[FOM] What's going on with the Poincare conjecture?
Timothy Y. Chow
tchow at alum.mit.edu
Mon Jun 19 09:50:38 EDT 2006
Joe Shipman wrote:
> This month, there has been a flurry of news stories that two Chinese
> mathematicians, Cao and Zhu, have provided the "final brick" in the
> proof, and that the Poincare conjecture is now proved. However, there
> are two disturbing things about this:
For some additional perspective on this matter, one should consult the
200-page manuscript by Bruce Kleiner and John Lott, "Notes on Perelman's
papers." This was posted on the Los Alamos ArXiv shortly before the
announcement by Cao and Zhu.
http://arxiv.org/abs/math.DG/0605667
This manuscript is the culmination of years of work by the experts in the
field, as those who have been tracking Kleiner and Lott's webpage know:
http://www.math.lsa.umich.edu/~lott/ricciflow/perelman.html
Note that, apart from presenting his papers and his lectures series at MIT
and Stony Brook, Perelman himself has not been significantly involved in
these efforts.
Here are a couple of relevant quotes from the Kleiner-Lott paper.
"Pereleman's proofs are concise and, at times, sketchy. The purpose of
these notes is to provide the details that are missing in [40] and [41],
which contain Perelman's arguments for the Geometrization Conjecture."
"Regarding the proofs, the papers [40, 41] contain some incorrect
statements and incomplete arguments, which we have attempted to point out
to the reader. (Some of the mistakes in [40] were corrected in [41].) We
did not find any serious problems, meaning problems that cannot be
corrected using the methods introduced by Perelman."
The dichotomous question in many people's minds---namely, "Is there a gap,
or has the proof been checked and found to be correct?"---is never
directly answered by Kleiner and Lott in so many words; the above quotes
come closest.
Tim
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