An excellent review article about the state of the proof of the Poincare conjecture by my colleague John Morgan has recently appeared. For more background on this, see an earlier posting. Morgan is a topologist, and his article contains an excellent survey of what this all has to say about the topology of three-manifolds. This past semester he has been teaching a course in which he has gone through Perelman’s proof very carefully. So far it all holds together.

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Peter,

I sure wish I had the time to drop by and ask Hamilton and Morgan more questions about this.

You indicate that the crucial gaps are analytic.

This surprises me. It’s not the geometric or

topological arguments that are unclear?

Any chance you could be more specific about what

kinds of techniques in analysis that Perelman is using and that people are uncomfortable with?

After further consultation with experts I think it would be fair to say that the situation is something like this:

Parts of Perelman’s outline have been filled in and people agree on them. But the status of other parts is still unclear and no one is about to claim yet that a complete proof exists. One problem is that Perelman is using some difficult techniques, for which it is not clear that there are adequate references in the literature. So filling in the details of his proof is a very non-trivial business. To start with you have to become an expert in the techniques he is using, and then you may need to do some hard work checking that these techniques really do work exactly the way they are supposed to.

So the problem is that the work that needs to be done is quite difficult and requires some real expertise in some little-known techniques in analysis. And some of the experts may not be too motivated to do this, since if they invest the time needed there is a serious danger that either Perelman will scoop them by producing his own full proof (no one knows if he plans to do this and he is not saying), or their work will not be very valued by anyone, since people will think that all they did was work out details already known to Perelman.

Problem is that at this point if X writes up a detailed proof no one is likely to refer to it as the “Perelman-X” proof, especially since so many people have now been going through the process of working out the details for themselves.

If X were to do this, I think it would be viewed more as a piece of exposition, rather than original research, and the reward structure of the field is such that expository work is not very highly valued.

Peter wrote:

….I don’t think the math community will be completely satisfied that there’s a proof until someone (or a group) finally writes out a real, detailed proof, and I don’t know if anyone intends to do this anytime soon….

Hmmm, that seems funny to me. Since Perelman only produced an outline, I would think some one would want to do the legwork and share the glory. I would have guessed mathematicians would be interested in filling the “whoever” slot in the Perelman-whoever proof of the Poincare conjecture.

After all, that’s a hell of a paper for a struggling post doc looking for a job.

ksh95

Hi Deane,

I don’t know of anyone now willing to stand up and make that claim. The problem is that Perelman has really only produced an outline of a proof, so a lot of work is necessary to fill in the details. Some people like Morgan are working on it and they haven’t found any problems. I haven’t talked to Hamilton about this, but I’d suspect he’s more interested in trying to do something new than in filling in the details of Perelman’s outline.

I don’t think the math community will be completely satisfied that there’s a proof until someone (or a group) finally writes out a real, detailed proof, and I don’t know if anyone intends to do this anytime soon.

Peter

I do not mean to interrupt thought processes here in regardng Poincare conjecture, but a little history was importnat for the commoners(me) to be exposed to a way of thinking that few can concieve of. I assumed all mathematicians and physics people are blessed with such insight?

But that’s not the point I wanted to raise. It had to do what might be concieved of this wonderful world abstractually creating a road to GR which continues to expand. I know I am probably wearing peoples patience thin who are well educated and very articulate in the mathematical realms, and realms of physics, but has no one consider the context of the way in whch we can now see, in relation too, Einstein’s realization of that extra dimension? Could have led to a more comprehensive view of where Einstein left off?

So one saids, “extra dimensions,” one would have to engage the new world of non-euclidean with some good understanding to meet these new mathematics I am seeing developed here. By others along side of that physics.

I hope this post is acceptable here, as a form of a question, about why Peter you will not accept those extra dimensions?

Peter,

This article gives the status of Perelman’s

proof as of last August. Is anyone

(Hamilton, in particular) ready yet to proclaim

Perelman’s proof complete and correct?