Grigory Perelman, the maths genius who said no to $1m

Invisible Fan · Mar 26, 2010

lpbman said:

I wanted some mention of what his prize was for, and while Perelman's behavior is unusual, what he accomplished is much more important than the prize.
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This New Yorker article shed some (sensationalized) light on the controversy.
http://www.newyorker.com/printables/fact/060828fa_fact2

It started when Perelman gave a round of early lectures for the proof but didn't spell things out for his colleagues. 3 teams of mathematician picked it up, but one group tried to horn in on some credit by claiming they "improved" Perelman's proof.

In science and math, sharing and borrowing information is sometimes crucial. Stealing credit can make or break researchers in a competitive field. Perelman's disgust is that he thinks his colleagues condones and accepts the shadiness even if they don't take part in it...and that he's the weird one for taking a stand against it.

In a sense, this is related to that Cheating thread...

REEKO_HTOWN · Mar 26, 2010

Ignorance is Bliss.

Air Langhi · Mar 26, 2010

Invisible Fan said:

This New Yorker article shed some (sensationalized) light on the controversy.
http://www.newyorker.com/printables/fact/060828fa_fact2
Click to expand...

It sounds like some of those Chinese guys had no problems with cheating even at the post graduate level

Qball · Mar 26, 2010

lpbman said:

Perelman's proof based on Hamilton's Ricci flow

The first step is to deform the manifold using the Ricci flow. The Ricci flow was used by Richard Hamilton as a way to deform manifolds. He used it to prove that many compact manifolds were diffeomorphic to spheres. However, he did not prove they were all diffeomorphic to spheres. The Ricci flow is an imitation of the heat equation which describes the way heat flows in a solid. Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow, the Ricci flow could run into singularities and stop functioning.

Hamilton was able to list a number of possible singularities that could form but he was concerned as to whether he had found all possible singularities. He wanted to cut the manifold at the singularities and paste in caps, and then run the Ricci flow again. But he needed to understand the singularities. Grigori Perelman examined the singularities and discovered they were very simple manifolds: essentially three-dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking circles stretched along a line.

This was proved using something Perelman called the "Reduced Volume" which is closely related to an eigenvalue of a certain "elliptic equation". Eigenvalues are difficult to describe without calculus but they are part of a famous problem: Can you hear the shape of a drum?. Essentially an eigenvalue is like a note being played by the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helped him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the cigar solution, which looked like a strand sticking out of a manifold with nothing on the other side. In essence Perelman showed that all the strands that form can be cut and capped and none stick out on one side only.

Completing the proof, Perelman takes any compact, simply connected, three-dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them. He cuts the strands and continues deforming the manifold until eventually he is left with a collection of round three-dimensional spheres. Then he rebuilds the original manifold by connecting the spheres together with three-dimensional cylinders, morphs them into a round shape and sees that, despite all the initial confusion, the manifold was in fact diffeomorphic to a sphere.

Two immediate questions were then: how can one be sure there aren't infinitely many cuts necessary? That the cutting does not progress forever? Perelman proved this using soap films on the manifold and showing that the areas of the soap films decreases as the manifold undergoes Ricci flow. Eventually the area is so small that any cut after the area is that small can only be chopping off three-dimensional spheres and not more complicated pieces. This is described as a battle with a Hydra in Szpiro's book cited below.
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the answer is 4, 8, 15, 16, 23, 42...

Surfguy · Mar 26, 2010

Yonkers said:

It's not your fault.
It's not your fault.
It's not your fault.
It's not your fault.
It's not your fault.

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You were too smart for me then...and you are too smart for me now.

You know what you can do with your Fields Medal?

Oh...never mind. I forgot...you refused to collect it along with your $1 million.

This time...it is your fault. Why don't you become a sheperd, son?
...
I smell a sequel.

DonkeyMagic · Mar 26, 2010

Billy Bob said:

or, in other words, the answer is 4.
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i agree with that answer...

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Grigory Perelman, the maths genius who said no to $1m

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Grigory Perelman, the maths genius who said no to $1m

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REEKO_HTOWN I'm Rich Biiiiaaatch!

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