The Poincarė conjecture has been in the news lately, with an article in the Science Times today. So I’ve been getting lots of mail from people asking me to explain what the Poincarė conjecture is, and why it’s a big deal lately?
I’m definitely not the best person to ask; the reason for the recent attention to the Poincarė conjecture is deep topology, which is not one of my stronger fields. But I’ll give it my best shot. (It’s actually rather bad timing. I’m planning on starting to write about topology later this week; and since the Poincarė conjecture is specifically about topology, it really wouldn’t have hurt to have introduced some topology first. But that’s how the cookie crumbles, eh?)
So what is it?
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In 1904, the great mathematician Henri Poincarė was studying topology, and came up with an interesting question.
We know that if we look at closed two-dimensional surfaces forming three dimensional shapes (manifolds), that if the three dimensional shape has no holes in it, then it’s possible to transform it by bending, twisting, and stretching – but *without tearing* – into a sphere.
Poincarė wondered about higher dimensions. What about a three dimensional closed surface in a four-dimensional space (a 3-manifold)? Or a closed 4-manifold?
The conjecture, expressed *very* loosely and imprecisely, was that in any number of dimensions *n*, any figure without holes could be reduced to an *n*-dimensional sphere.
It’s trivial to show that that’s true for 2-dimensional surfaces in a three dimensional space; that is, that all closed 2-dimensional surfaces without holes can be transformed without tearing into our familiar sphere (which topologists call a 2-sphere, because it’s got a two dimensional surface).
For surfaces with more than two dimensions, it becomes downright mind-bogglingly difficult. And in fact, it turns out to be *hardest* to prove this for the 3-sphere. Nearly every famous mathematician of the 20th century took a stab at it, and all of them failed. (For example, Whitehead of the infamous Russell & Whitehead “Principia” published an incorrect proof in 1934.)
Why is it so hard?
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Visualizing the shapes of closed 2-manifolds is easy. They form familiar figures in three dimensional space. We can imagine grabbing them, twisting them, stretching them. We can easily visualize almost anything that you can do with a closed two-dimensional surface. So reasoning about them is very natural to us.
But what about a “surface” that is itself three dimensional, forming a figure that takes 4 dimensions. What does it look like? What does *stretching* it mean? What is does a hole in a 4-dimensional shape look like? How can I tell if a particular complicated figure is actually just something tied in knots to make it look complicated, or if it actually has holes in it? What are the possible shapes of things in 4, 5, 6 dimensions?
That’s basically the problem. The math of it is generally expressed rather differently, but what it comes down to is that we don’t have a good intuitive sense of what transformations and what shapes really work in more than three dimensions.
What’s the big deal lately?
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The conjecture was proved for all surfaces with seven or more dimensions in 1960. Five and six dimensions followed only two years later, proven in 1962. It took another 20 years to find a proof for 4 dimensions, which was finally done in 1982. Since 1982, the only open question was the 3- manifold. Was the Poincarė conjecture true for all dimensions?
There’s a million dollar reward for answer to that question with a correct proof; and each of the other proofs of the conjecture for higher dimensions won the mathematical equivalent of the Nobel Prize. So the rewards for figuring out the answer and proving it are enormous.
In 2003, a rather strange reclusive Russian mathematician named Grigory Perelman published a proof of a *stronger* version of the Poincarė conjecture under the incredibly obvious title “The Entropy Formula for the Ricci Flow and Its Geometric Application”.
It’s taken 3 years for people to work through the proof and all of its details in order to verify its correctness. In full detail, it’s over 1000 pages of meticulous mathematical proof, so verifying its correctness is not exactly trivial. But now, three years later, to the best of my knowledge, pretty much everyone is pretty well convinced of its correctness.
So what’s the basic idea of the proof? This is *so* far beyond my capabilities that it’s almost laughable for me to even attempt to explain it, but I’ll give it my best shot.
The Ricci flow is a mathematical transformation which effectively causes a *shrinking* action on a closed metric 3-surface. As it shrinks, it “pinches off” irregularities or kinks in the surface. The basic idea behind the proof is that it shows that the Ricci flow applied to metric 3-surfaces will shrink to a 3-sphere. The open question was about the kinks: will the Ricci flow eliminate all of them? Or are there structures that will *continually* generate kinks, so that the figure never reduces to a 3-sphere?
What Perelman did was show that all of the possible types of kinks in the Ricci flow of a closed metric 3-surface would eventually all disappear into either a 3-sphere, or a 3-surface with a hole.
So now that we’re convinced of the proof, and people are ready to start handing out the prizes, where’s Professor Perelman?
*No one knows*.
He’s a recluse. After the brief burst of fame when he first published his proof, he disappeared into the deep woods in the hinterlands of Russia. The speculation is that he has a cabin back there somewhere, but no one knows. No one knows where to find him, or how to get in touch with him.
Good Math/Bad Math
The beauty of math; the humor of stupidity.
If anyone is interested here are some of his papers, including the one MarkCC mentioned.
Actually, the reason the Poincare conjecture is hard to prove is not that high dimensions are hard to visualize. The Poincare conjecture was first proved by Smale for manifolds of dimension 4 or more. Later Freedman was able to prove the 4-dimensional case, which is much harder. The 3d case – the one Poincare originally thought up – is the hardest of all!
Many things about the topology of manifolds are hardest to understand in dimensions 3 and 4. There’s less “wiggle room”, so all sorts of tricks that work in higher dimensions fail in these cases. As a result, completely different ideas are needed; a lot of them come from physics.
So, if you ever meet a mathematician and they tell they’re a “low-dimensional topologist”, it doesn’t mean they’re a stick figure or a cardboard cutout – it means they work on manifolds of dimensions 3 and 4.
Perelman sounds interesting. He seems to be following the tradition of Alexander Grothendieck, arguably the most visionary mathematician of the 20th century. In 1991 he walked out the door of his house and… disappeared! People say he now lives at an undisclosed location in Southern France, and entertains no visitors.
Not only do ideas come from physics but as I understand it this is actually very relevant in String theory.
Hi Mark,
The Whitehead who attempted to prove the conjecture in 1934 was Henry Whitehead, not Alfred North Whitehead of Principia fame.
There is a nice write up here:
http://www.nytimes.com/2006/08/15/science/15math.html?_r=2&8dpc=&oref=slogin&pagewanted=print&oref=slogin
A slightly more technical one in the September issue of Notices of American Math.soc. One George Szpiro is writing a popular book on the Poincare conjecture ( he has written one on the Kepler Problem). I have seen parts of the book in March and it seems good. He plans to include a description of the work of Hamilton, Perelman and others.
Actually Perelman proved something much more fundamental, that effectively killed a large part of three manifold topology research.
Thurston formulated a conjecture, around 1980, that every 3 manifold (a space that locally looks like a 3d space) can be cut up into “simple” pieces such that each piece has a definite geometry. Since there are only eight possible geometries for these simple pieces, this conjecture is extremely strong.
What Perelman did was to prove this conjecture.
Hmm – this reminds me of the famous inventor of translucent aluminium, he just showed up, kidnapped a whale and disappeared.
Has anyone thought that perhaps these mysterious disappearing mathematics are send back from the future, to influence our primitive society?
I guess I’ll be the one to complain– it’s really not right to say that a 3-manifold is something imbedded in 4-space. The actual definition is mentioned in passing in one of the above comments– a 3-manifold is a space that looks ‘locally’ like euclidian 3-space. The point is that being a 3-manifold is an ‘intrinsic’ property of a space & describing one doesn’t require bringing in a reference to some higher-dimensional enclosing space.
Hmm – this reminds me of the famous inventor of translucent aluminium, he just showed up, kidnapped a whale and disappeared.
Has anyone thought that perhaps these mysterious disappearing mathematics are send back from the future, to influence our primitive society?
Either that or they went to join the inventor of the perpetual motion machine, a female railroad executive, and a controversial Argentine celebrity in a secret compound in the Rockies, there to await the end of socialism as we know it? (LOL)
That second paragraph should be in the quote too.
Indeed. And it’s always worth noting that, in general, you can’t embed an n-manifold into n+1-dimensional Euclidean space anyway.
Hmmm, speedwell just may have an idea here. After all, to make the GaltCo Static Energy Motor actually work, you’d need to break the Second Law of Thermodynamics, and maybe an “entropy formula for the Ricci flow” is just what you need to do that. . . (-;
Actually, I think “Grigory Perelman” is just a pseudonym for Thomas Pynchon.
“Not only do ideas come from physics but as I understand it this is actually very relevant in String theory.”
Yes. I’m not familiar with any of this, but the wiki page was actually quite informative.
“Hamilton’s idea was to define a kind of nonlinear diffusion equation which would tend to smooth out irregularities in the metric. Then, by placing an arbitrary metric g on a given smooth manifold M and evolving the metric by the Ricci flow, the metric should approach a particularly nice metric, which might constitute a canonical form for M.”
So I think this and similar flows are used to study metric spaces and if they are the same solutions, modulo kinks and perhaps other singularities.
“The Calabi flow is important in the study of Calabi-Yau manifolds and also in the study of Robinson-Trautman spacetimes in general relativity.” ( http://en.wikipedia.org/wiki/Ricci_flow )
Speaking of Thurston’s canonical forms, I’m reminded of the characterisation of minimal surfaces I happened on today.
“Little to nothing was known, however, about the characteristics of myriad other, more complicated minimal surfaces until Minicozzi and Massachusetts Institute of Technology colleague Tobias H. Colding broke another “minimal surface code,” revealing that pieces of planes, catenoids and helicoids are the building blocks of all minimal surfaces, and not merely the less complicated ones.
Their article (“Shapes of embedded minimal surfaces”) appeared in the July 25 issue of the Proceedings of the National Academy of Sciences.” ( http://www.sciencedaily.com/releases/2006/08/060815162038.htm )
“study metric spaces”
Duh! Study manifolds, rather.