### 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 $S^3$.

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

which you’ll recognize as the 1-loop renormalization-group equation for a nonlinear $\sigma$-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 $M$ and after a suitable rescaling of the overall volume of $M$).

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.

## 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?)