Two postings: News about a possible proof of Poincare Conjecture, and Mimura and Toda on the naming of the elements in the homotopy groups of spheres............DMD _________________________________________________________ Subject: proof of Poincare/Geometrization Conjecture? Date: Mon, 18 Nov 2002 18:26:05 -0500 From: Zbigniew Fiedorowicz My colleague, Dan Burghelea, believes that there is a major development, if not a complete proof of Thurston's Geometrization Conjecture for 3-manifolds (and hence also of the Poincare Conjecture). The paper in question is by Grisha Perelman (of the Steklov Institute in St. Petersburg), entitled "The entropy formula for the Ricci flow and its geometric applications", which was deposited in the Math. Archive on Nov. 11 and is available via the link http://front.math.ucdavis.edu/math.DG/0211159. The paper does not make any explicit claims about proving the above conjectures, but does claim to prove the conjectures in section 6 of Richard Hamilton's paper "Formation of singularities in the Ricci flow" in Surveys in Diff. Geom. 2 (1995), (which unfortunately is not available in our library). Moreover Perelman refers to another 1999 paper of Hamilton on the latter's program to prove the Geometrization Conjecture and concludes his introduction with the following tantalizing statement: "We have not been able to confirm Hamilton's hope ... ; still we are able to show that ...; by our earlier (partly unpublished) work this is enough for topological conclusions." Perelman is a well respected differential geometrer who is regarded as an expert on Ricci flow. I wonder if anyone has comments or further information. Actually I overlooked the following explicit statement in Perelman's paper: "Finally in Section 13 we give a brief sketch of the proof of the geometrization conjecture." Zig Fiedorowicz --