Perelman and the Poincaré Conjecture

Interview with maths communicator Katie Steckles for the Naked Scientists. You can listen to the audio here.

Grigori Perelman is a quiet unassuming mathematician from Russia, who took the world of maths by storm in 2010 when he not only solved the Poincare problem but then refused the $1 million reward! Tom Crawford went along to the Millennium Bridge in London to meet mathematician Katie Steckles to shed some light on Perelman’s story and to find out why the Millennium Bridge was in fact its own millennium maths problem…

Katie – Turns out when this first opened, you might remember that there was some issues with it being wobbly. They’d forgotten to take into account something called resonance, resonant frequencies, and each object has its own resonant frequency and it just happened that the frequency that the bridge like to resonate at best was about the same as the frequency of people walking. So they had to close the bridge and they put in some kind of dampening supports to stop it from doing that, but it’s still a little bit wobbly. I think most suspension bridges are anyway, so it’s fine.

Tom – It’s quite windy today, but I’m feeling quite safe underfoot. Which of the Millennium Problems are we looking at today?

Katie – The Poincare Conjecture which is the first of the Millennium Prize Problems to actually get solved and I’m especially excited because it’s in the area of maths that I studied which is topology.

Tom – The main thing I remember about the very minimal topology I’ve done is that a donut and a teacup are the same thing mathematically.

Katie – That’s true. So, there’s basically a concept in topology where you can consider things to be equivalent if you can get from one to the other by doing a smooth change. So, if you have something made out of blue tack or something you can squidge around, if you can take one and deform it into the other one, but in a kind of very gradual way, you will consider those two things to be equivalent. And you can take a donut made out of plasticine and then squidge it around into a cup made out of plasticine. That’s why there is this joke about, you call a topologist when you can’t tell the difference between his donut and his cup of tea. It’s interesting how the typology view of things interacts what the real world view sometimes, things that you wouldn’t expect to be able to do, you can do. So, I can be wearing a waistcoat, take the waistcoat off, turn it inside out and put it back on the other way around while my hands are handcuffed together.

Tom – So, Katie has got her waistcoat on and now, she’s attaching the handcuffs, voluntarily, I’d like to add. Handcuffs are on, waistcoat is on, let’s see what you can do. And it’s beautiful. It’s all inside out. It looks great as well. It’s covered in stars. We’ve got someone clapping as they walk past us.

What Katie’s handcuff waistcoat trick has shown us is how simplifying shapes down to their basic structure allows us to see them in a different light and perhaps do new things with them that we previously would’ve thought impossible. In some sense, this is what the Poincare Conjecture is all about. The conjecture states that any shape satisfying a set of three conditions can be deformed into a sphere. I know this sounds a little abstract but just bear with me. Any shape that is smooth, finite, and without any holes can be deformed into a sphere. For example, Maths says I can squash a banana into an orange. This not only holds true in 3-dimensions, but in higher dimensions as well. Like there are other dimensions that we can’t see such as time, in maths there are in fact an infinite number of dimensions. The Poincare Conjecture had been shown to be true in every dimension except the fourth and proving this was the Millennium Problem. I say ‘was’ because as Katie mentioned earlier, it has now been solved by a man called Grigori Perelman.

Katie – Perelman was from Russia. He was a fantastic mathematician and he started working on this particular problem in about 1995. So, before it even became a Millennium Prize problem and it was in 2002, he basically put up what he’d done on the internet. Perelman didn’t even publish it. He didn’t even submit it to a journal. He just put it on the internet. It was kind of a bit out of nowhere so it was a really exciting time. It became a really, really big story.

Tom – So, I guess the big question is, what did he do with his prize money?

Katie – Well, that’s the interesting thing because Perelman didn’t actually want a million dollars and it’s one of those things that kind of the official line is that he didn’t want the publicity, didn’t want a massive change in his lifestyle. But it turns out, the best way to get loads of media attention is to refuse a million dollar prize. So, that kind of backfired for him. It’s one of those weird stories because he’s so reluctant to do any press about it, talk to people, but he is such a giant of mathematics and I’m really glad he’s proved it because that means typology has almost won the Millennium Prize race. I guess we got the first one in.

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