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Poincaré

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Today was a ‘big day’ in mathematics because the ICM opened in Madrid. It takes place once in four years and it is the occasion on which the Fields Medals (aka ‘maths Nobel prizes’) are awarded. Being the masters of drama that they are, the mathematicians decided to award the prizes right at the beginning, so they can move on and do serious stuff for the rest of the week.

Four Fields medals were awarded to Andrei Okounkov, Terence Tao, Wendelin Werner and Grisha Perelman. To be honest I didn’t even know about the first three, but Perelman has been well known since 2002 when he put a few preprints on the arXiv which were said to essentially prove Thurston’s Geometrisation conjecture and – as a consequence of that – the famous Poincaré conjecture. I think pretty much right away everybody who is proficient in that area of mathematics and could understand the arguments quickly was amazed and pretty confident that the techniques he used will actually do the job. Apparently it was really solid work.

And thus, he was a pretty hot candidate to be awarded a Fields medal. Not only does it look like he has solved a fiendishly hard problem, but he also turned 40 this year – the maximum age at which you can receive a Fields medal. Actually he was such a hot candidate that even the mainstream press managed to write about in advance. Of course the didn’t use the opportunity to say what Perelman prove but rather used choice quotes from other mathematicians like It’s a central problem both in maths and physics because it seeks to understand what the shape of the universe can be. (so what exactly is that supposed to tell us?!) and elaborate on the freakiness of Perelman as a person. While I don’t think that it’d be strongly misleading to portray the world of mathematics as something resembling a freak show, it still seems inappropriate to me to report the social skills – or lack of – of the protagonists in a way that exceeds the reporting of what they actually did and may make people dismiss the whole achievement because the people in question are odd.

One thing that makes Perelman newsworthy is that he declined accepting the Fields Medal and likewise he didn’t make an effort to turn the solution he came up with into a ‘proper’ paper with all the details worked out which can be published in a journal. Doing this could be worth a million Dollars as the Poincaré conjecture is one of the milennium problems set out by the Clay Maths Institute. But Perelman doesn’t seem to be interested in that either.

So what is the Poincaré conjecture, you may ask – a point that seems to be mostly ignored by the mainstream media. The fun thing is that it can be stated in a short sentence. The not-so-fun thing is that there are some non-trivial words in the sentence Every simply connected compact 3-dimensional manifold is homeomorphic to the 3-sphere. I’ll try to explain some of these words and make a few remarks on them. Let’s start with the easy bits:

manifold
That’s maths talk for ‘object’. Err, that didn’t help! So what’s an object then? A reasonably well behaved shape. Like a point. Or a line. Or a circle. Or a sphere. Or a torus, to name the favourites. Those are simple examples. A manifold has a defined dimension n (i.e. you can’t have something like a sphere and a line together) and indeed once you ‘zoom in’ on it, it looks like the well-known n-dimensional real space everywhere. E.g. just looking at a part of a circle will give you a line which in turn is 1-dimensional real space.
homeomorphic
But stop you say, the little bit of the circle is round while a line is straight, so they aren’t the same. That’s what it looks like for sure. Particularly as I didn’t mention yet that we are doing topology, aka bendy mathematics, where we take the liberty to consider all things ‘the same’ which differ from one another only by bending and squeezing (but not by tearing!). To avoid having to say ‘the same’, there’s a special word for that which is ‘homeomorphic’. A round line is homeomorphic to a straight line and – the mathematicians’ favourite example – a doughnut is homeomorphic to a coffee cup.
3-sphere
That’s a 3-dimensional sphere. It’s a bit hard to imagine, though. Just consider the 2-dimensional sphere (e.g. a football): While it is only 2 dimensional, you cannot place a 2-sphere into a plane. It just won’t fit in. That’s why it’s hard for us to imagine a 3-sphere. But luckily there are a number of tricks to think about it. One is to start off with a 3-ball (i.e. a 2-sphere that is ‘filled’ which makes it 3-dimensional). And then you use the 2-sphere that is the ball’s outside and glue it all together into a single point. Well, you can’t really picture that either, but you can do it for the 2-sphere (start with a circle, which is the outside of a 2-disk, then lift that circle up and shrink it down to a single point, voilà you’ve got a sphere) and then at least get a good technical idea about how things work in higher dimensions.
compact
Let’s just say that means ‘small’. Not in the everyday sense of size, though, as we can stretch things arbitrarily but in the sense that it’s not infinitely large. For example the line of real numbers would not be compact but the circle is.
simply connected
A manifold being simply connected means that if you have any circle in the manifold you will be able to contract (not cut or tear!) it to a single point. This is a great way to detect holes. For example, the 2-sphere is simply connected (just try playing with an elasting band on the surface of a football) but the torus isn’t simply connected (when the circle goes around a ‘hole’ of the torus there’s no way to contract it).

All right, I hope that wasn’t too bad.

The key point there somehow is the following: It turns out that there are loads of manifolds and that many of them are homeomorphic. But actually proving that they are can be quite a pain. So people came up with really clever ideas (homotopy and homology, say) which let you reduce the things you have to look at to much fewer and technically easier situations, e.g. by looking at some properties of circles which you can embed in your manifold. This is extremely powerful. But it simplifies things slightly and it can happen that manifolds look ‘the same’ from the point of view of these techniques and yet they aren’t homeomorphic. And for the special case of simply connected compact 3-dimensional manifolds, the Poincaré conjecture asserts that we don’t need to fear: Whatever looks reasonably like a 3-sphere actually is a 3-sphere.

The strange thing about the Poincaré conjecture is that in the century of its existence many topologists gave it a try. And each of them failed and gave up after some years. It looks like a reasonably easy problem but it proved to be so subtle that it defied being proven.

And in the proof – if we may call it a proof already – we have now goes beyond topology. In fact, Perelman proved Thurston’s Geometrisation Conjecture which is a much more technically difficult conjecture that classifies the geometrical building blocks of three manifolds. Here ‘geometrical’ means that unlike in topology things like size and things not being straight do matter. Having this should be a great help for three dimensional geometry.

Further Reading

August 23, 2006, 0:16

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