview like Puancares had, you've got your fingertips on these techniques and you can spot these connections. And so I think his breadth, and as you're saying, not just his breadth within mathematics itself, but contacts with things in the outside world. You can get ideas from the outside world. So he was tremendously learned and so and so forth.

But he had a theory about, or an idea about inspiration too, didn't he? Which I found fascinating. He has one of the best descriptions of what it's like to come up with new mathematical ideas and how you think about it that I've ever seen. he arrived at it completely by introspection. Basically what he says is if you're going to solve a mathematical problem, the way the mind works is firstly you work very hard on the problem. In a conscious way, you attack it from all sorts of different directions,

you learn what other people have done, you really do serious conscious work on it. That usually doesn't get you anywhere, except it stirs up your subconscious. And subconscious mind is busy churning away, thinking of all sorts of things that you would never have come up with, Most of them totally stupid. And if you then stop and go away and do something else, the subconscious is still working on this.

And it will come up with ideas maybe that seem to be progress on the problem. And then it sort of tap you on the shoulder, wake you up and say, hey, what about this? And you then start consciously thinking about this idea again. And then if it's the right idea, all you have to do is the technical job of kind of polishing it up a bit, making sure it works. So he tells the story about stepping off a bus and as he's getting off the bus suddenly the solution to a problem he's been thinking about for months

just flashes across his mind and within two days he's solved the whole thing So we come back to the career, Marcus Gisotoy As June said He became a lecturer at Parrish University in 1881 And he becomes interested in Newton's theories of how bodies move in the solar system Can you tell us what implications that had for him

Yes, well he actually went in for a prize which was set up for Oscar the second in Sweden's The King there was having his 60th birthday And actually that was how to make your name at this sort of time Was these sort of prize problems Solving a particular problem We went back to really sort of in France In the 18th century these prize problems to solve particular things

This problem was about trying to understand Whether the motion of the planets was actually stable or not so had quite serious implications for the whole of society, whether actually at some point the Earth might suddenly spiral off into space and disappear from our solar system. Now, the equations that Newton set up showed us that we should be able to sort this out. We know where all the planets are, we know what direction they're going in, so solve the equations and actually calculate whether the thing is stable or not.

So, Juan Coray thought, okay, it's a little clockwork sort of universe, as we say. let's try and have a go at solving this thing. And as he worked on it, he put in his theories on the thing. And actually, he'd made a mistake. And this is the discovery of chaos theory, actually, because chaos theory is all about this thing called the butterfly effect. Puancair made an interesting. Quankaray made a mistake, yeah.

He'd actually thought a small approximation wouldn't really matter and wouldn't cause too many problems on the way the planets would actually evolve over time. But actually, when he re-looked at this, he discovered that actually a small change could have a very dramatic effect where something which is very stable, if you change the initial condition of where the earth is, the thing could go in a completely different direction. And this is actually, we talk about the butterfly effect in the weather, that a butterfly flapping its wings in Brazil could actually cause a tornado in Tokyo. So that's why we call Puancared the father of chaos theory because of this small mistake he made. And he had to withdraw. It was all a bit embarrassing because, you know, this was meant to be a celebration of the king's birthday. And there was a little mistake.

But the interesting thing is this theory of topology that he started to develop was actually developed in trying to answer this problem. Because if you think of the two planets traveling around each other, that's very stable. we understand that very well. They just travel in two ellipses around the sort of center of gravity of both. And they just repeat themselves again and again and again. So you can think of the path of the planets a little bit like if they had paint behind them. You'd see this sort of nice little two little ellipses. And once they got back to the beginning again, they would just repeat the pattern.

So Poincoray was interested in if you have three bodies, okay, if you want to know the thing stable, is there some point where these sort of paints, pots, the trails of paint actually join? up and it's again just repeats itself. So he actually, it's interesting that we heard that Puan Cray wasn't very good at drawing because actually Puan Cray was interested in when things closed up, not in the actual paths themselves. And this is the sort of beginning of topology where actually the actual dimensions of the paths are not important. What's important is do they join up or do they never join up and repeat themselves? Before we go to Petrae, just briefly, there was another side of, well, many sides of Pankerre,

but one side was his determination to make signs available to a bigger public and write books of that effect. Can you just mention that and then I'll get back to Topology? Yes, certainly. He clearly, although it's quite interesting because there's this sort of dichotomy in a way, his peers found a lot of his work very difficult to follow because he was so brilliant that he would do things

that one equation followed by another and other people needed lots of work in between. But the other side of him, was that he was very keen for his work to be understood and for the public in general to actually understand science. And he wrote three books of popular science in the first decade of the 20th century, which sold incredibly well,

were translated into a number of languages. And these books had essays on all kinds of different subjects, ranging from geometry and electromagnetic, magnetism, theories of space, all sorts of things. And it was said that you couldn't go into a park in Paris at that time without coming across someone who was reading one of Poinclair's books. Yes, so we have a very extraordinarily rounded man in many ways and also his belief in the aesthetic value of mathematics

as from the quotation I stumbled over at the beginning of the programme and so on. But let's go for topology markers, you so to. Right, here we go. Can you explain what topology is and why we should all know about it? In a sense, it was a new way of looking at shapes and geometry. In the past, geometry actually means sort of measuring the earth. And what was important to the ancient Greeks was measurements and angles and the distance between things.

But Poincoray really initiated a new way of looking at shapes where perhaps the actual distances were not so important, but it was sort of how the thing was connected. together. In fact, anybody who's traveled on the London underground will have experienced a sort of topological view of London rather than a geometric view of London, because the wonderful thing about the map of the underground is, of course, it's not a physical representation of where all the stations are. I mean, it's a wonderful thing to look at where the physical, you get all these concentration of stations right in the centre of London and all these big spaces on the outside. But that's

completely useless. What's important about the London Underground map is that you're interested in the connections, how to get from one place to the other. And so you can sort of pull apart the geometric map and you just need to make sure you keep the connections. And you've now got what's called a topological map of where the stations are in the London Underground. So somehow not the actual measurements in the thing, but the way the thing is connected together. And we call this sort of bendy geometry because you're allowed to push and pull shapes until they look. And they look easier for you to deal with, as long as you don't cut them. So, for example, a rugby ball in topologically would be the same as a football because I can sort of squash it to one from one

to the other. Whilst a bagel, for example, or Homer Simpson donut, well, this has got a hole in the middle and I can't sort of deform that in a smooth way into a sphere. But I can deform it into a smooth way into a coffee cup because a coffee cup has a little handle on the side. And so actually, you know, a topologist is somebody who can't tell the difference between a coffee cup and a bagel. So it's all about, and what's important about some mathematical problems, for example, when Van Quay was trying to solve how

the planets were travelling and whether there were periodic orbits, sometimes it's not important what the geometry is, it's about the connections between the paths they make. Ian, can you develop this? I'm not developed it, it's pretty developed from me already, but I mean... Can you take it on?

I think in mathematics, it's always important to realize that it's not a case sometimes of having too little information about what's going on. It's about having too much. You've got so much fine detail about the thing you're looking at and trying to understand. And topology is one of the ways of throwing away a lot of extraneous information, which in certain circumstances really isn't very important. And say, now let's focus on the deep structure of this, the simple structure. This is a point about simplicity.

Simple things, there is simplicity, simple things are not necessarily. necessarily easy. In fact, the simple things are often very, very hard, as we all learn in our lives. It's the simple observation everyone's missed that catches us out. So topology focuses on the sort of basic properties of shapes, as Marcus says, how they're connected together. The kind of things topologists worry about are things like knots. Is an overhand not the same as a figure eight knot or is it different? How can you prove they are actually knots and you can't untimely? them. They focus on surfaces and the important thing about surfaces is not how long it is, what the area is, what the angles on it are, but how many holes has it got? Bagel's got one hole, a sphere has no hole, a pretzel has, depends on your variety of pretzell,

but it has two or three holes in it. And these are topologically distinct properties. What was the thinking behind that? Why is he driving in that direction? I'm sorry to be so simplistic. He's driving this direction Because when he's trying to understand his three-body problems, so he's coming from a real world, okay, it's astronomical. Can you go back to three-body problem, sorry, yeah. Okay, three-body problem. Three-body problem, sun, earth and moon. How do they move under gravity?

And what the mathematicians discovered to their horror is that three-bodies is really, really hard, whereas two bodies are pretty straightforward. And the reason we now know, thanks to Pankaraya and what followed, is that in the three-body problem, you can get extraordinarily complicated, orbits for the planets. They don't need to close up. They don't need to repeat. They have aspects of randomness. Now, what this means is that if you try to write down mathematical formulas for how those

bodies move, you're not really going to get anywhere. There aren't any formulas. There aren't any nice formulas like there are for two bodies. So Poincere said, let's ditch the formulas, or at least we can use them as an aid, but they're not going to solve the problem. and let's think a little bit about the geometry. For example, if you want to know that something follows a closed path, all you have to do is take a little cross-section to the orbit, start on that cross-section, follow the orbit round until it comes back close to where it was

and hits that cross-section again, and see if it hits in the same place or not. If it is in the same place, it's closed up. If it hits in a different place, you can say, how does the first hit relate to the second hit? can we extrapolate that? Can we predict where it will come around the third time? Is there somewhere in this structure a point where if we started from that and went all the way around, we come back to that exact point, fixed point?

Yeah, can I just bring you in it for one second? Just so we clarify, how radically different was this from the thinking that went into Euclidean geometry? Oh, very different. I mean, Euclidean geometry, as some of us learned at school, is really to do very, very, much to do with proof and it's in the plane or in three-dimensional spaces as we know it. And you're working with, in a way, in a familiar world. Here, Poincere is theorizing completely. And I think it was this idea of being able to work with only two dimensions, looking at something

that was going on in three dimensions. So he was making a representation. So he wasn't actually dealing with the thing as itself in the real, as it was really happening. It was a representation. And that was a big move away. I think there's also, I mean, we say Pank-Rae is the father of topology. Well, there is actually a grandfather. I mean, we should perhaps go back to Euler, who solved a mathematical problem in a topological way

and really started this new way of looking at problems. and this was a famous problem, the bridges of Kernigsburg. In Kernigsburg, there's seven bridges which connect an island in the middle of the town, and it was a classic problem of sort of Sunday afternoon for the residents to try and go across all the bridges once and once only and return back to the beginning again. And every Sunday they try this and they could never do it. But they couldn't prove that there wasn't some way that they'd missed, which they could find their way around this path.

What Euler did was to solve this problem and put them all out of their misery and show that that actually was, it was impossible to do this. And of course, what he did was to say, well, just to like London Underground, the physical dimensions of Kernigsberg are unimportant. What's important is the connections between one island and the riverbank and the bridges. And so he crystallized it into just a very simple graph, which he could analyse and show that because certain points in the graph had odd number of bridges coming out of it, you couldn't get round this thing.

And this is the first, it's sort of a new, I think it's very, really important to stress. It's a new way of looking at things and sort of saying when two things are identical or not. True. The other thing I think is important about Puancoray is that he was really interested in looking at how things worked from a qualitative

perspective as opposed to quantitative. So he was actually interested in the properties of things like we were talking about periodic orbits. Whether an orbit is periodic was important rather than actually precisely what its orbit was in numerical terms.

Yeah, that's right. In fact, most of the current research in that area of mathematics uses this qualitative approach as a complete framework on which to understand everything else. You can stick your sums on a computer, put the equations, do the calculations, get very, very complicated pictures of how things move. And for certain problems like landing on the moon or going to Mars or something, that's quite a good way of getting the answer.

But if you want any understanding of why that's the answer, and what you're really seeing, it all goes back to this qualitative approach of Juan Correy. And in fact, very recently, the NASA and the European Space Agency and other such agencies have been focused on rather unusual orbits for transferring a space probe from Earth orbit to somewhere near the moon or sending it out to go and look at the comet.

You want low energy, you want to use as little fuel as possible for those missions, and it doesn't matter if it takes 10 years. if you use less fuel, it's cheaper. And they've been using basically Poincere's qualitative ideas to map out a kind of skeleton of where these highly efficient orbits will go. Right, June, quickly. A little aside to that,

I always thought it was rather nice when I discovered that, in fact, the first people to translate Puancoré's work from the French into the English, his work on Celeste Mechanics, was NASA in the 1950s when they started their satellite program. Marcus, can you tell us about microis conjecture published in 1904?

Is it possible to summarize it in words that even I will understand? Absolutely. But we're going to go in stages, okay? So what's important for mathematicians is to classify what's possible. So, for example, what possible shapes are there? What's possible?

What does nature allow us? What does mathematics allow us? Well, first of all, let's look at what sort of two-dimensional shapes there could be. So I'm going to take a two-dimensional piece of rubber, and I'm interested in how can I wrap it up? What sort of different shapes can I wrap up this two-dimensional piece of rubber into? So one of the shapes is into the shape of a football.

I can close the thing up and make a football, my favourite mathematical shape and others. But another possibility is to wrap it up into a cylinder, roll it round, and to make a bagel shape. Well, what other shapes are there possible? Well, Ian's alluded to the fact you could have a bagel with several holes. in it so I could wrap it up in such a way that I've got two, three, four. Now, is that all that's possible? You know, you might have some weird shape, but what Poincerey and the scientists in the beginning

of the 20th century showed was actually any other shape could be deformed gradually into one of these shapes, either a ball or a bagel with one hole, two holes or more holes. Yeah, any shape, if you're just looking at them and it's closed up in this piece of rubber is closed up in some weird way you'll always be able to deform it so for example a rubber teapot can be deformed it's got two holes in it it's got a handle and where the tea goes in and out that could have a spout it goes in and out the spout yeah exactly so if you had a rubber teapot

as you moulded it you find that you could actually make it into a bagel with two holes in so those are the only possible shapes that can exist there's something very special about the football of course which is that you can tell whether you're on a football on... I mean, actually, if you think about before we could get outside the earth, what possible shape could we live on if we were on the surface of the earth? Well, first of all, people thought it was sort of flat earth and it just went on forever, but then they realized it closed up.

But how could you tell if you were on that surface of the earth without going out into space what sort of shape you were on? And there were actually people who believed that perhaps we were on a bagel-shaped earth. and we go around and actually there wouldn't be a pole, we'd find that we could go around and find this hole in the middle. But you can tell whether you're on a ball or a bagel in the following way. If I take a lasu and wrap the lasu around the earth, and then I tie, so it's sitting on the surface of the earth,

this piece of string, and I pull the lasu too, then however I string it around the earth, it will always, I can always pull it until it vanishes to a single point. On a bagel, that's not true, because I can take the lasu inside the bagel and back around the outside, so it's sort of hooked on to, it's caught the bagel in the lassoe. If I pull that tight, then it doesn't come to just vanishing to a point. It sort of gets snagged on there.

So actually you can tell what sort of shape you're on. So Puancaray then moved up a dimension. Okay. So now I'm going to take you into the fourth dimension. Are you ready? Okay. So we're living a three-dimensional universe. So how is our three-dimensional universe wrapped up?

It's, it might be, we might be on a sort of flat earth. It might be a flat universe where it just goes on forever. I think that's what most people's impression is, that it just goes on out into infinite space. But actually, most people think it's probably closed up in some way, such that if I keep on looking, I'll actually eventually see the back of my head. So it's sort of closed up in the same way as the surface of the earth is closed up. Now, Punk Ray was interested what possible shapes could the universe be wrapped up in.

could it be a big hyper football, a hyper bagel, and what other shapes are there? And if it is a hyper football, how can we actually tell whether it is? Is this fact that if I tie a lousseau around the universe and pull it to, and it always comes two, does that mean it must be a sort of four-dimensional football, or possibly, is there another shape which is possible in this higher dimension,

which has the same property as the ball? It sort of looks like a ball, smells like a ball, feels like a ball, but it actually is something topologically completely different and can't be moulded into this sort of hyper football. And that was Poincere's conjecture. Is the fact that the globe, a two-dimensional surface wrapped up into a football has this unique property that Lesous pull off it? Is that the same when you move up a dimension

and talk about how the universe is wrapped up? Or is there some weird shape that's out there that's possible to wrap up space in sort of higher dimensions that we just don't know about? Well, I'll tell you something. At this moment, I understand. but whether I'll remember it on Saturday morning, I don't know.

Ian. Yes, Melvin. Do you want to develop that little further? Because in your note you use some interesting analogies. And then you say analogies that don't really work. It's been very helpful for something, you know. Yeah, I think it's important to realise we live in a three-dimensional space,

a particular three-dimensional space. We actually live in a fairly small bit of it. We don't explore huge amounts of this space. And we have a rather naive view that the model of space we have in our head, Euclidean three-dimensional space, is really all there is. I mean, Marcus is talking here about ways of bending three-dimensional space, and I'm sitting there thinking, now I understand this stuff, but that's a pretty strange thing to want to do. How can you bend it? Where can it go?

Now, the way math petitions bend three-dimensional space is actually they don't bend it. What they just do is they slice it into bits, and then they tell you how conceptually. to glue the pieces together again. So the way that this proof that surfaces are spheres or bagels or two-handled bagels or bagels with 17 holes or some specific number of holes, and that's all there is. The way that works is basically you chop the surface into triangles, you work out how all the triangles fit together edge to edge,

and then you do a massive mathematical simplification of this sort of huge digsaw puzzle, and you end up discovering, that you can simplify the structure down until you can actually count how many holes there are and that really is the only thing that's going on. You can do this in three dimensions. You can chop space into, let's say, I mean, one of the simplest three-dimensional curved spaces

to, well, curved in a sense to understand, is called the flat torus. It's a sort of hyperbagle. You just take a cube and you have a rule which says whenever you go off the edge of the cube, you immediately come in again on the corresponding space. position on the other side. It's like these video games where where something goes off the edge of the television screen,

it comes back, goes up on the right, it comes back on the left, as if the screen wraps round. You can wrap the faces of a cube round. And it's that representation that suddenly opens up a huge pile of different, weird, fascinating, three-dimensional shapes. From the inside, it's just lots of little solid lumps. You know, if you looked around in a limited region, it would look as if you were just in our ordinary three-dimensional shapes.

three-dimensional space. But as Marcus says, if you look far enough, you might discover you're looking at the back of your head. June, Barry Green, Pai Carrey himself, as I understand it, warns other mathematicians to beware and being seduced by the conjecture. Why did he say that and why was it saying that?

Actually, I'm not really sure myself about that. But I can imagine that it would be because it's easy to state. I mean, it's a natural progression from the two dimensions. And he himself had actually, in 1900,

come up with what he thought was a proof of something comparable only to discover that he'd made another mistake. And actually, I find it very reassuring to know that someone of Poincere's stature can make mistakes. And I'm sure most of us do too. So I suspect that it was because he himself knew that it wasn't going to be. a straightforward thing to do as it had been in two dimensions.

Now for people, non-mathematicians, he having made that conjecture, why has it taken, why do people be pursuing it for over 100 years to try to solve it? I mean, you'd say, well, he's made, there it is, let's get on with it. So what's been going on, Marcus? Well, the exciting thing is it might have been wrong, you see. There might have been an interesting shape out there, which looked very much like a hyper football,

and it was something completely different, I mean, that's funny things happen when you move that, Donation. Well, and in fact, there was a very similar problem. June's already been talking about it. Poincerey, there's a statement which, if you're a topologist, sounds very similar to this conjecture of quencarus about the three-dimensional wall. And it says there's another set of properties that should characterize the three-dimensional ball. And it looks just as plausible, just as likely,

and it's just as difficult to think of anything else. But there's a thing called the Poincerey dodecahedral space, which is a dodecahedron whose opposite faces are twisted together with a two-fifths turn. And it proves... You're going to have to on, right. You're just trying it on, Stuart. I just want to show you how difficult...

You're going to have to... I'll show how difficult it is to come up with this idea. There is a three-dimensional space constructed from a dodecahedron. This is 12 pentagons fitted together. There's a pattern like this on footballs. We're back to footballs. It's a sort of hyper-football.

But when you... The rule for going out through a face where you come back in again to glue it together is not just opposite faces are glued together but you twist them. And the result of this, this is a space that's not a sphere but in many, many topological respects it looks like a sphere.

Once you've had your fingers burn with this kind of example you realise that nothing in this subject is obvious. Can we just get back? June, you want to say something? I was just going to say that, I mean, and of course, I mean, after Juan Correa, we were going to go into that. But, I mean, there was another classic proof

in inverted commas in 1930 by a mathematician called Whitehead from Oxford and he thought he'd come up with the answer but in fact for a sharper statement than the conjecture and like Poincerey himself found the mistake and found a counter-example and that is often actually what happens with these discoveries of mistakes

I mean you just need one counter-example and the whole thing gets blown away It can be very disappointing. There's something called quoncaritis, actually, where so many people think they've proved this thing, and that it turned out, I mean, it's had so many false proofs, probably more than anything else,

because it's such a slippery, slippery subject. But behind it there must be, obviously you mathematicians have been intrigued by it and still are intrigued by it. Because as it goes along trying to be, sorry, John, I just finished my finger a bit of off, as it's trying to be proved, people are driving because they're finding out more and more

as they're getting towards it, or maybe not getting towards it. Absolutely. Jerry June had a... Well, I'm just going to say, I mean, one of the things that was very curious about the Poincre Conjecture is the dimension aspect of it, because it's the three-dimensional case. And you would think naturally, well, okay, you can do two. Three is turning out to be pretty tough.

Well, obviously four, five and, you know, 17 are going to be horrendous. But actually, no, it turns out that the other cases, the high dimension cases, have been solved, which was one of the things, and of course, which really, put the focus onto the three-dimensional case, the Ponkeric architecture. Marcus, and then I want to get on to this Russian mathematician. You were going to say so. Well, I was going to say exactly that, actually. How helpful.

But I mean, it is kind of curious that, in a sense, three-dimensional space is a little bit tight, and you can't do the same tricks that you can do in four or five and higher-dimensional spaces. So we're now talking about wrapping up spaces that obviously we can't see. I think that's another important thing. because listeners are probably saying, well, what is this four-dimensional football or something? And the important thing is we can't, yeah, mathematicians, we can't see these things,

but we produced a language to be able to talk about them, which is basically numbers and coordinates. So, and that's how we see four-dimensional objects is to change the shape into numbers. So in two dimensions, three dimensions, the thing in numbers also has a picture, but when we move up into four dimensions, the numbers still work. the picture's gone away because we can't see what a hyper football looks like. It's like the LLA, isn't it? I've discovered the book of the universe. It's written in the language of mathematics.

That's exactly right, yeah. So, I mean, we use it. We talked about the London Underground. Well, the A to Z is about numbers. You know, if you want to find a particular place, you've got two numbers to go somewhere. In four dimensions, you've just got four numbers which tell you where to go.

I've got to come to this man, Gregory Perilman, a Russian, who a few years ago, just four or five years ago, posted something saying, which led people to believe. that he had solved this conjecture. He was awarded the equivalent of the Nobel Prize. He refuses to take it, a million pounds prize.

He refuses to come and accept that. He hasn't been offered a million dollars yet, actually, but he was offered the Fields Medal in the summer and turned it down. Now, what is he said that convinces many people, and it isn't quite proved yet, he's taking an awful long time. Three teams are trying to prove whether his proof is provable. At least. Okay, it's over to you, Ian.

Okay. Perroman is clearly a very, very clever mathematician, slightly eccentric, becoming more and more reclusive, possibly as a result of the publicity for his work. And he picked up on a particular approach to these problems that was introduced by an American called Richard Hamilton, which was effectively to relate it to,

it's related to an idea from general relativity, from Einstein's relativity. And this is the idea of curved space, and there is a thing called the Ritchie Flow, named after an Italian mathematician Ritchie. And the Ritchie flow is a lovely gadget. You take a complicated curve shape, and you let the curvature slowly redistribute itself around the shape. So you can imagine you take a balloon and you twist it into some very complicated sort of shape, but basically it's just a round balloon.

And if you slowly relax the forces that are twisting it, it will simply twist itself back into a nice round spherical balloon shape. The richy flow does this for any shape whatsoever, and it kind of tries to simplify it as much as possible. And basically, Perlman's idea is, let's take any old three-dimensional shape, let's follow it through the richy flow and see where it goes, and let us hope that if this Poincerey condition about,

as Marcus says, every lasso closes up, that we can use that to show that this means the richy flow is constrained in certain ways, it must behave in certain ways, and the only thing it can end up with is a nice round hypersphere. So this is the idea, and it's a beautiful idea. It's got a nice pedigree because it works in two dimensions,

and Perrauman has found a whole lot of interesting and unexpected properties of this richy flow, which mean that a lot of nice things work in three dimensions. And he sort of stopped there with his work, which was never refereed. So where is it now? I mean, we're getting towards the end. This programme is closing up.

Yes, we're going to tie the list. We're going to become a sphere, quite soon. Well, this summer was quite a key moment because we had the International Congress of Mathematicians, which happens every four years when these field medals are given out. And so everybody was looking to see whether Perlerman, you have to be under 40 to get one of these. This is the last time he could win it.

So the fact that he was awarded this Fields Medal was felt to be some sort of recognition of whether he'd actually proved it or not. The Fields Medal Committee was still a little bit hesitant about saying whether it had been completed. But I was at the ICM and I heard several talks, one by Hamilton, declaring that, yes, it now has been proved, that they really do feel that although it was an incredibly condensed proof, it was posted on the internet. He hasn't submitted it to any journal. But the interesting thing is taken four years for us to go through this proof.

I mean, this is the, you know, mathematics has become very sophisticated and quite tough. But I think these three groups have now stabilized and really feel, in fact, he's proved a lot more than just the Poincorre conjecture, because we like, as I said, to classify things. Actually, what Parliament has proved is something called the Thurston Geometrisation conjecture, which tells you the building blocks of all the shapes are possible in three dimensions. June, do you want to carry that for? No, not particularly. It's a bit too hard for me. Well, that's a relief. And so where does that take us now, then? Is this the equivalent of a Newtonian leap forward?

If this is proved, and there's still a slight doubt, and the Chinese are saying they did it really, and there's all that. Well, you know, there's a lot of work that these other groups have been done, and they should be recognised for it. But I think most of the groups are saying that, no, the essential proof is in Perlman's papers. Sometimes there look like gaps, but actually it's just using an idea he's used before. So most people feel that it's really Perlerman's proof. And it is completely fundamental.

These are the building blocks of space and therefore, you know, it's something that we're going to be using all of us because we're all interested in the nature of space and shapes. Briefly, Jo. Just one thing I wanted to say was that in a way, Perilman is very like Poincerey in the way that he works. You know, this business of being able to actually write down mathematics where mere mortals need to put in, you know, 17 pages of their own working

to get from one line to the next. And Puancray was like that for his peers, and it seems like Perilman is the same. So I think it's rather a nice... Time and assume. All right. Well, thank you all very much indeed. I feel as if I don't have had one heck of a workout. I'm exhausted.

Right, thank you very much to June Barrow Green. And with pleasure and delight, that's my question. In Stuart and Marcus de Sootoy, and next week we'll be talking about Alexander Pope, back to the 18th century. Thanks for listening. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programmes

about history, science and philosophy, at bbc.com.com.uk forward slash radio four.