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Russian May Have Solved Poincare Conjecture
nev4 writes "Reuters (via Yahoo News) reports that Grigori Perelman from St. Petersburg, Russia appears to have solved the Poincare Conjecture. The Poincare Conjecture is one of the 7 Millenium Problems (another is P vs NP, also covered on /. recently). Solving a Millenium Problem carries a reward of $1M, but apparently Perelman isn't interested..." nerdb0t provides some background in the form of this MathWorld page from 2003.
True math genius and the desire for money (and fame and babes, etc.) seem to be mutually exclusive traits and I think that's rather inspiring (and damned practical).
/. come form "anonymous cowards" sitting in their offices at MIT. What a god.
Take the case of Paul Erdos who was essentially homeless, but published over 1500 papers and is considered one of the all time greats in the field.
Perelman just casually posted his solution out to the web in much the same way that some of the most brilliant posts on
"...all the labours of the ages, all the devotion, all the inspiration, all the noonday brightness..." yada yada
"There is good reason to believe that Perelman's approach is correct. But the trouble is, he won't talk to anybody about it and has shown no interest in the money," said Keith Devlin, Professor of Mathematics at Stanford University in California.
I'm always amazed how much free stuff is on the internet. Free million dollar solutions! Good luck with em!
Open Source Sushi
He's trying to integrate homeomorphic convergence using a Baxter-Bates supermodality, which Krause clearly explained is impossible for T(s) in a non-linear progression. Fantastic thought process on this complex differential geometric problem.
Just kidding! I have no clue what the hell this is. I got lost after the word conjecture.
His answer to the problem was "42".
- Greg
Start a happiness pandemic
From the article:
...)
A reclusive Russian may have solved one of the world's toughest mathematics problems and stands to win $1 million (560 million pounds) -- but he doesn't appear to care.
Heh. Last I checked, $1 million dollars was not quite equal to 560 million (British) pounds. (560 thousand, sure
In an article on mathematics. Of all things.
Whocarés Conjecture If we stretch a g-string around the surface of somebody's buttocks, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same g-string has somehow been stretched in the appropriate direction around someone's face, then there is no way of shrinking it to a point without breaking either the g-string or suffocating the person. We say the surface of the buttocks are "simply connected," but that the surface of the person's face is not. Whocares knew almost hundred years ago, knew that a well shaped pair of cheeks is essentially characterized by this property of simple connectivity, and asked the corresponding question for the rest fo the people still reading this, as to why they were doing so. This question turned out to be extraordinarily difficult, and slashdotters have been struggling with it ever since.
READY.
PRINT ""+-0
- sm
I'm joking, but you're still an idiot.
"A language that doesn't affect the way you think about programming, is not worth knowing" - Alan Perlis
For those too lazy to click:
Think nothing is impossible? Try slamming a revolving door.
Place a 2 by 4 on the floor in the door.
Slam the revolving door.
Another impossible problem solved.
"Grigori Perelman May Have Solved Poincare Conjecture"
I've noticed that these kinds of announcements often make a point of appending a nationality to the name of the person involved in the discovery. Surely this proof builds on mathematical knowledge from around the world. Or was Grigori Perelman standing solely on the shoulders of "fellow Russian" mathematicians? I highly doubt it...
Perelman was unemployed for 10 years while he worked on the problem. His last job was in the States in the early 90s, where he saved enough money to live in Russia for the whole time he worked.
So think about his perspective: he's a complete loner who was ignored by the mathematical community for 10 years! Now that he's going to be a "certified" genius (with the $1M prize) why exactly should he care.
Also, it's worth pointing out that like Wiles (who solved the Fermat Conjecture), Perelman's work develops a theory that has the Poincare conjecture as a corollary which is interesting but not of central importance.
I go to Purdue, and de Branges is unable to explain himself at all. He has attempted to explain his process to other professors at a seminar here, and has only confused them. He also kicked first year grad students out of his seminar, stating they were to inexperienced. From these grad students, I have learned that he is pretty much and hotshot and an asshole. I'm thinking about going to his seminar on wednesday just to see how long it takes him to kick me out. (I'm a first year undergraduate). A note about his proof of the Bieberbach Conjecture. While de Branges did prove the conjecture, he overcomplicated it, as he does many things, and everybody and their thesis advisor has simplified his proof in some way. Mathworld really discredits his "proof" for one, it contains no proof, and his method was proven flawed by counterexample in 1998.
A better analogy would be to continuously move a circle on the surface until it becomes a point. In the case of a donut, you could draw the circle through the middle hole and around again, so you can't "shrink it to a point" my continuously moving it anywhere; it goes around the donut anywhere you put it. With a sphere, though, you can continuously move the circle to a "pole," where it becomes a point. This property is called simple connectivity.
It's pretty easy to see that all simply connected 2-manifolds (in 3 dimensions, at least) are homeomorphic to the shell of a sphere, i.e. they may be stretched and contorted to look like it. The question answered here is whether the same is true in the next dimension.
Karma: Bad (mostly due to all those "In Soviet Russia" jokes)
The n-sphere (which mathematicians generally denote by S^n) can be thought of as `all points in (n+1)-dimensional space which are at unit distance from the origin'. So S^2 is the surface of a solid 3-dimensional ball. This sometimes surprises people, who expect this to be S^3 but the key observation here is that the 2 refers to the intrinsic dimension of the object, rather than the extrinsic dimension of any space you might happen to put (`embed') the object in. The fact that we often think of the 2-sphere as being embedded in 3-dimensional space doesn't change the fact that it's inherently a 2-dimensional object. An ant wandering around on it still only has two degrees of freedom.
The 3-sphere (S^3) locally looks like ordinary, flat, Euclidean 3-space, but on a larger scale it kind of doubles back on itself - if you keep walking (or floating) in a `straight line' (well, actually the 3-dimensional analogue of a `great circle', but never mind) in any direction, then you'll eventually get back to where you started.
The Poincaré Conjecture says
This, by itself, isn't particularly enlightening to the non-topologist, but what it actually boils down to is:
What does this mean?
Well, an `n-manifold' is a space which locally looks like ordinary, flat, Euclidean n-dimensional space. So a 3-manifold is a space (like S^3) which locally looks like ordinary 3-space (but which might twist back on itself in a peculiar way on a larger scale).
`Closed' means that the 3-manifold doesn't have a boundary - no matter how far you walk, you're not going to run into a brick wall, or fall off the end. `Compact' is a bit more technical, but in this context essentially means you don't get odd shooting-off-to-infinity stuff you have to deal with.
And `simply-connected' means that the first homotopy group (the `fundamental group' of the space) is trivial. What that means is that any closed loop (of string, if you like), in the manifold, can be continuously shrunk down to a point. Here `continuous' means that you're not allowed to cut or glue the string while you're doing it.
To use a 2-dimensional analogy, the 2-sphere (the surface of the 3-dimensional ball, remember, or alternatively a British doughnut) is simply-connected, because given any closed loop in the surface, you can shrink it down to a point without it getting snagged on anything. Whereas the 2-torus (the surface of an American doughnut) isn't, because you can't shrink all closed loops down to a point - one which goes all the way round the central hole, for example, can't be shrunk.
Finally, `homeomorphic' is basically a technical word for `topologically equivalent' - we allow continuous deformations (stretching, twisting, etc, but not cutting or pasting), rotations, reflections, or any combination of these.
So, the (classical) Poincaré Conjecture is essentially a technical way of saying ``If it looks like a 3-sphere then, basically, it is''. (For certain definitions of `is', and `looks like'.)
The analogous conjecture in n-dimensional space is known to be true for n=1 (trivial), 2 (pretty simple), and 5 and above (the 5-dimensional case was proved by Zeeman, who is my PhD grandsupervisor - my supervisor was one of his students). The 4-dimensional case is weird, and there are three different forms to consider - the `piecewise linear' and `topological' cases have been proved, but the `smooth' case is still unproven.
As I understand it, what Perelman claims to have done is prove Thurston's Geometrisation Conjecture, which implies the Poincaré Conjecture as a special case - rather lik