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."

Y'know

[DarenN]

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

:)

Re:Y'know

[kvn299]

I actually thought the article did a great job at explaining the problem. Did you read it?

Re:Explanation

[Vann_v2]

That's not really fair. There is a lot of mathematics that is useful, especially to scientists, but something like Fermat is just one of those mathematical problems which are interesting because 1) they look very simple, but 2) turn out to be maddeningly difficult to prove. To say Fermat, which is basically a mathematical problem akin to getting Linux running on your toaster, is indicative of the field of mathematics is unfair.

sigh

[danro]

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.

Wait for it wait for it....

[I Want GNU!]

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.

Re:Oh no..

[pVoid]

When I was doing high level math at university (which I'm not doing anymore - so don't get me wrong, I'm more or less in the same boat as you), every problem I read at first sounded like klingon to me... And I was good at math.

Math is one of those disciplines where you just can *not* skim the problem and expect to understand it... you have to load into memory every word that is in the text (like 'manifold' etc), and create a working instance of that object in your brain...

It's basically like launching a heavy app like Photoshop.

So yeah, to answer you: even when I was right in the middle of studying this stuff, there were moments when I would think I was stupid too... but if you concentrate *and* you know what they're talking about, it makes sense.

Conclusion: it's knowledge, not intelligence.

In Squarepoint's own words

[stud9920]

In Henry Sqarepoin's own words :

"La pensée ne doit jamais se soumettre, ni à un dogme, ni à un parti, ni à une passion, ni à un intérêt, ni à une idée préconçue, ni à quoi que ce soit, si ce n'est aux faits eux-mêmes, parce que, pour elle, se soumettre, ce serait cesser d'être."

(thinking should submit itself neither to a dogma, to a party, a passion, an interest, a

prejudice, neither to whatever, because , for thinking, submitting would mean the end of being)

It's the motto of my university actually.

Practical Applications?

[lildogie]

> 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 ;-)

Re:Explanation and George Boole

[SystematicPsycho]

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?

"Useless" mathematics that we use

[Len]

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?

Re:Donuts, apples, I'm hungry

[AxelBoldt]

A woman is topologically equivalent to a torus.

Have you ever drunk something, started to laugh, and have the stuff come out your nose? That proves that nose and mouth are connected, and the topology of a person is therefore more complicated than a torus. Because of the two holes in your nose, we're talking at least genus 3. I think the ears are connected to the nose/mouth system too, which would make it genus 5.

Re:Y'know

[uberdave]

It isn't Calculus. It's Topology.