Tom Rocks Maths: S02 E05

Tom Rocks Maths is back on Oxide Radio for Hilary Term 2019 with the usual eclectic mix of maths and music. Learn more about the only million-dollar Millennium Problem that’s been solved so far, fun facts about the number 6, and a nursery rhyme themed puzzle. Plus, music from Bring me the Horizon, Queen and Papa Roach. This is maths, but not as you know it…

Perelman and the Poincaré Conjecture

Grigori Perelman is a quiet and 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! I 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…

  • When the Millennium Bridge opened its resonant frequency matched that of walking pedestrians which caused it to vibrate massively as seen in the video below

  • In the field of topology things are considered equal if you can get from one to the other by doing a ‘smooth and gradual change’
  • The Poincare Conjecture states that any shape satisfying a set of three conditions can be deformed into a sphere, and this will hold true in any number of dimensions
  • It had been proved for all dimensions except 4, which was shown to be true by Grigori Perelman in 2002
  • He published his proof on the internet and then refused the $1 million prize money, instantly becoming a sensation

You can listen to the full interview for the Naked Scientists here.

The Poincare Conjecture

I’ve saved the best until last because this one’s been solved! Hallelujah! Praise the Lord! God save the Queen! Slap my thighs and serve me a milkshake! And the story of the man that did it is fascinating. But we’ll get to that… first up I’d better tell you what the problem is/was.

For the Poincare Conjecture we venture into the shape-shifting world of topology. This is a real favourite amongst mathematicians because it’s great for blowing people’s minds. The classic example: in topology a donut and a teacup are the same item. Yes, you did read that correctly. The reason? They both only have one hole: in the centre of the donut and in the centre of the handle of the teacup. It’s the number of holes that’s key. If you have a muffin and a donut how would you tell them apart? And no you can’t eat them. In topology what you would do is take an elastic band and put it around each object. Then you squeeze it in tightly until the object becomes one ball of mass. Well the muffin does, but the donut doesn’t. Not without breaking it at least – that’s the key. The hole in the donut means that you can’t shrink it and squeeze it down into one little ball without breaking it somehow. There’s no way to remove the hole.

If you’re still struggling to get to grips with topology, think of it like this: you have a donut made out of Playdoh and you need to mould it into a teacup with the only rule being that you cannot create or destroy any holes. You can do it. It’s a little fiddly yes, but it can be done. Now imagine you need to make a muffin with the same rule. No hole destroying. You can’t do it. You’ll always be left with a loop of Playdoh hanging off your perfectly crafted muffin.

The Poincare conjecture is based on this same idea: imagine you have a smooth shape made out of Playdoh and it has no holes, then the question is: can we mould it to make a sphere? Sounds easy enough in our three dimensional world, you can literally get a ball of Playdoh and make any shape you want – you can always get back to a sphere. But what happens in higher dimensions? Time is often referred to as the fourth dimension in Physics, but what comes after that? We as humans are not programmed to visualise it, but in maths higher dimensions exist. The interesting thing with this problem is that we know you can make a sphere in any number of dimensions except four and this is what the Poincare Conjecture asks. Can you take any four-dimensional smooth object that doesn’t contain any holes and turn it into a sphere? Turns out you can, just ask Gregori Perelman (photo credit: George M. Bergman – Mathematisches Institut Oberwolfach (MFO), GFDL, https://commons.wikimedia.org/w/index.php?curid=11511619).

Perelman,_Grigori_(1966)

Perelman is a Russian mathematician and he is quite the character. He showed that the Poincare Conjecture was true and then without really telling anyone just posted his solution online in 2002. No big event, no announcement, just a casual ‘oh here’s what I’ve been working on the past few years – I solved a Millennium Problem’. You have to love him for it. And it gets better. It took the Clay Institute eight years to verify that his solution was indeed correct and Perelman did not like this, not one bit. He couldn’t understand why they had to check his work – he is a mathematician; he doesn’t make mistakes! When the time came around for him to be presented with his money he declined, flat out turned it down. He was also awarded the maths version of the Nobel Prize, the Field’s Medal, and didn’t want that either. He was so annoyed at the way in which he was treated following his work that he gave up the subject and it is rumoured that he now works in Computer Science. Get your bets in now that he solves the biggest problem in that subject within the next few years…

This is a nice story for me to end on as it brings me back full circle to my starting point for these articles. The Millennium Maths problems were the first set of problems to get me really excited about maths. Whether it was because of the money or just the idea that these things even existed, I don’t know (it was probably the money), but what I do know is that Gregori Perelman is the perfect example of everything that is great about mathematicians. He started working on the Poincare Conjecture in 1995 before it was even a Millennium Problem, and then he turned down all of the fortune and fame that came with his solution. He simply wanted to be left alone to ‘do the maths’.

If after reading my articles you were thinking of attempting to solve one of these problems yourself by all means get stuck in, but as I started with a word of warning about the difficulty of these problems, let me end with another. Estimates of the number of hours spent by Perelman in solving the Poincare Conjecture actually put the $1 million prize money at less than the minimum wage. You have to love the subject to tackle these problems and I hope that I have and will continue to help you do exactly that.

 

You can listen to me talking to mathematician Katie Steckles about the Poincare Conjecture here.

I’ve written a series of articles on each of the 7 Millennium Problems which can be found here.

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