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

I'm sure his mother appreciates his scruples.

 

Ricci flow ftw.

 

Ted Kaczynski would understand....

 

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.

 

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.

 

Spoiler

 

I don't think he was just picking those shrooms....

 

Spoiler

 

If you cut and paste from Wikipedia, you should cite your source rather than present it in such a way that one might think you are trying to pretend that you wrote it.

 

or, in other words, the answer is 4.

 

Yes, when I found the summary I just control c/v'ed it without thinking. My apologies.

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.

There was a book on the Millennium Prize Problems that I enjoyed, and I read much of Perelman's proof... but I get lost early and often. To be extra clear on the matter, I absolutely could not have given that summary.

 

How did Hamilton find the time to become such a math genius and so good at basketball?

 

He didn't. But he did well in grammar.

 

What would the world be like without fruits and nuts?

 

Can I get a WTF?

 

Ha....I was right!

 

You are disturbing me. I am picking mushrooms.

 

Worst flow ever. Ricci needs to listen to some Bone Thugs and try again.