There's an interesting article in the Times today about Grisha Perelman and the Poincare conjecture:
Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.
After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world's mathematicians to pick up the pieces and decide if he was right.
Of course, in the classic manner of such articles, you've got to go down to the bottom of the first page to find out what the conjecture is, and to the bottom of the second before you find out what Perelman did to prove it. Even then the details are a little sketchy, but they have a lot of quotes from excited mathematicians about how cool this is.
(Don't expect a detailed explanation from me, by the way. I know next to nothing about topology-- you want Mark Chu-Carroll for that...)
They also have a fair bit of stuff about how weird Perelman is. Why are so many brilliant mathematicians so goofy, anyway?
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Why are so many brilliant mathematicians so goofy, anyway?
Because math is hard.
Why are so many brilliant mathematicians so goofy, anyway?
Possibly also because they like to talk about bunnies without holes. Unholey bunnies, if you will. For some indescribable reason, even though I knew what they were talking about, the quote from the sidebar just cracked me up:
To a topologist, a rabbit is the same as a sphere. Neither has a hole.
A rabbit isn't a sphere. It's a donut with two hollow cavities...
Actually, Hamilton's Ricci flow is best thought of as an analogy of the heat equation for 2-tensors, which is math that many a physicist would be comfortable with. Perelman's proof that you can remove the singularities as the flow evolves is topological, though.
I don't know why mathematicians are so goofy, but I never met one who wasn't. Perhaps it's because they do not speak a human language as their native tongue. (I've always marveled at Vernor Vinge's writing considering that English is not his native tongue.)
MKK
Poincare conjecture? OK, imagine you've drawn a circle on the surface of a balloon. Now imagine slowly distorting the balloon so that the circle shrinks to a point. You can always do that with a 3-dimensional balloon.
Now try it with a 4-dimensional balloon. It's fairly obvious that it still works but, in 4 dimensions, there's another problem of interest: is that the only 4D shape with this property? Are all 4D shapes with this property equivalent to the 4D sphere? This conjecture of Poincare is surprisingly hard to determine.
The reason it's important is because, in algebraic topology, objects are classed in terms of whether various loops can be shrunk to a point. This allows us to distinguish between, say, a donut and a balloon - a loop that runs right round the edge of a donut can't be shrunk to a point under any circumstances.
The ultimate goal of algebraic topology is to eventually be able to classify all shapes in this fashion. To this end, it's necessary to know whether every hole-less 4d shape is indeed equivalent to a 4D sphere. If true, this would automatically allow us to appropriately classify a massive number of shapes.
That's not quite right. First of all, S^2 x S^2 is simply connected, but not a four sphere; you need further conditions for higher dimoensions. In three dimensions, simple connectedness (ie, that all circles can shrink) is enough for the conjecture, and that is the last case to be solved. 5+ dimensions was proven by Smale, and 4 dimensions was proven by Freedman.
On another note, you can also prove that it is impossible to classify all possible four dimensional manifolds. It turns out (IIRC) to be equivalent to the halting problem.
Aaron,
Is it obvious that S^2 x S^2 is simply connected?
Michael
Aaron: Sorry, when I said stuff like "4D balloon" I meant a 3-manifold embedded in a 4D space. Apologies for sloppy phrasing.
Michael -- It's an easy theorem that pi_1(X x Y) = pi_1(X) x pi_1(Y).
One more:
The reason so many excited mathematicians were saying how cool it is, apart from it being a conjecture that had gone unproved/unrefuted for over a century, is that the proof, in fact, ranges over a great deal of mathematics. Despite the conjecture itself being purely topological (recall: pi_sub_1 of a closed 3-manifold M is trivial iff M is diffeomorphic to S_sup_3), this proof starts out by considering the geometries that can be imposed on a manifold, goes on to set up a non-linear differential equation based on those geometries and then, to solve the differential equation (or rather, to get round the awkward fact that solutions of the differential equation will in general become infinite), reintroduces topology. The ways in which the solutions of the differential equation can become singular are limited by the topology of the underlying manifold. What Perelman did was give a list of them and show that the singularity can be removed in all cases, by "surgery" on the manifold: cut away the portion that gives rise to the singularity and then restart the differential equation with new initial conditions given by the restructured manifold.
What's cool about this is the way in which the topology and the analysis based on geometry interact.
And, of course, not an exact sequence in sight.
John Morgan wrote a nice article on the proof for the Bulletin early last year:
http://www.ams.org/bull/2005-42-01/S0273-0979-04-01045-6/home.html
And what's wrong with exact sequences?
(He says with his current paper drowning in derived categories....)