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Has The Poincare Conjecture Been Solved?
Zack Coburn writes "An article in the Boston Globe alludes to the Poincare Conjecture being solved, possibly. For those who are unfamiliar with the conjecture, the article gives a brief description: "To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes. Though abstract, the conjecture has powerful practical implications: Solve it and you may be able to describe the shape of the universe." Apparently Grigory Perelman may have proved it, which would mean a $1 million award from the Clay Mathematics Institute." We've previously discussed other possible Poincare proofs.
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Clay Atkins Diet Award Face huhhhhhhh
I saw something like this recently over at tubgirl tech archive
(First off, remember that us MATHEMATICANS DO IT SMOOTHLY AND CONTINUOUSLY.) Hehehe, wow, too many New Year's drinks. Anyway, on to the story.
Last year I assisted with some research involving Poincare along with four other professors. We studied weak wide-angle temperature correlations in the cosmic MICROWAVE background.
There exists a simple geometric model of a NON-INFINITE and NON-NEGATIVE curved space, which we call the POINCARE space.
This may sound foreign to you, and I'd probably be worried if it didn't, but this POINCARE space can account for these observations with no fine-tuning.
From our "Nature" (425 2003 593) article: "If confirmed, the model will answer the ancient question of whether space is finite or infinite, while retaining the standard Friedmann-Lemaitre foundation for local physics."
So, yes, Poincare is VERY important, and this postulate as well as the query as to whether it's been appropriately solved has a HUGE impact on all kinds of other research (math, physics, computer science, etc.) such as this very research that I participated in.
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Could I have a different story? One that I even remotely relate to?
But I've been too busy trying to get first post to tell someone. I wonder how many other huge discoveries are stopped by the same problem. It's a good thing Einstein didn't have Slashdot.
No.
(It even says in the freaking article stub that the proof is merely alluded to, for crying out loud.)
Slashdot: when news breaks, we give you the pieces.
welcome our new topological overlord.
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But in all seriousness, if he really solved it, then this could mean all kinds of "kewl" stuff will be coming out of the field of topology...
I remember seeing a (webcasted) talk given by the Clay Institute about their $1M math prizes, in particular, the one about P=NP. In it, the speaker said that "if P=NP is proven, then all the others are going to fall in short time, making that solution worth $8M" (or $1M * the number of problems).
I was really hoping that that kind of money would get the P=NP results first...
my sig's at the bottom of the page.
Let me guess.. he says that the new topological object is universe-shaped?
Karma: SELECT `karma` FROM `users` WHERE `userid`=138474;
I was planning on proving it for my Phd Thesis... :/
Now, what am I supposed to talk about ?
in case of slashdotting...
BERKELEY, Calif. -- A reclusive Russian mathematician appears to have answered a question that has stumped mathematicians for more than a century.
After a decade of isolation in St. Petersburg, over the last year Grigory Perelman posted a few papers to an online archive. Although he has no known plans to publish them, his work has sent shock waves through what is usually a quiet field.
At two conferences held during the last two weeks in California, a range of specialists scrutinized Perelman's work, trying to grasp all the details and look for potential flaws.
If Perelman really has proved the so-called Poincare Conjecture, as many believe he has, he will become known as one of the great mathematicians of the 21st century and will be first in line for a $1 million prize offered by the Clay Mathematics Institute in Cambridge.
Colleagues say Perelman, who did not attend the California conferences and did not respond to a request for comment, couldn't care less about the money, and doesn't want the attention. Known for his single-minded devotion to research, he seldom appears in public; he answers e-mails from mathematicians, but no one else.
"What mathematicians enjoy is the chase of really difficult problems," said Hyam Rubinstein, a mathematician who came from Australia to attend meetings at the Mathematical Sciences Research Institute in Berkeley and the American Institute of Mathematics in Palo Alto, Calif., hoping to better understand Perelman's solution. "This problem is like the Mount Everest of math conjectures, so everyone wants to be the first to climb it."
The Poincare Conjecture, named after the Frenchman who proposed it in 1904, is the question that essentially founded the field of topology, the "rubber-sheet geometry" that looks at the properties of surfaces that don't change no matter how much you stretch or bend them.
To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes. Though abstract, the conjecture has powerful practical implications: Solve it and you may be able to describe the shape of the universe.
Dozens of the best mathematicians of the last century tried with all kinds of approaches to solve the conjecture. Some thought they had it for months, even years, but counter-examples and flaws just kept springing up. Simply-stated but elusive to prove -- like Fermat's Last Theorem -- this conjecture has spurred the development of whole branches of mathematics.
A decade ago, after some work in the United States that colleagues described as "brilliant," Perelman gave up a promising career to work in seclusion in St. Petersburg. Although he appears occasionally, most recently for lectures at the Massachusetts Institute of Technology and several other US schools last spring, he keeps a very low profile.
Even in mathematical circles, surprisingly little is known about him, and those who know him often don't want to speak publicly about his work.
At any rate, he seems to have used his time alone wisely. While working out the Poincare Conjecture, Perelman also seems to have established a much stronger result, one that could change many branches of mathematics. Called the "Geometrization Conjecture," it is a far-reaching claim that joins topology and geometry, by stating that all space-like structures can be divided into parts, each of which can be described by one of three kinds of simple geometric models. Like a similar result for surfaces proved a century ago, this would have profound consequences in almost all areas of mathematics.
As the foundation for his proof, Perelman used a method called Ricci flow, invented in the mid-1980s by Columbia University mathematician Richard Hamilton, which breaks a surface into parts and smooths these parts out, making
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I wish they'd help me fold my one dimensional shape!
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...if your past posts weren't virtually all trolls.
um, so you can't create a hole just by bending stuff. I have been going about hole creating all the wrong way. How will I plant this tree?
Perelman used a method called Ricci flow
Now, I'm as much a fan of Christina Ricci as the next guy, but knowing about her flow is beyond the limits of what I'd like to know about her. I do wonder, however, how that guy from "Beauty and the Beast" is involved.
Imagine a square sheet of rubber (so we can stretch, bend as we like). It has a finite area, and four edges. We choose one edge and glue it to its opposite edge. Now if you start from one point and draw a line in the right direction, you'll get back to where you started. Otherwise you'll just spiral around until you hit an edge.
... which is identical to the last one.
...
Now we take the two circular edges and we glue them together, giving a donut (a torus). Now if you go in [what you see as] a straight line in any direction, you'll never reach an edge. The surface of the donut doesn't have any sides in the way the original sheet of rubber did, but it still covers a finite area.
N.b. The problem with this example is that it's difficult to think of just the surface of the donut, without imagining it being 'in' some larger space such as the 3D world.
Now if you want a headache, try to imagine doing this starting not with a square, but rather a cube, and joining opposing faces together. The first pair is easy - you get a sort of square donut shape. The second pair gives you a donut with an inner donut removed - something like the inner tube in a tyre.
The third one is the real bugger - you have to imagine joining the inner surface of the tube to the outer one, without going through the tube. I've seen a video [uiuc.edu] that included a representation of what a similar manouvre (sp?) would look like in the 3D world that the cube started in, and I still can't fully get my head around it.
No matter what direction you moved in this weird twisted-cube-thingy, you'd never see an edge. It would give you the same effect as if there were an infinite array of cubes , with the exact same thing happening in each one. When you reach the edge of one cube, you ust move into the next one
This article says that the Universe is doing the same sort of thing, only starting with a dodecahedron instead of a cube (i.e. 6 pairs of faces instead of 3). Don't seriously try to picture this, or your head'll explode
-----
What Happened to the Censorware Project?
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Surely they mean obloid? An egg doesn't have holes. Can anybody provide a better description?
ah, yes, the red-headed stepchild of the conjecture family - Collatz!
People have been going at Collatz Conjecture For Years, and maybe this recluse is giving that a swing next time.
For Information regarding Collatz Conjecture seek The Collatz Conjecture
This proof has been out for about 9 months, and so far has stood up to intense scrutiny. Perelman is considered one of the top mathematicians in his field, and other mathematicians believe his proof is likely correct, although it is still being scrutinized. I recently attended a lecture by Richard Hamilton, who has been leading a team going through the proof, and he showed the method used and which sections of the proof had already been verified. It appears that the Poincare Conjecture finally has been solved.
If you are interested in the method of proof, Perelman used the Ricci Flow, blow-up arguments, and surgery to prove the Thurston Geometrization conjecture (a theorem far more powerful than the Poincare Conjecture alone).
There's no sig like SIGSEG
"To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes ... And while the equivalent of the Poincare conjecture has already been proven for dimensions four and up..."
:-)
Being a non-math person, it seems to me if it has been solved for two dimensions (has it?) and four and up, wouldn't three dimensions just be a special case of the many (four and up) dimensions proof? Or is there something special about that proof that limits it to four and up? Or perhaps something in a form like the two dimension proof?
Perhaps my simple understanding of proofs in euclidian geometry doesn't scale up like this
There are 1.1... kinds of people.
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Pure science is pure science. All great discoveries that have ever existed have been because of small, previously unrelated pure mathematical works (or other pure true sciences), which when done on their own seem to have no superficial meaning to someone such as an engineer or common layman, but pure mathematics is akin to pieces of a grand puzzle. Each piece is intrinsically linked to the whole picture. Looking at each piece will not reveal the puzzle, although solving each piece on its own will. This proof need not prove anything to an engineer, a computer scientist, a ballerina, or the mailman, but to a mathematician and others who understand its significance (among others) this proof advances the pure science of mathematics...and by that the world will eventually be forever changed.
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Happy new year, and to robbIE too, while you're at it.
Seen this exact post before.
...but there is not enough space in this Slashdot comment to write it.
How is this a troll? And what does an AC have to gain anyway?
POST!
like the fucking loser faggot he is
Known troll. If you look at his posting history, he posts total BS to every thread he can get his unwashed claws on.
There is more coverage at the Smegma Research Institute's information page
How do you know that the shape of the universe does not include holes?
Informatus Technologicus
Comment removed based on user account deletion
For those of you wondering "who cares" or "what's the point", well there really is none. Poincare's conjecture has no immediate practical application, or even in the near future. However, proving Poincare's conjecture is a shot in the arm for Special Relativity, which is still a "theory" (much like I have a theory that slashdot exists).
This is an exciting time to be alive. The Riemann hypothesis has been proven.The 16th Hilbert problem has been solved (by a student no less - proof that important discoveries in science are still an individual sport). After thousands of years, Archimedes Loculus has been solved. While these are airy egg-head endeavours, so was once the notion of Diracs Quantum Electrodynamics. Today, the antimatter particles predicted by QED are used to image and diagnose diseases of the brain (Positron Emission Tomography), produce light (Light Emitting Diodes), and they make transistors and diodes work. Having a mathematical proof for Poincare's conjecture could lead to new ways of structuring matters behaviors, including time-dependant transformations. For instance, shorter crumple zones which absorb more energy in automotive collisions.
Mod parent down. He's claimed to be all sorts of things and is just a pain in the ass. He's not even funny. Jesus man, at least be a funny troll. Or good. You're neither.
Here is an article from the current issue of Discover magazine on the state of the Poincare proof, and mathematical proofs in general. Sorry not a full text. Go to your library.
m at hematics/
http://www.discover.com/issues/jan-04/features/
But our universe is not 3-dimensional... it is 4-dimensional(so far as we know):
Physical dimensions 1.length, 2.width, 3.height
AND 4.time
So it would seem to me that Poincare would describe only the physical aspects of our universe, but not the universe as a whole.
One other thing, we don't know for sure that there is only one way to bend 3 dimensional space into a shape with no holes', the dimension number/num. shapes could be related to a different pattern, such as the fibbonachi sequence (0,1,1,2,3,5,8...)
I am only a high school math student, so if there are any other mathemeticians out there that can disprove any of my 'conjectures', please post.
ac is a loser faggot. check his other posts
In what way would this proof be applied outside the realm of the mathematica and theory?
Grand Unified Theory? Time Travel? Big Crunch?
Dan East
Better known as 318230.
Century-old math problem may have been solved
By Jascha Hoffman, Globe Correspondent, 12/30/2003
BERKELEY, Calif. -- A reclusive Russian mathematician appears to have answered a question that has stumped mathematicians for more than a century.
After a decade of isolation in St. Petersburg, over the last year Grigory Perelman posted a few papers to an online archive. Although he has no known plans to publish them, his work has sent shock waves through what is usually a quiet field.
At two conferences held during the last two weeks in California, a range of specialists scrutinized Perelman's work, trying to grasp all the details and look for potential flaws.
If Perelman really has proved the so-called Poincare Conjecture, as many believe he has, he will become known as one of the great mathematicians of the 21st century and will be first in line for a $1 million prize offered by the Clay Mathematics Institute in Cambridge.
Colleagues say Perelman, who did not attend the California conferences and did not respond to a request for comment, couldn't care less about the money, and doesn't want the attention. Known for his single-minded devotion to research, he seldom appears in public; he answers e-mails from mathematicians, but no one else.
"What mathematicians enjoy is the chase of really difficult problems," said Hyam Rubinstein, a mathematician who came from Australia to attend meetings at the Mathematical Sciences Research Institute in Berkeley and the American Institute of Mathematics in Palo Alto, Calif., hoping to better understand Perelman's solution. "This problem is like the Mount Everest of math conjectures, so everyone wants to be the first to climb it."
The Poincare Conjecture, named after the Frenchman who proposed it in 1904, is the question that essentially founded the field of topology, the "rubber-sheet geometry" that looks at the properties of surfaces that don't change no matter how much you stretch or bend them.
To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes. Though abstract, the conjecture has powerful practical implications: Solve it and you may be able to describe the shape of the universe.
Dozens of the best mathematicians of the last century tried with all kinds of approaches to solve the conjecture. Some thought they had it for months, even years, but counter-examples and flaws just kept springing up. Simply-stated but elusive to prove -- like Fermat's Last Theorem -- this conjecture has spurred the development of whole branches of mathematics.
A decade ago, after some work in the United States that colleagues described as "brilliant," Perelman gave up a promising career to work in seclusion in St. Petersburg. Although he appears occasionally, most recently for lectures at the Massachusetts Institute of Technology and several other US schools last spring, he keeps a very low profile.
Even in mathematical circles, surprisingly little is known about him, and those who know him often don't want to speak publicly about his work.
At any rate, he seems to have used his time alone wisely. While working out the Poincare Conjecture, Perelman also seems to have established a much stronger result, one that could change many branches of mathematics. Called the "Geometrization Conjecture," it is a far-reaching claim that joins topology and geometry, by stating that all space-like structures can be divided into parts, each of which can be described by one of three kinds of simple geometric models. Like a similar result for surfaces proved a century ago, this would have profound consequences in almost all areas of mathematics.
As the foundation for his proof, Perelman used a method called Ricci flow, invented in the mid-1980s by Columbi
I'm so drunk I can't s up strait and we're asking if some mathematical conjecture has been proved? Is this really the right storey for New Years Eve? Lets go with stories about things that are bright and shiny.
I do security
GO FUCK A RAT VALLON OR ONE OF HIS FAGGOT LITTLE COCKSUCKS, LIKE IT FUCKING MATTERS. YOU ARE ALL JUST A BUNCH OF AIDS-RIDDEN TEENAGE ZITFACED LOSERS. I'D LOVE TO RAPE YOUR WHORE MOTHER IN FRONT OF YOU THEN MAKE YOU CLEAN OUT HER AIRPLANE HANGER SIZED PUSSY WITH YOUR LOSER TONGUE. GO FUCK A RACCOON. *_g_o_a_t_s_e_x_*_g_o_a_t_s_e_x_*_g_o_a_t_s_e_x_*_ _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __
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Really. Somebody should go to Russia and rough him up. Why can't he just drink beer and watch football like everybody else?
So where are the documents/papers by Dr. Perelman describing his proof of the Poincare Conjecture? Or are they on purpose not being put available for the grand public?
Robert
OK, a fairly unfunny introduction. Fair enough.
There's no evidence of this; we don't even know who this person is. There's very little research done merely 'involving' Poincare, and this claim is just so nonspecific it could mean anything. 'Poincare' could mean anything of his, not necessarily his infamous Conjecture.
This has nothing to do with the Poincare Conjecture at all. Nor mathematics in general. This makes little sense, and is totally offtopic.
This is the only ontopic sentence here, and it's just been copy-and-pasted from the article and capitalised strangely.
The reason it sounds foreign is because it makes no sense. "I'd probably be worried if you didn't" is just message padding, and the final clause of the sentence refers to 'observations' which no one, not even the poster himself, mentioned. "no fine-tuning" is just more message padding.
I can't find any such quote on Google. The "425 2003 593" is simply a US court case reference number. Friedmann-Lemaitre is just two random names stuck together. "foundation for local physics" means nothing.
Sweeping into the conclusion in response to a nonexistent question ("Is Poincare important?")
Why does he refer to it as a postulate and not 'Conjecture' all of a sudden?
This very research which you just made up out of thin air, yes. And while Poincare's Conjecture is quite important in number theory, topology and consequently numerical cryptography, it has little relevance to physics or other sciences. He's just listed these to sound credible.
And there you have it. One of the most effective trolls today, and you all fell for it. *Sigh.*
Slashdot: when news breaks, we give you the pieces.
MARK OWEN, AOSM - Anti-Open Source Movement.
/whois'd
/whois mark /whois scottk /whois kc /whois on them.
[22:33] @Mark: Time to build in a new mode to CR - I'm sorry but this person doesn't want to be
[22:33] @masked: doesn't he know -y does the trick?
[22:33] @masked: idiot
irc.webchat.org
they have a mode +y that lets them know when someone does a
HI!
MARK OWEN, future GNAA member?s id=91172& cid=7850841. pl?sid=91172& cid=7850834
http://science.slashdot.org/comments.pl?
http://science.slashdot.org/comments
NUFF SAID.
irc.webchat.org 6667 #spiderslair, #help.services
say hi to Mark, Scottk, Thunderwoman, KC and his sluts.
all statements fictional unless proven guilty buy the first anonymous poster to reply to this.
I thought this was easy in oragami class?
My UID is prime is yours?
YOU INSENSITIVE CLOD!!!!
I went there about 4 years ago, and it was a great college.
Stop Trolling.
"-1, Wolfram Troll"
I heard Al Gore solved this years ago.
John Kerry is a Joke!
Anybody who goes out of his way to spend a half hour researching, has by definition, LOST. And the troll is laughing his ass off. Capiche?
CAN you prove this or are you just dying with karma envy?
Gosh, and I thought the Poincare Conjecture was a book by Umberto Eco!
god...how is that infomative? I already know what a penis looks like.
The parent's link is to a porn site, as you probably guessed.
If you are interested in the method of proof, Perelman used the Ricci Flow, blow-up arguments, and surgery to prove the Thurston Geometrization conjecture (a theorem far more powerful than the Poincare Conjecture alone).
It's kinda like Fermat's Last Theorem... when they finally manage to prove it, it's like a "trivial consequence" of some vastly more fundamental and powerful theorem. While it's cool and all that they can solve it now, it's quite frankly fucking annoying to know that this super-duper difficult problem, which you might have tried to bang your head against in the past, is nothing but a mere collorary to something else.
Personally, I got that relevation when I thought I'd "discovered" something real but obscure, only to find out Leonhard Euler had figured out the same 250 years ago. And with some additional stuff I didn't think of either. One moment you feel real smart, the next "that guy with an abacus in the 'stone age' figured it out long long time ago".
It's rarely that you get it so "in your face" as you do it in maths. There's no historical relativity, no real defense. They were smarter than you, plain and simple. If this guy really has figured out something that no other mathematician in all of history has figured out, I applaud him. That is not a small feat in itself.
Kjella
Live today, because you never know what tomorrow brings
Come on, what's all this science crap? Let's get back to rumor and innuendo.
I have a proof of Poincare's Conjecture, but it is too big to fit in the margins of this Slashdot post.
Equivalent in difficulty to Equine Kamasutra: Find all possible positions of a male human and female equine in standard gravity conditions, which gives pleasure to both.
transistors dont involve antimatter particles.
:-)
I don't think CPUs emitting hundreds of rads of gamma rays would go over very well anyway
Does that even work?
Did you hear about the constipated mathematician?
He worked it out with a pencil.
When you look at the state of the world, how can you not become a radical, liberal anarchist?
YHL HAND lol get a life loser
but you're right on, dude.
Has anyone figured out how to solve this equation:
1. Underpants.
2. ???
3. Profit.
From the article:
Even in mathematical circles, surprisingly little is known about him, and those who know him often don't want to speak publicly about his work.
Oh boy. People who know him won't talk about his work. That means bad news, I'm sure. Like... the proof solves the Poincare Conjecture, but as a byproduct it also proves that Cthulhu's going to wake up in 2005, and that he's really pissed.
You're the most insightful post in this thread!
*cue music*
You win the Daily Calling Bullshit On The Various Incarnations Of James Joyce On Slashdot Award!
It's very prestigious.
Look, you're smart! You don't have to make armchair observations on every physics and math problem out there.
-Libertarian secular transhumanist
last time i checked.
Proofs have reached such a level of complexity that I really have my doubts that mathematicians can verify them reliably.
It's rather like writing a 50000 line program from scratch, without ever running it through a compiler, and then having a dozen people look it over for whether it would compile. Do you really believe that a dozen people looking at a 50000 line program would be able to find all the syntax and type errors contained in it just by eye? And, if anything, mathematical proofs are more complex and subtle. With type checking and syntax, there is at least something where people have years of experience with an unforgiving "proof checker", whereas (most) mathematicians have never had to face the rigor of a formal, automated, unforgiving proof checker.
For any proof of this complexity, I think the proof needs to be formalized and the checked by computer. Even then, there is a big risk that there is some bug in the formalization of the proof.
I was wondering if the concepts in the Proof can be used to User Interface (UI) Design because the User Interface is really a surface when viewed, but 3D (plus time) (plus nD) when acted upon.
The article says that space-like structures can be divided into three parts, each of which can be described by one of three simple geometrical model? Can you say a little more about the nature of these "three kinds of simple geometrical models?"
To see a world in a grain of sand, and then to step back and see the beach where the sand lies
Over at the site of the AMS, there is an interesting overview article by J. Milnor on the ideas behind the Poincare hypothesis and Perelman's proof. You don't have to be an expert in low dimensional topology to read this...
Milnor's article
If you imagine traveling along a dimension, you see that the amount of total energy in each plane you pass though can change. But if you travel in time you can clearly see that the total amount of energy cannot do so.
So we must conclude that time is different from our three dimensions, and hence is not.
FRA: STFU GTFO
Hats off to Perelman for reminding us that money has never been a mathematician's incentive. The whole Clay thing is a travesty and not the right way to help mathematics.
(Contrast: this sort of snake-oil merchant, who puts money over truth.)
The conjecture is state wrongly. Whoever wrote it should be banned to hell, or SCO. :o). You state that a 2D simply connected closed surface is always homotopic to a S^2(regular sphere).
:o) .
First of all, you will NEVER bend a plane into a sphere. If you do so, you have solved our problems with mapping OUR planet,
The right conjecture is : any simply connected(no holes) 3 dimensional closed surface is homotopic to S^3. Simple hum?
But it seems the Russian professor did it, so I heard in the halls
A promising career in what?... Mathematics?
I am not a mathematician, but would not either a mobius strip or a tauroid be without holes, smooth and continuous?
People who think they know everything really piss off those of us that actually do.
you are a fucking idiot.
not everything relates to computers, you fanboy fucking troll. get out of your computer room a little more often.
- see subject -
-- Fighting mediocrity one bad post at a time.
This link from back in May purports the same news, not sure why its so delayed.
Its one damn thing before another. (Dick Bird 1999)
"a different piece of proof verifying software."
It's probably best to just prove the proof verifier manually. Clearly if you "proved" it with a different proof verifier, how do you verify that proof verifier? With the first one? I think there is actually a famous problem about this having to do with orders of logic and completeness, maybe Godel's problem.
Travis
http://mathworld.wolfram.com/news/2003-04-15/poinc are/
This story came out several months ago and I believe it was on slashdot incompetent fucks...
Travis
nt
The Ricci spacetime curvature tensor is a contraction of the general Riemann spacetime curvature tensor. A contraction here just means a special case of Riemann. Basicly one has :
Ricci (Rij) = Riemann (Riajb) with "slots" 1 and 3 "contracted".
Perelman and Hamilton (correct me if mistaken) tried to do a opposite contraction of the Ricci spacetime curvature by making either "slot 1" or "slot 3" variable again. And of course also prove that Ricci Flow is Homeomorphic. Hamilton proved it for some relaxed Ricci Flow conditions, Pavelman took the full scale curvature to the test and apparently succeeded.
For some details read page 218 onto 224 and page 289,290 in the black book called "Gravitation". Those last 2 pages show how by applying the simplification of Riemann to a Ricci spacetime curvature in the case of a Euclidian/Newtonian metric (no special relativity) F = m.a = m.d2x/dt2, which is our daytime geodesic path on earth, the Newton law of gravitation shows up:
Fgrav = G.(m1.m2)/r^2
Searching for "Gravitation" on www.bn.com/ will show that book. The papers of Perelman can be found like this:
checkout http://eprints.lanl.gov/lanl/ and fillout "Perelman" in the Author Field and "Ricci Flow" in the Title/Subject/Abstract field
Robert
I think there is actually a famous problem about this having to do with orders of logic and completeness, maybe Godel's problem.
No, this has nothing to do with Godel, or with anything complicated logical puzzle. It's a simple software engineering and risk management question.
It's probably best to just prove the proof verifier manually.
As opposed to what? Proof verifiers take as input manually constructed proofs. So, a proof of the proof verifier's correctness is, of course, a manual proof. And you might as well feed your manual proof to the proof verifier: if it finds bugs in it, that is useful information and if it doesn't find any bugs in your proof, you didn't lose anything.
Haven't they seen the goatse man?
'To be stupid, selfish, and have good health are three requirements for happiness, though if stupidity is lacking, all is lost.' - Gustave Flaubert
As opposed to what? Proof verifiers take as input manually constructed proofs. So, a proof of the proof verifier's correctness is, of course, a manual proof. As opposed to verifying the proof verifier with another proof verifier. You clearly didn't read my entire post carefully. I was responding to greenrd's post which reads: It's the proof verifier which really needs to be gone over with a fine tooth-comb - which is why I'd advocate that proof verifying software should be itself proven correct, and checked by a different piece of proof verifying software. He is suggesting that the first proof verifier be checked by another. I was suggesting an alternative to this. This was what my post was about in case you were unable to understand it. And I would strongly encourage you to explain to everyone what Godel's incompleteness theorem is about. And you might as well feed your manual proof to the proof verifier: if it finds bugs in it, that is useful information and if it doesn't find any bugs in your proof, you didn't lose anything. What if the proof verifier is faulty? If it finds bugs, they may not be bugs at all and vice versa. I think Godel would have something to say about your "suggestion"... Travis
As opposed to verifying the proof verifier with another proof verifier.
You are making a type error here. A proof verifier does not "check other proof verifiers", as you state, a proof verifier verifies manually constructed proofs.
What if the proof verifier is faulty? If it finds bugs, they may not be bugs at all and vice versa.
Then you do what you always do when a proof verifier finds bugs in your manual proof: you fix your manual proof. That's the whole purpose of having a proof verifier: they help you find conceptual bugs in your manually constructed proofs. You still have to fully understand your manually constructed proofs. It's roughly like "lint" or compile-time type checking. And like a good compiler, a proof verifier doesn't just say "yay or nay", it gives you specific error messages with line numbers. You don't just say "oh, it says there's a bug, darn, I guess I'll just move on to collecting butterflies", you fix the bug and try again.
And if you come to the conclusion that there is no bug in your manual proof, then there obviously has to be a bug in your proof verifier and you fix that. Either way, you win.
I think Godel would have something to say about your "suggestion"
He probably would, but it would have nothing to do with his incompleteness theorem.
i would be inclined to disagree: it would have everything to do Godel's Incompleteness Theorem. The work of Church/Turing/Godel points to the impossibility of having a function (in this case our proof verifier) that can determine whether every statement given is true or false (within an axiomatic system, which this would be). This would mean that there will always be things which cannot be proven. And in fact there are some examples of this already in mathematics. However....it is unlikely that this would pose a problem with an automated proof verifier, unless the proof happened to be one such statement (such as feeding in the statement "The proof verifier will never say that this statement is false"). WE would know that the statement is true, based upon the non-response of the proof verifier...but the machine would never know it.
A proof verifier does not "check other proof verifiers", as you state, a proof verifier verifies manually constructed proofs.
Sure you could do this. If you constructed it properly you could use the verifier to check the code of the other verifier to make sure that it is in fact correct. But anyways, I was against this use of a proof verifier; it is greenrd who suggested using the proof verifier in this way. Why don't you bother him.
What if the proof verifier is faulty? If it finds bugs, they may not be bugs at all and vice versa.
Then you do what you always do when a proof verifier finds bugs in your manual proof: you fix your manual proof.
Excuse me, are you mentally retarded? This case I'm examining is when the proof is correct but the verifier isn't, so "fixing your manual proof" would be the wrong thing to do, because it's already correct.
I think Godel would have something to say about your "suggestion"
He probably would, but it would have nothing to do with his incompleteness theorem.
Sure it would. We are checking one logical system with another. The complications that arise in that situation are what the incompleteness theorem is all about, I think you are talking out of your ass alot on this, please go back to your compiler programmer boy.
Thank god someone understands the incompletness theorem and how it relates to this topic! Thank god!!