To Pantazis and to the other mathematicians...

http://abcnews.go.com/International/wir ... 460&page=1

What is this Theorema bout the nature of multidimensional space that has stumped people for 100 years?

How did he solve it? Where can you see this? Ugh, I hate when news articles don't go into detail, they just make it a 'story' and don't give the actual details

They do mention this

But how did he prove that?

Maybe he has contacted with Homer Simpson, cause he's expert about the doughnuts....... :):):)

I am also quite shocked about the new...... How can he prove it? Well, I have to say that I cannot understand perfectly the Poincare conjeture......

It says that a doughnut shape can't be converted to a sphere while other without holes yes? But they are talking about withouth changing their propertys? Without changing their gravity centers or what? Really cannot understand it.......

Well... this guy was rejected from taking a university math job long time ago, then he dedicated the last ten years in proving Poincare's Conjecture.

They say he lives in a humble home surviving with his mother's pension. Currently he is jobless, but maybe not for long!

Not accepting the prize, made him one of my idols.

Just when you think materialism has taken over this world, a fine example like this prevails. A few days ago, he said that the reason he does not accept the prize, is because he was only interested in solving the problem and nothing else!

He is such a pure (regarding his character) pure (regarding the math field) mathematician!!!

Some references in mathworld (a very nice site regarding math stuff)

http://mathworld.wolfram.com/news/2003-04-15/poincare/
http://mathworld.wolfram.com/PoincareConjecture.html



Pantazis



PS The article pretty much explains the problem: "in three dimensions, a doughnut shape cannot be transformed into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere". Like saying that in topology, a doughnut is exactly the same thing with a tea cup that has a handle!
Regarding the proof now.... it may take a while to explain!

thanks for the replies

but YES... what is this proof?

I can easily say, yeah, just fold space together and bam! easy travel to the next galaxy.
sure I came up with that theorem. now where's my prize? lol


What I'm getting out, is how the heck can he prove that it won't rip, or will?

What kind of equipment do you have to prove/test that theory out?



---i agree Pantazis. math/science shouldn't be dealt with by giving out $$$/awards. it's about our human knowledge; our drive to learn and expand on our brains and what we can think of and carry out.
but it does help winning the $$$ when it can help you back and put more funding into your research. i'll admit that stuff ain't cheap. not that i know first hand.

No equipment (besides a few heavy books) is needed.
We are talking about generic and theoretical stuff here. A few axioms, a few definitions, and a lot of theorems with their proofs.

And I can't say I have read anything that he has, because I am no expert in topology, as I expect he has not read much of Algebraic Graph Theory LOL (that was the topic of my thesis)

Imagine now, he is a top mind and spent ten whole years!
I am pretty sure that for me to understand his proofs I would have to read a lot more "basic" stuff!







For me money is secondary stuff, sometimes even knowledge. But all can exist in harmony as long as we are happy!


Pantazis

I think that those kind of demonstrations are purely mathematicians no?

So he only needed a hundreds of pieces of papers, a few pens and a lot of imaginations, and of course, amazing maths skills no?

Just my theory. I am not mathematician, of course. But the topology studies the changes of the elements when deforming without changing their properties. So he studied mathematically the transformation from a doughnut to a sphere (dunno what kind of math operators does he use...) and looked if the new elements keep the original properties.......

Well...... as I have said, I am talking WITHOUT having any idea and inventing that.........

If that is true, why does he bother to publish papers about it? Plus, your idol could at least give the prize money to charity.

Well by no means am I doubting his finds or saying that it's not true. Yes, I know he may have something considering he took 10 years of his life to prove it.

I just wish they could put it in laymen's terms for all

I've always been into Astronomy/Physics/Quantium mechanics. I just want to be able to understand his finds more thoroughly. I guess I'm just still at the Pre-Basic stage

Actually, this is the best part. He didn't publish the particular proof in a journal. He placed it online "for anyone interested".




Well... different people will do different things (in a hypothetical situation, I would probably do exactly what you said, but too bad it will remain a hypothesis...).

And I believe that living in a topological world is not a bad thing!



Pantazis

I think he should take the money. There's nothing wrong with that in my opinion.

I think it's great that prizes are offered to encourage people to make discoveries to further mankinds understanding of science. Although I have no clue how this "Poincare conjecture" could affect my life, I have no doubt that someone has something in mind or soon will.

Keep it up super-brainiacs! I want a hover skateboard like in "Back to the Future II" already!

Poincaré Conjecture



In its original form, the Poincaré conjecture states that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, where a three-sphere is simply a generalization of the usual sphere to one dimension higher. More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes. This conjecture was first proposed in 1904 by H. Poincaré Eric Weisstein's World of Biography (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 incorrect proof, 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 (and was known to 19th century mathematicians), n==3 (the original conjecture) appears to have been proved by recent work by G. Perelman (although the proof has not yet been fully verified), n==4 was proved by Freedman (1982) (for which he was awarded the 1986 Fields medal), n==5 was demonstrated by Zeeman (1961), n==6 was established by Stallings (1962), and n>=7 was shown by Smale in 1961 (although Smale subsequently extended his proof to include all n>=5).

The Clay Mathematics Institute included the conjecture on its list of $1 million prize problems. In April 2002, M. J. Dunwoody produced a five-page paper that purports to prove the conjecture. However, Dunwoody's manuscript was quickly found to be fundamentally flawed (Weisstein 2002). A much more promising result has been reported by Perelman (2002, 2003; Robinson 2003). Perelman's work appears to establish a more general result known as the Thurston's geometrization conjecture, from which the Poincaré conjecture immediately follows (Weisstein 2003). Mathematicians familiar with Perelman's work describe it as well thought-out and expect that it will be difficult to locate any substantial mistakes (Robinson 2003, Collins 2004). In fact, Collins (2004) goes so far as to state, "everyone expects [that] Perelman's proof is correct."


----The definition of topology leads to the following mathematical joke (Renteln and Dundes 2005):

Q: What is a topologist?
A: Someone who cannot distinguish between a doughnut and a coffee cup.

HAHAHA that's a good one

That's probably for the topologist what "I always peeled the stickers off" is for cubers...

Hi guys,

I also learned from some people at the International Congress of Mathematics that Perelman will refuse the Fields medal as well. (It is the equivalent of the Nobel prize for Mathematics, only it is only given every four years and you have to be 40 or younger to get it).

Bernardo

hmm another article explains it pretty well...

[quote="Century-old brain-twister now solved"][/quote]

MADRID, Spain - The Poincare conjecture involves topology, a branch of math that studies shapes.


It essentially says that in three dimensions you cannot transform a doughnut shape into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere.

There is a catch: the space has to be finite. Imagine an ant crawling on an apple in a straight line. It can only walk so far before it's back where it started.

Even though the apple has three dimensions, its surface is two-dimensional. The ant can walk backward, forward and sideways on the surface but not up and down. In three dimensions, shapes are harder to determine because people cannot directly 'see' them and there are many more possible types of holes.

The conjecture is named for French mathematician and physicist Henri Poincare, who proposed it in 1904.

An analogous conjecture was proved for spaces of more than three dimensions over 20 years ago. But the specific 3-D case flummoxed mathematicians for years.

In 1982, Columbia University's Richard Hamilton developed a technique called Ricci flow that mathematically ironed out wrinkles in 3-D surfaces and provided a blueprint for cracking the Poincare conundrum.

A problem was posed by puzzling, dense spots called singularities, which exhibited sudden, uncontrolled change.

Perelman's breakthrough was to understand how to analyze these singularities, essentially neutralizing them for a while and allowing the Ricci flow to proceed smoothly and show what a given space is really like, topologically speaking.

One of the problems with these kinds of math problems is that by the time people usually find one they no longer speak a language other than math.

Poincare's conjecture is the same problem as that posed by the construction of a Klien bottle.

DJ

AHA! Yes indeed. Interesting shapes but with only one surface area.

May this would come in handy for those bottles. I got this free with a pair of diagonal pliers a couple of years ago.

The same way there is nothing wrong with not taking the money. You must remember, that when he really needed encouragement, that was when he was let down by his fellow scientists.



Pantazis

Hi!

If anybody needs some more detailed information on the Poincaré problem or wants to win another million dollars (there are still six chances left ), here is the original site of the Clay Mathematics Institute:

http://www.claymath.org/millennium/

Unfortunately in German, but including a picture of Gregori Perelman:

http://www.spiegel.de/wissenschaft/mens ... 10,00.html


Best regards

Frank

LMAO!!! Funny how we just had a topic about that on the forum

Frank: Thanks for the links, i'll check them out

--------

Wow, that's alot of head exercises to think and read through that!

http://www.claymath.org/millennium/Poincare_Conjecture/

yeah, it very well expands on the Klein Bottles! haha
Click the link and then click on the link - Official Problem Description — John Milnor... it'll open up a PDF file describing just that

Well, I guess he does not want to become the 21 century Fermat