Slashdot Mirror
Poincaré Conjecture May Be Solved
Flamerule writes "The New York Times is now reporting that Dr. Grigori (Grisha) Perelman, of the Steklov Institute of Mathematics of the Russian Academy of Sciences in St. Petersburg, appears to have solved the famous Poincaré Conjecture, one of the Clay Institute's million-dollar Millennium Prize problems. I first noticed a short blurb about this at the MathWorld homepage last week, but Google searches have revealed almost nothing but the date and times of some of his lectures this month, including a packed session at MIT (photos), in which he reportedly presented material that proves the Conjecture. More specifically, the relevant material comes from a paper ("The entropy formula for the Ricci flow and its geometric applications") from last November, and a follow-up that was just released last month."
for the first time in ages, I'm looking forward to the discussion on this, in the hope that someone explains it in a manner I can understand
:)
Rational thought is the only true freedom
Only two years more of eating noodles before he's rich!
The link to mathworld.wolfram.com from the post says:
So, why the excitment about this later Perelman paper? Has the Dunwoody paper been debunked?
You win again, gravity!
Trick question: Green!
The subject of 3 dimensional objects with holes is quite fascinating... wouldn't it be awesome if it was discovered that toroids are actually some extradimensional manifestation... Or even that Toroids have special properties allowing FTL travel...
Food not Bombs is a nice platitude but it breaks down when you notice that the Bombees are usually well fed
What exactly is the Poincare Conjecture anyway?
Now, can someone tell me what practical applications there might be of this? Or is it strictly an abstract concept?
Game dev and music blog
For the lazy/paranoid.
I hate stories like this on /. - they bring back memories of highschool maths classes.
I try, God knows I try, but after about thirty seconds' worth of attempting to read the explanations ("homeomorphic", "closed manifolds", "simply connected") something in my brain goes "Pfffft" and I have to give up.
In short, these articles make me feel very, very stupid. Is it just me?
For those who do not know about the Poincare Conjecture, copied from http://www.claymath.org/Millennium_Prize_Problems/ Poincare_Conjecture/
If we stretch a rubber band around the surface of an apple, 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 rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.
The Solution: Yes, I do have Grey Poupon.
This is my sig. Its pathetic.
but what do i know, i'm just a model.
Easy, i shall explain
The conjecture that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. This conjecture was first proposed in 1904 by H. Poincaré (Poincaré 1953, pp. 486 and 498), and subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere iff it is homeomorphic to the n-sphere. The generalized statement reduces to the original conjecture for n = 3.
The Poincaré conjecture has proved a thorny problem ever since it was first proposed, and its study has led not only to many false proofs, but also to a deepening in the understanding of the topology of manifolds (Milnor). One of the first incorrect proofs was due to Poincaré himself (1953, p. 370), stated four years prior to formulation of his conjecture, and to which Poincaré subsequently found a counterexample. In 1934, Whitehead (1962, pp. 21-50) proposed another theorem which proved to be incorrect, then discovered a counterexample (the Whitehead link) to his own theorem.
The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical, n = 3 (the original conjecture) remains open, n = 4 was proved by Freedman (1982) (for which he was awarded the 1986 Fields medal), n = 5 by Zeeman (1961), n = 6 by Stallings (1962), and by Smale in 1961. Smale subsequently extended his proof to include
you see ?, its all quite clear if you think about it
Entropy, how can I explain it? I'll take it frame by frame it,
to have you all jumping, shouting saying it.
Let's just say that it's a measure of disorder,
in a system that is closed, like with a border.
It's sorta, like a, well a measurement of randomness,
proposed in 1850 by a German, but wait I digress.
"What the fuck is entropy?", I here the people still exclaiming,
it seems I gotta start the explaining.
You ever drop an egg and on the floor you see it break?
You go and get a mop so you can clean up your mistake.
But did you ever stop to ponder why we know it's true,
if you drop a broken egg you will not get an egg that's new.
That's entropy or E-N-T-R-O to the P to the Y,
the reason why the sun will one day all burn out and die.
Order from disorder is a scientific rarity,
allow me to explain it with a little bit more clarity.
Did I say rarity? I meant impossibility,
at least in a closed system there will always be more entropy.
That's entropy and I hope that you're all down with it,
if you are here's your membership.
Seems likely. Googling reveals:
http://mathworld.wolfram.com/news/2002-
Somebody mirror this before it gets /.ed
--Joey
I distinctly remember not understanding what the fuck I was reading about the first time it was posted.
I don't need no instructions to know how to rock!!!!
"Though his early work has earned him a reputation as a brilliant mathematician, Dr. Perelman spent the last eight years sequestered in Russia, not publishing."
"However, according to the rules of the Clay Institute, the paper must survive two years of academic scrutiny before the prize can be collected."
So, all told, Perelman is going to wait a total of 10 years from the time he started to work on the solution to the Conjecture, to the time where the scientific community lets him know if his answer is correct. Wow.
Complex mathematics? Looks like its time for Matt Damon and Pretty-Boy Affleck to write Good Will Hunting II.
http://www.theinformationminister.com/press.php?ID =612212491
we got this ages ago. i swear
Y'know - if there's ampty seats, then it can't really be described as packed. I remember the day when people sat on the floor in the aisles to receive words of mathematical wisdom from Dmitri [www.bath.ac.uk].
an article about a FRENCH mathematician ?
are you some sort of unamerican antipatriot ?
better change his name to "squarepoint" before this site gets banned...
Now that Poincaré's Conjecture has been solved, let's get going on Portnoy's Complaint.
Silly people... this is TOPOLOGY! It's not meant for people to USE it! It's just for mathematicians to RUMINATE UPON!
Has Fermat's Last Theorem actually been used in practical applications? I don't think so...
If everyone thought like you we'd still be living in caves.
Just because practical applications aren't totally obvious for a layman (or even a matematician) doesn't mean this will never be of practical use.
Even if no practical applications are ever found, this proof (if it survives peer review) may well pave the way for something else that is immensly useful.
There's just no way to tell right now.
"First lesson," Jon said. "Stick them with the pointy end."
Nerd Alert!
translation to make it easier.
basically all the poincare conjecture says is that if you have a 3 dimensional figure which is closed (therefore, it it bounded (doesn't go off to infinity in either direction), and doesn't have any "holes" in it (like a donut)) then you can take every point and map it to a point in an equivalent sphere without losing continuity (therefore, everypoint will have the same "neighbourhood" of points as it had in the initial shape.)
ie. You can map a cube into a sphere, or a dodecahedron, or a weird globlike thing that doesn't fold back on itself, or a whole piece of paper (without holes), or a pencil, or a lot of different figures.
As well, this conjecture also handles figures with holes in them (like donuts), and maps them all to simpler figures.
It's a very simple concept, but has been incredibly hard to prove, and what makes this conjecture even more frustrating is the fact that 1 and 2-dimensional forms of this conjecture were incredibly easy to prove, as well as 4 and up have been solved, and were reasonably easy as well. Yet for some reason the 3 dimensional version does not lend itself easily to a simple proof.
Everyone generally believes this is true, but no one has been able to prove or disprove it.
If proven, this is an important aspect of topology, because then we can map all n-dimensional figures to a simpler form (like a sphere) and know that the continuity and general structure of the figure will remain the same.
~ kjrose
Mathematical rigor demands we see the proof first. One decade ago Wiles thought he solved Fermat's Last Theorem but a mistake was found and worked again for several months before ultimately solving it. Faulty proofs are made all the time. Until it undergoes peer review I will be very skeptical.
Good Will Hunting...
"The faculty have answered (a long-since-dead guy), and answered with vigor"
It appears most people are spelling incorrectly! Including the sites included in the post!
..... its "mathemagician"
:)
It is not "mathematician"
Please make the appropriate corrections.
[I can picture a world without war, without hate. I can picture us attacking that world, because they'd never expect it]
Me. Hammer. Pliers. Every available 3-manifold. Can I have my $1 million please?
(This of course assumes that 3-manifolds are malleable.)
Note to M1-ers: a curt but otherwise insightful message is not "Flamebait" or "Troll".
Defining entropy as disorder's not complete,
'cause disorder as a definition doesn't cover heat.
So my first definition I would now like to withdraw,
and offer one that fits thermodynamics second law.
First we need to understand that entropy is energy,
energy that can't be used to state it more specifically.
In a closed system entropy always goes up,
that's the second law, now you know what's up.
You can't win, you can't break even, you can't leave the game,
'cause entropy will take it all 'though it seems a shame.
The second law, as we now know, is quite clear to state,
that entropy must increase and not dissipate.
Creationists always try to use the second law,
to disprove evolution, but their theory has a flaw.
The second law is quite precise about where it applies,
only in a closed system must the entropy count rise.
The earth's not a closed system' it's powered by the sun,
so fuck the damn creationists, Doomsday get my gun!
That, in a nutshell, is what entropy's about,
you're now down with a discount.
From here
I thought that this Wolfram guy was the smartest man in the universe and had all the answers. Now some brie-muncher comes along and proves something in math that Wolfram couldn't? This can only be due to one of three reasons:
1. When Wolfram and Hart were all killed by the Beast, Wolfram was in the house.
2. Wolfram is human and isn't as smart as the papers say.
3. He stepped up to MCHawking and is now hanging from a tree with a sign pinned to him that reads: WHACK EMCEE.
If Slashdot were chemistry it would look like this:Cadaverine
I swear that looks like perl.
By reading this comment, you immediately waive any and all rights regarding it.
Is that like Shrinkage?
Anyway, if true, this is kind of like Wiles proof of Fermat's Last Theorem -- proving an old conjecture by proving a more general (and more modern) one (in Wiles case, it was proving part of Taniyama-Shimura).
> Now, can someone tell me what practical applications
;-)
> there might be of this? Or is it strictly an abstract concept?
Speaking as a layman, the practical application of these sorts of proofs is that you can use them to prove equivalent, more practical questions.
One of the references in another comment explained that this conjecture has been proved for all other dimensions, and this 3-sphere seems to be a special case, as far as proof is concerned.
If the Poincare' conjecture were proved, then the general case could be solved. After that, "simply" proving that another hard problem is equivalent to the Poincare' conjecture is enough to prove that other problem.
Now, I've heard the problem described with a lasso instead of with a rubber band. I can imagine times when I'd really like to know when my lasso is going to close around something or if it's just going to slip off
This picture.
;)
Well, made me laugh anyway...
Why is a good-old friendly sphere called a 2-dimensional sphere?
Did I miss something? Are we living in Flatland?
I am fascinated by the idea of multiple geometrical dimensions.
Congratulations! Now we are the Evil Empire
Are you on a computer right now? Ever heard of a guy called George Boole? Does a "boolean" sound familiar? Well you see, this guy called George Boole he hated mathematicians so much he decided to invent this thing called Boolean Logic. You know the, 1 & 1 == 1, 1 || 0 == 0 stuff? As it turns out it was totally useless and that's what he intended, to invent something mathematically correct that is totally useless. So thanks to George Boole for accidentally inventing the foundation of computer architecture, logic gates and boolean logic - and he has something to do with you being on the computer right now. Indeed he is pissed off as he intended it to be useless.
Give maths time and it will applicable to your everyday life. What has been going on for the past 3,000 years?
Analytic & algebraic topology of locally Euclidean meterization of infinitely differentiable Riemmanian manifold
I have a most enlightening decoding of this proof which unfortunately the lameness filter will not allow me to post.
I mean seriously, its not like this guy is starting a book tour or series of $200 seminars or anything. He's presenting his proof to the community so that it can be studied and discussed; that's what this is about.
So what the hell are you talking about... was it supposed to be tounge-in-cheek humor?
Fuck Beta. Fuck Dice
what people keep seeming to misunderstand, is that that a 3-manifold is NOT a 3 dimensional object. when we speak of manifolds, we only care about the surface of the object, not its volume. therefore a 3-manifold is one with a 3 dimensional surface, i.e. a 4 dimensional object.
another thing is interesting to note: there are a LOT of problems in mathematics in which n=1 is trivial, n=2 is hard but straight forward. n>3 is not too hard and usually falls under one proof, and n=3 is EXTRAORDINARILY difficult. poincare, random walks, you name it.
BSD is for people who love UNIX. Linux is for those who hate Microsoft.
now I need to get a new hobby.
The Kruger Dunning explains most post on
that much Jenna Jameson earns just on one anal scene.
Moral of the story..
Stick to porn business, much better buck..and you dont have to think so much
>> Since the 19th century, mathematicians have known that the sphere is the only bounded two-dimensional space with this property, but what about higher dimensions?
Do they mean the sphere is 2-dimensional?
100 years ago a proof of the difficulty of factoring large numbers might only have been interesting to mathematicians. Now that we use encryption based on the difficulty of factoring products of large primes, it's very important.
Galois fields are used for checksum algorithms, something I'm sure Galois never thought of.
Fourier transforms are used for image compression (JPEG).
Who knows what Poincaré's topology might be used for in the future?
Was the answer 42?
Ok, there, I said it. Please mod me +5:Funny now, as everyone else who says this shit gets it. I feel dirty.
actually, 1||0==1
Another recently ( last 30 years) solved problem involving 2d-spheres is the 4 colour problem. Since you can project a 2d sphere onto an infinite plane a lot of work on polygons helped with the 4 colour problem. Like the fact a polygon with just hexagons and pentagons always has exactly 12 pentagons.
11:15, restate my assumptions:
1. Mathematics is the language of nature.
2. Everything around us can be represented and understood through numbers.
3. If you graph these numbers, patterns emerge. Therefore: There are patterns everywhere in nature.
Hey! That sounds like the old tri-sexing an angle problem: the 1-, 2- and 4- problems are easy, but tri-sexing is really hard.
I should know. I tried. Dislocated my hip.Who says math can't hurt ya?
You were 80% angel, 10% demon. The rest was hard to explain. - Over The Rhine
"Math in a song is good."-Linford
They want me to type something in this box.
that's the best dept. you could come up with Taco?
As a mathematician, I can assure you that not a single mathematician in the world considers a cube to be the 'same' as a sphere. Merely topologicaly equivalent.
As a computer scientist, I must inform you that all small integers (anything less than +/- 2^23 for IEEE) are accurately represented, and specifically that 10 is NOT equal to 9.9 (recurring). Yes, (10/3)*3 will yield that value - as will decimal arithmetic to any precision.
Oh well, I guess that makes you a troll.
How Eric can keep from writing "Fuck CRC Press" on every page, I do not know.
(BTW, what a nice way to discover my ISP has a transparent proxy.)
Fuck the system? Nah, you might catch something.
The AES encryption algorithm uses GF(2^8). The Galois field of order 2^8.
...interesting if true.
Well, IAAT so I might as well try to put in a 3-manifold topologist's perspective. First: the Poincare conjecture. Instead of talking about "holes" and whatnot, I would have put it like this:
First, what is a manifold? Well, take a bunch of tetrahedra, and start gluing their sides together in pairs. Start with finitely many, pair the sides up, and say how the sides are matched. What you have at the end will be a closed 3-manifold, providing every face gets glued to exactly one other face, and providing some number you can calculate called the "Euler characteristic" is equal to zero. The Euler characteristic is just
the number of vertices, minus the number of edges, plus the number of triangles, minus the number of tetrahedra *after* you have glued it all up. You now have your manifold M.
Simply connected has "something" to do with holes, but I think it's easy enough to say exactly what it is. Think of the unit circle in the plane, the set of points which are distance 1 from the center of the plane. Then think of a continuous function from the circle to the manifold. That is, for every point in the circle, you get a corresponding point in M such that when you vary the point you choose in the circle continuously, the corresponding point in M moves continuously. Since a circle is 1-dimensional, we can wiggle the image a little bit so that it doesn't cross itself; we call this image a *knot*. "Simply-connected" means that every continuous map from a circle to a knot in M extends to a continuous map of the unit disk. So the knot "bounds" a disk in M (which *is* allowed to intersect itself, and probably has to) which gives a way of shrinking the knot down to a point continuously.
The Poincare conjecture claims that the only 3-manifold M with this property is the 3-sphere.
Well, what is the 3-sphere? Take two solid balls and completely glue their boundaries together. What you get is the 3-sphere. Another description is as the set of points in 4-dimensional space which are distance 1 from the origin, just like the circle was the set of points in 2-dimensional space (i.e. the plane) at distance 1 from the origin. How would you prove such a thing? Well, one way is to use a criterion of Bing, who showed that any closed manifold with the following property is S^3: Bing's property says "every knot in M is contained in a solid ball in M". There are other criteria, but part of the problem is that they are very hard to check or verify, and the hypothesis (that S^3 is simply connected) is hard to use.
So, what does Perelman do? He actually proves not just the Poincare Conjecture but a much stronger conjecture called Thurston's Geometrization Conjecture. Unlike the Poincare Conjecture, which is a conjecture just about S^3, Thurston's conjecture is a conjecture about *every* 3-manifold. It says, roughly speaking, that every closed 3-manifold which is *irreducible* (i.e. every sphere bounds a ball) can be cut up into a finite number of pieces which have a canonical "geometric structure". What is meant by a geometric structure? Well, a football and a soccerball are both *topologically* spheres, but they have different *geometries*; the football is pointy at both ends, but the soccerball is perfectly round everywhere. So the soccerball is "geometric" since it doesn't have odd bumps or lumps, but looks the same everywhere. It turns out in 3 dimensions there are 8 different ways that a small piece of a geometric manifold can look, and one of these is the geometry of S^3. It's not hard to show that any *simply connected* 3-manifold with the geometry of S^3 is actually equal to S^3, so Thurston's conjecture implies the Poincare conjecture, and that's what Perelman proves.
How does he do it? Well, he starts off with the manifold, and if it's not geometric, he starts to deform it so that it looks more and more geometric. That's the "Ricci flow" bit - it means that if you have some direction that looks pointy, you stretch it out, and if you have some direction which looks more stretched, you squash it do
In Soviet Russia, the conjecture is YOU!
Who made me the genius I am today,
The mathematician that others all quote,
Who's the professor that made me that way?
The greatest that ever got chalk on his coat.
One man deserves the credit,
One man deserves the blame,
And Nicolai Ivanovich Lobachevsky is his name.
Hey!
. .
I am never forget the day I am given first original paper
to write. It was on analytic and algebraic topology of
locally Euclidean parameterization of infinitely differentiable
Riemannian manifold.
Bozhe moi!
This I know from nothing.
But I think of great Lobachevsky and get idea - ahah!
I have a friend in Minsk,
Who has a friend in Pinsk,
Whose friend in Omsk
Has friend in Tomsk
With friend in Akmolinsk.
His friend in Alexandrovsk
Has friend in Petropavlovsk,
Whose friend somehow
Is solving now
The problem in Dnepropetrovsk.
And when his work is done -
Ha ha! - begins the fun.
From Dnepropetrovsk
To Petropavlovsk,
By way of Iliysk,
And Novorossiysk,
To Alexandrovsk to Akmolinsk
To Tomsk to Omsk
To Pinsk to Minsk
To me the news will run,
Yes, to me the news will run!
And then I write
By morning, night,
And afternoon,
And pretty soon
My name in Dnepropetrovsk is cursed,
When he finds out I publish first!
And who made me a big success
And brought me wealth and fame?
Nicolai Ivanovich Lobachevsky is his name.
Hey!
Tom Leher Revisited
Grigoriy Yakovlevich Perelman is the son of Yakov Isidorovich Perelman.
You guys do not know who is Yakov Isidorovich Perelman.
Every scientist in Russia knows Yakov Perelman.
He wrote about 100 popular books about mathematics and physics.
He was the russian Martin Gardner (i can assure you that every mathematician in Russia knows who is Martin Gardner too).
http://www.amazon.com/exec/obidos/tg/detail/-/0
http://www.amazon.com/exec/obidos/tg/detail/-/8
Cool !!!
--
anton
Mod up parent .... and grow a sence of humor people! This is funny!
HallmarkOrnaments.Com
In Soviet Russia, donut break you!!
To make laws that man cannot, and will not obey, serves to bring all law into contempt.
--E.C. Stanton
In the first half of the last century, Hardy was very proud for being a "pure mathematician", whos results would never ever have practical implications. He worked on number theory.
1975, Rivest, Shamir and Adleman realized that the results of Euler, Fermat and Lagrange (see above) are not only true, but can be used as an asymmetric cipher. It's known as the "RSA algorithm". I think You can call that "a useful application".
It may take centuries until a mathematical truth finds it's useful application. If You only demand results with obvious practical use, you won't get far.
You have no idea what you are talking abot. Closed and bounded are not equivalent. "map a cube into a sphere, or a dodecahedron, or a weird globlike thing that doesn't fold back on itself, or a whole piece of paper (without holes), or a pencil, or a lot of different figures." Some of these things aren't even homeomorphic, you moron. "As well, this conjecture also handles figures with holes in them (like donuts), and maps them all to simpler figures." no!!!! "that 1 and 2-dimensional forms of this conjecture were incredibly easy to prove," FUCK, YOU DO IT, SMART ASS. " 4 and up have been solved, and were reasonably easy as well" THIS GOT A FUCKING FIELD'S MEDAL. TWO!!! "an important aspect of topology, because then we can map all n-dimensional figures to a simpler form (like " DAMN IT. THIS IS JUST WRONG. FUCK>,
Your idea that Boole predated computers is also wrong, since Boole was a contemporary of Babbage. Which is not to say that Babbage's computers were digital, but the idea of encoding problems as calculations, and munging on them with machines, was definitely there.
My maths lecturer for data communucations told us. He also said "George Boole would be rolling over in his grave if he found out what we were using boolean alegebra for". I can't find any immediate links on this (got an exam in 8 hours).
Analytic & algebraic topology of locally Euclidean meterization of infinitely differentiable Riemmanian manifold
doh
Analytic & algebraic topology of locally Euclidean meterization of infinitely differentiable Riemmanian manifold
Aha... so you're saying that something useful might spring from hatred of this conjecture thingamagig. Now we're getting somewhere.
My Blog
The parent is a known idiot who routinely pretends that he knows about topics about which he knows absolutely nothing (if you couldn't guess that from his sig).
No it's not you big fucking loser.
OTOH, taken as the border of a set of points on the surface of the apple (closed if you include the points of the border line, open if not) I can accept that that set of surface points could be shrunk to a single point.
Can this result be used in cutting edge grand unified theories like brane theory?
"sweet dreams are made of this..."
Sorry Marvin, this is not the first time you write more about maths than you actually know.
Firstly, the Poincare Conjecture states that every compact 3-dimensional manifold is homeomorphic to the 3-sphere. So we're dealing with manifolds, not just any 3-dimensional figures. This means that every point of the figure is completely surrounded by a small region which looks like 3-space. In particular, any manifold is open. We also want our manifold to be compact (not just closed, inaccurate claims on Mathworld notwithstanding), which here means closed and bounded. Note further that closed does not imply bounded.
The following 3-dimensional objects are simply connected, but not homeomorphic to the 3-sphere:
(a) A closed ball in 3-dimensions. This is what most laymen would consider a "sphere". It consists of all points in 3-space at a distance of less than or equal to 1 from the origin. It is simply connected, closed, bounded, but not open, hence not a minifold.
(b) 3-space itself. This is a closed manifold, but not bounded, hence not compact.
Marvin seems to have misunderstood what a 3-sphere is. By "sphere", mathematicians generally mean the surface, not the inside. So a 3-sphere is the set of points in 4-space at a distance of exactly 1 from the origin. It is not the same thing as a closed 3-dimensional ball.
Oh, and Smale and Freedman's proofs of the Poincare Conjectue in higher dimensions were not easy. In fact, they both won Fields Medals for their efforts.
"...Look on my works, ye mighty, and despair!"
Firstly, the Poincare Conjecture states that every compact 3-dimensional manifold is homeomorphic to the 3-sphere. So we're dealing with manifolds, not just any 3-dimensional figures. This means that every point of the figure is completely surrounded by a small region which looks like 3-space. In particular, any manifold is open. We also want our manifold to be compact (not just closed, inaccurate claims on Mathworld notwithstanding), which here means closed and bounded. Note further that closed does not imply bounded.
I do thank you for being more technical for the readers who want it. I was trying to be less technical (and therefore using as simple of terminology as I possibly could to make it easier for most of slashdot to understand.)
I am well aware that it must be a compact space.
Yes, I also know that closed does not imply bounded. Rather closed only means that all limit points exist within the space, while bounded is completely different. Again, I sacrificed accuracy for simplicity to try to make it easier for people not in university math to comprehend.
(a) A closed ball in 3-dimensions. This is what most laymen would consider a "sphere". It consists of all points in 3-space at a distance of less than or equal to 1 from the origin. It is simply connected, closed, bounded, but not open, hence not a manifold.
Correct, again, I was sacrificing accuracy for simplicity. In hindsight, it may have been better for me to specifically discuss this simpler version of this. Yet, hindsight is 20/20.
(b) 3-space itself. This is a closed manifold, but not bounded, hence not compact.
Naturally, because 3-space is infinite, yet contains all limit points.
Marvin seems to have misunderstood what a 3-sphere is. By "sphere", mathematicians generally mean the surface, not the inside. So a 3-sphere is the set of points in 4-space at a distance of exactly 1 from the origin. It is not the same thing as a closed 3-dimensional ball.
Again, I probably should've been more specific and dealt specifically with the simpler version of this problem, and just then given a minor technical transfer to this version.
Oh, and Smale and Freedman's proofs of the Poincare Conjectue in higher dimensions were not easy. In fact, they both won Fields Medals for their efforts.
True, but it is still interesting that this proof in this dimension is so incredibly difficult. Like a joke I heard once from a math professor:
"A professor was showing his students a problem related to complex analysis. He wrote a theorem on the board and said, 'I am not going to prove this, because this is trivial.'
One of his students put up their hand and said, 'Are you sure?'
The professor stopped, looked at the theorem, sat down at the desk in the classroom, wrote down 10 pages of proof, let the class go, went back to his office, worked for a week on the problem, and came to the class the following week with the proof.
'Yes, it is trivial.' The professor said."
I am not putting down Smale and Freedman's proofs. They are excellent pieces of mathematical work. I am just saying that in this dimension, the proof has been found to be so incredibly elusive, that it is fascinating to me. Even though, the other proofs were difficult and well thought out, this proof still hasn't been found. That alone makes this interesting to me.
I will apologize though for stating they were easy proofs. (A bad habit I have picked up regarding mathematics.)
But regarding my accuracy. Perhaps I should put a quick disclaimer before hand in the future and note that I purposefully remove the rigor and accuracy to these "dumbing down" of mathematics to make it easier to understand.
As well, I should probably proofread what I type in before hand as well. (I made some typos that read really wrong.) Yet, since this is slashdot, and not a peer-reviewed journal, I usually don't care and just post it hoping that more peo
~ kjrose
You aren't even a good liar.
A good mathematician does not have to sacrifice ACCURACY in order to be less RIGOROUS. The two are separate. If you really understood the math, you could express it in simplier terms that might not be as PRECISE, but not ENTIRELY WRONG.
And why talk about limit points to define closed? Shit, did you just finish a real analysis course?
" am not putting down Smale and Freedman's proofs. They are excellent pieces of mathematical work. "
You are not remotely qualified to pass judgment on their work. Their work, FWIW, is brilliant. But you can't even understand it.
"Yet, since this is slashdot, and not a peer-reviewed journal,"
You will never publish in a peer-reviewed journal. Frankly, you don't have the ability or the temperment to do good mathematics.
Dear "Marvin Mouse,"
would you PLEASE PLEASE FOR THE LOVE OF GOD spare us your extremely misinformed "translations" of every math result that ends up as a Slashdot story? This is just one time too many. The explanation of the Riemann proof was really hideous and I prayed that that would be the end of your stinking career as a math lecturer, but alas - it was modded up! - and now here you go again. Why don't you don't just link to the page where you find that little story instead of trying to pretend you understand the conjecture itself? I'm not even going to point out all the severe errors in your short post, but only note that the closed-compact error you made is enough proof that you stole it all from mathworld.
Listen, if you can't even STEAL without making gross errors, maybe you shouldn't post at all. And putting down work that won the Fields medal as "relatively easy" is just ridiculous.
Marvin, you write on math every single time you have the opportunity, yet you never have any idea what you're talking about. Please shut up. I'm not these rude normally, but you had it coming.
I do thank you for being more technical for the readers who want it. I was trying to be less technical (and therefore using as simple of terminology as I possibly could to make it easier for most of slashdot to understand.)
You made a lot of very grave errors in your post. Are we supposed to believe you did that in the name of simplicity? Simplicity and accuracy are not mutually exclusive concepts. Do you think that Perelman would have needed to make your errors in order to explain his results to laymen? Allow me to be doubtful.
I am not putting down Smale and Freedman's proofs. They are excellent pieces of mathematical work.
You don't know that. You just like to pretend that you know things you don't. Before you dismiss that as mere trolling, maybe you should ask yourself: "Do I understand these proofs? Have I even read them? Am I competent enough to pass judgment on them?" A sincere answer to those questions could do wonders for your knowledge of who you are, as opposed to who you'd like to be.
I am just saying that in this dimension, the proof has been found to be so incredibly elusive, that it is fascinating to me. Even though, the other proofs were difficult and well thought out, this proof still hasn't been found. That alone makes this interesting to me.
A proof you wouldn't understand of a conjecture you don't understand hasn't been found, and this in spite of a similar proof being so easy as to merely be awarded a Fields medal. Must really keep you awake at nights.
Perhaps I should put a quick disclaimer before hand in the future and note that I purposefully remove the rigor and accuracy to these "dumbing down" of mathematics to make it easier to understand.
"In the future"? Can't this be your last "explanation" of a difficult mathematical result? Your posts are modded up because people who don't know anything about these things assume you're not just a hack and that you give a decent explanation. It's not fair to them and it's terribly irritating to the rest of us.
I usually don't care and just post it hoping that more people will be interested in the mathematics and go and learn more. (and perhaps even discover that my translation wasn't very accurate, just accurate enough to get the idea across.)
I should apologize to fellow mathematicians though. I understand the need to keep rigor and logic formal in mathematics, I just also like to every once and a while introduce someone to the basic concepts without having to teach them first year real analysis.
"Go and learn more"? It is plain to see that you like to think of yourself as some kind of mathematical prophet, teaching the dumb masses about the wonders of math. It's more than a little pathetic.
And as far as mathematical rigor is concerned, you are to that as CmdrTaco is to grammar.
"If you think education is expensive, try ignorance" - Derek Bok
I, too, have made a dumb mistake. In fact, the mistake we both made was to think of the manifold as embedded in some R^n, in which case "closed" means closed in the topology of R^n. However, manifolds should rather be regarded intrinsically, and in its own topology any topogical space is closed. This is a tautology. When people (e.g. at MathWorld) talk about "closed manifolds" they actually mean compact manifolds.
It's always nice to come across an enthusiastic maths undergrad, and I certainly don't want to dampen your enthusiasm. But please, be careful not to mislead less informed readers with unaccurate posts. Somebody else has pointed this out in a less polite way, but I don't think you deserve any flames. Just be a bit more careful, and good luck with your studies.
"...Look on my works, ye mighty, and despair!"
I happened upon your unhelpful trolling, and would like to paraphrase and translate one of your statements. You also have not offered any useful discussion to this topic, so remember when you accuse Marvin of failing to help, you have not even tried.
You're trying to teach the dumb masses like some kind of math prophet when you aren't qualified!
= Damn, if you make it that easy then I'm going to be out of a job scaring the shit out of kids!
As far as helpfulness is concerned, you are to that as Marvin is to generosity.
Why slashdot? Why not?