Wednesday, January 03, 2007

Breakthrough of the Year

It is often asked whether pure mathematics is discovered or invented. I think the answer is both. Pure mathematics is an exploration of the operating space of the human mind, and because products of the human mind are considered to be inventions, pure mathematics is both invented and discovered. There is no need to postulate a 'Platonic realm' of timeless mathematical forms to accept that pure mathematics is discovered, but pure mathematical discovery is not the discovery of something which exists independently of the human mind. How are we to explain the effectiveness of mathematics in the natural sciences? Well, the mind supervenes upon the brain, and the brain is the outcome of evolution by natural selection, hence the operating space of the human mind has been shaped by the structure of the natural world. Appropriately enough, the December 2006 issue of the journal Science declared the 'breakthough of the year' to be Grigori Perelman's proof of the Poincare conjecture.

This famous conjecture asserts that the 3-dimensional sphere is the only simply connected, compact, 3-dimensional manifold-without-boundary. In simple terms, a compact manifold is one which will always be of finite volume, irrespective of which geometry you place upon it. A simply connected manifold, in simple terms, is one which has no holes passing through it (when that manifold is embedded in a Euclidean space of higher dimension). A sphere in any dimension is simply connected, but the 2-dimensional surfaces of doughnuts and pretzels, for example, clearly have holes passing through them, hence they are non-simply connected. As another example, there is a 3-dimensional version of the doughnut, called the 3-torus, which is compact, but non-simply connected.

Perelman appears to have proved the Poincare conjecture, and a more general statement called the Thurston geometrization conjecture, in three papers which were submitted, not to an academic journal, but to the arXiv electronic preprint repository, between November 2002 and July 2003. In the three years since, other mathematicians have been attempting to verify Perelman's claims, and once a consensus was reached, this culminated in Perelman being awarded the Fields Medal this year.

Intriguingly, Perelman has refused to accept the award. Since producing his proof, Perelman has abandoned his academic post, and is reportedly jobless, and living with his mother in St Petersburg. Perelman appears to be disgusted by a lack of ethics in the mathematics community, and, in particular, by claims in 2006 that two Chinese mathematicians, Cao and Zhu, had found the first complete proof. According to the article in Science, two other mathematicians, Kleiner and Lott, who attempted to expound and elaborate Perelman's proof, complained that Cao and Zhu had copied a proof of theirs and claimed it as original, and "the latter pair grudgingly printed an erratum acknowledging Kleiner and Lott's priority. This fall, the American Mathematical Society attempted to organize an all-star panel on the Poincaré and geometrization conjectures at its January 2007 meeting in New Orleans, Louisiana. According to Executive Director John Ewing, the effort fell apart when Lott refused to share the stage with Zhu."

The Poincare conjecture was one of the million-dollar Millennium Prize problems posed by the Clay Mathematics Institute in 2000. Perelman has apparently said that he will not decide whether to accept this prize until it is offered. This, of course, is exactly the right thing for Perelman to say: if he were to say now that he would reject the prize, then it might well not be offered to him, or perhaps offered jointly; by waiting to be offered the prize, he can then reject the offer when it comes.


Anonymous said...

Discovered or invented - yes, that has often stopped me for a minute. Now I know the answer - both. Thank you, Gordon.

However I'm afraid 'In simple terms, a compact manifold is one which will always be of finite volume, irrespective of which geometry you place upon it.' is not quite simple enough for me! I think I need pictures.

It's very interesting that the question of priority is so important to such intelligent minds - in some ways they work on the same level as the rest of us.

Gordon McCabe said...

Intelligent, but not always mature minds, Clare.

In differential geometry, you basically have four levels of structure: (1) the level of the point set, where the only structure is the cardinality of the set, i.e. whether it has a finite number of points, or a countable infinity of points, or an uncountable infinity etc; (2) the level of the topology, where the point set is equipped with continuity and connectedness; (3) the level of the manifold, where the set is equipped with coordinates and a dimension (this level of structure enables one to introduce vector fields on the manifold, and their generalisation, tensor fields); and (4) the level of the geometry, where the set is equipped with shape and size.

You can keep the topological and manifold structure fixed, and vary the geometry. See, no pictures necessary at all!

Anonymous said...

I have been thinking about this and I have an idea in my mind now - and I think that will have to do. It is just a very vague idea of what this is all about, I'm afraid. I think I need more, far more, maths than I have - and it's too late now, I fear - I've spent too many years in the concrete world.

But thank you very much for taking the trouble to explain. Even if my idea is just vague it is much better than nothing at all.

Anonymous said...

Gordon, you have a way with words. The only problem is, they are not words I am familiar with. Or should I say, I am familiar with all the words, but I have never come across them placed together in such a way. So I'll take your word for it that you are making sense. I will certainly never look at a doughnut in the same way.

Gordon McCabe said...

The concrete world Clare? Well, at least you've cemented your reputation in that field.

Jeffrey Weeks writes some very nice non-technical stuff on the topology of space, with lots of diagrams and pictures, and I definitely recommend his book:

Gordon McCabe said...

I'm using all the right words Shifty, but not necessarily in the right order.

Mmmmmmm, doughnuts.

Anonymous said...

strange, i understood your post but not your explanation to Clare.

Gordon McCabe said...

What ever happened to Shifty Loner? That's what I want to know.

Anonymous said...

After compactification of the real line, could we still assert that the resulting compact projective line is of finite size?