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April 19, 2003

Poincaré Proven

The New York Times reports on Grisha Perelman’s papers (I, II) which claims a proof of Thurston’s Geometrization Conjecture, and hence of the celebrated Poincaré Conjecture.

The Poincaré Conjecture, you’ll recall, is the statement that any compact, connected, simply-connected 3-manifold without boundary is homeomorphic to S 3S^3.

Perelman’s proof involved proving properties of the “Ricci flow”

(1)ddtg ij=−2R ij\frac{d}{dt} g_{ij}=-2R_{ij}

which you’ll recognize as the 1-loop renormalization-group equation for a nonlinear σ\sigma-model on this manifold. The idea is to study the long-time behaviour of this flow (perhaps after repairing some singularities which might form — in finite time — locally on MM and after a suitable rescaling of the overall volume of MM).

I don’t understand any of the details, but if someone who does would like to chime in, that would be very cool!

Update: Paul Ginsparg pointed me to this pretty review by Milnor on the history of the Conjecture.

Posted by distler at April 19, 2003 2:19 PM

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4 Comments & 0 Trackbacks

Re: Poincaré Proven

According to s.m.r this is not a complete proof but part of an ongoing work.

Search google groups for “0303109 sci.math.research”. (can you do links in comments?)

Posted by: Volker Braun on April 21, 2003 4:49 AM | Permalink | Reply to this

Re: Poincaré Proven

“Can we do it? Yes we can!” (Sorry, my son is very much into Bob the Builder these days.)

Try this thread.

Posted by: Jacques Distler on April 21, 2003 9:31 AM | Permalink | Reply to this

Re: Poincaré Proven

The review link to Milnor is broken at the moment. I think you want this URL for the PDF review. Here’s a list of Milnor’s publications.

Posted by: Aaron Suggs on August 24, 2006 12:37 PM | Permalink | Reply to this

Bitrot

Thanks for pointing out the broken link. It used to work.

Anyway, it’s fixed now.

Posted by: Jacques Distler on August 24, 2006 3:21 PM | Permalink | PGP Sig | Reply to this

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