May 8, 2003 2:31 PM Subscribe

'The Poincare Conjecture' Solved? "Dr Grigori Perelman, of the Steklov Institute of Mathematics of the Russian Academy of Sciences, St Petersburg, claims to have proved the Poincare Conjecture, one of the most famous problems in mathematics. The Poincare Conjecture, an idea about three-dimensional objects, has haunted mathematicians for nearly a century. If it has been solved, the consequences will reverberate throughout geometry and physics."

Also of note is that Perelman's solution is only a benign side effect of his efforts toward defining all three-dimensional surfaces mathematically, which if successful would allow humanity to "produce a catalogue of all possible three-dimensional shapes in the Universe, meaning that [mankind] could ultimately describe the actual shape of the cosmos itself."

posted by eyebeam (13 comments total)

Also of note is that Perelman's solution is only a benign side effect of his efforts toward defining all three-dimensional surfaces mathematically, which if successful would allow humanity to "produce a catalogue of all possible three-dimensional shapes in the Universe, meaning that [mankind] could ultimately describe the actual shape of the cosmos itself."

posted by eyebeam (13 comments total)

Here's more info about The Poincare Conjecture. I confess that while I find this stuff conceptually fascinating, the actual mathematics involved make my ears bleed.

posted by eyebeam at 2:35 PM on May 8, 2003

posted by eyebeam at 2:35 PM on May 8, 2003

Well, delmoi, you're right in that according to the Wikipedia, the first rumors about his work surfaced late last year. But apparently the math is so advanced that his work is still being checked, and it has to stand up to two years of scrutiny to be considered okay. So, arguably, me and the BBC are behind the curve on the fast-breaking math news.

I just thought it was juicy brain food, and might make for some interesting (if perhaps uber-geeky) discussion around here. I thought the idea of being able to create a catalog of every possible three-dimensional object was a particularly mind-blowing concept.

Sorry if it's too "old news" to be interesting.

posted by eyebeam at 2:48 PM on May 8, 2003

(Intro) Topology is mostly definitions... Understanding those definitions is most of the early coursework. The subject is pretty mind blowing, and a lot of fun to wrap your head around. To show what the definitons are generally like, I'll dole out a few from the beginning of the MathWorld article. In topology, you're dealing with what are called 'topological spaces,' which include planes, spheres, donuts, and Moebius Strips. You can assume that they're just surfaces in n-dimensions.

Simply-Connected: If you take any two points in your space, you can walk from one to the other without leaving the space. For example, the Earth's surface is simply-connected, while Hawaii (if you leave out the ocean) is not.

Closed: A space that contains all of its boundaries. Most of the things we deal with in the physical world are closed: for example, a bag of skittles includes the bag. A line that includes its endpoints is closed, while one which doesn't is open. (For example, on a number line, if we take all numbers greater than 0 and less than 1, that is not closed.)

Three-Manifold: A 3-Dimensional space where Euclidean geometry works if you look at a small part of that pace. We live in a very good example of a manifold. Here on earth, if we draw a trangle, the angles will add up to 180 degrees. But if we draw a triangle between, say, New York, London, and Casablanca, the angles won't add up to 180 degrees because we're drawing on a sphere.

Most topological definitions are kinda like this, but there's a lot more meat to them in the textbooks 'cause they want us to be good formal mathematicians. They are tools for understanding the relationships between different spaces and for talking about spaces we can't visualize, essentially.

posted by kaibutsu at 2:57 PM on May 8, 2003

Simply-Connected: If you take any two points in your space, you can walk from one to the other without leaving the space. For example, the Earth's surface is simply-connected, while Hawaii (if you leave out the ocean) is not.

Closed: A space that contains all of its boundaries. Most of the things we deal with in the physical world are closed: for example, a bag of skittles includes the bag. A line that includes its endpoints is closed, while one which doesn't is open. (For example, on a number line, if we take all numbers greater than 0 and less than 1, that is not closed.)

Three-Manifold: A 3-Dimensional space where Euclidean geometry works if you look at a small part of that pace. We live in a very good example of a manifold. Here on earth, if we draw a trangle, the angles will add up to 180 degrees. But if we draw a triangle between, say, New York, London, and Casablanca, the angles won't add up to 180 degrees because we're drawing on a sphere.

Most topological definitions are kinda like this, but there's a lot more meat to them in the textbooks 'cause they want us to be good formal mathematicians. They are tools for understanding the relationships between different spaces and for talking about spaces we can't visualize, essentially.

posted by kaibutsu at 2:57 PM on May 8, 2003

For what its worth, either because I've had my head stuck too deep in the books, or I've been severed from the mathematical community for months, I hadn't heard about this yet. Thanks for the link, eyebeam.

posted by kaibutsu at 2:58 PM on May 8, 2003

posted by kaibutsu at 2:58 PM on May 8, 2003

Here are the rest of the million-dollar Clay Problems. Erm, um, rather, HERE are the other Clay Problems. The biggest, in my humble opinion, are the Riemann Hypothesis, which would cause a new revolution in number theory if its proven, and P-vs.-NP, which would either revolutionize our linear computing power, or (more likely) tell us that we really do need to find better ways around the big problems. Either way, the proof would probably be revolutionary.

posted by kaibutsu at 3:07 PM on May 8, 2003

posted by kaibutsu at 3:07 PM on May 8, 2003

Thanks for dropping some definitions, kaibutsu, that helps me get my brain around this stuff a little better. I think I find this stuff interesting because I'm a visual thinker, and geometry is a type of math that's easier for me to see in my head than, say, algebra.

Also, totally off topic, I loved all the selections from Eastern poetry in your profile. Thanks for that, as well!

posted by eyebeam at 3:16 PM on May 8, 2003

Also, totally off topic, I loved all the selections from Eastern poetry in your profile. Thanks for that, as well!

posted by eyebeam at 3:16 PM on May 8, 2003

I can walk between any two points on a circle by simply following the circle. However, a circle is not simply connected. The loop definition given here is probably a better one.

posted by joaquim at 3:29 PM on May 8, 2003

And that's probably why they threw me out of University. The surface reason, of course, was my Honor's thesis on philosophy and rock history entitled "You, Kant, Always Get What You Want." But the confusion of Path Connected and Simply Connected certainly played a good hand in the final decision.

I'm sorry.

posted by kaibutsu at 6:08 PM on May 8, 2003

I'm sorry.

posted by kaibutsu at 6:08 PM on May 8, 2003

If he's right, this could get ugly, and no one wants to see a repeat of the math riots that broke out 10 years ago after Fermat's last theorem was proven.

posted by planetkyoto at 6:46 PM on May 8, 2003

posted by planetkyoto at 6:46 PM on May 8, 2003

Math riots!

*Mounted police throughout Hyde Park kept crowds of delirious wizards at the University of Chicago from tipping over cars on the midway as they first did in 1976 when Wolfgang Haken and Kenneth Appel cracked the long-vexing Four-Color Problem. Incidents of textbook-throwing and citizens being pulled from their cars and humiliated with difficult story problems last week were described by the university's math department chairman Bob Zimmer as "isolated."*

Zimmer said, "Most of the celebrations were orderly and peaceful. But there will always be a few -- usually graduate students -- who use any excuse to cause trouble and steal.

gleuschk, can you account for your whereabouts during this time? Hmmmm?

Oh course, this was nothing compared to the Foucault disturbances in English Departments during the 1970s. Now those were ugly.

posted by jokeefe at 8:02 PM on May 8, 2003

Zimmer said, "Most of the celebrations were orderly and peaceful. But there will always be a few -- usually graduate students -- who use any excuse to cause trouble and steal.

gleuschk, can you account for your whereabouts during this time? Hmmmm?

Oh course, this was nothing compared to the Foucault disturbances in English Departments during the 1970s. Now those were ugly.

posted by jokeefe at 8:02 PM on May 8, 2003

Wow, I went to work after reading this thread and there was Poincare on my desk, being used in a paper to prove that snoring sounds show features of "chaos".

posted by planetkyoto at 1:33 AM on May 9, 2003

posted by planetkyoto at 1:33 AM on May 9, 2003

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Someone solved that like a year ago.

posted by delmoi at 2:33 PM on May 8, 2003