Complete Solution of Poincare Conjecture

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The forum discussion centers on the complete proof of the Poincaré and Geometrization Conjectures as presented by Huai-Dong Cao and Xi-Ping Zhu in their paper, which builds on the Hamilton-Perelman theory of Ricci flow. The proof is recognized as a culmination of thirty years of work by geometric analysts and is detailed across 200 pages. The discussion highlights the historical context and contributions of Richard Hamilton, who laid the groundwork for this solution but did not complete it himself.

PREREQUISITES
  • Understanding of Ricci flow theory
  • Familiarity with differential geometry concepts
  • Knowledge of three-manifolds
  • Awareness of the Poincaré and Geometrization Conjectures
NEXT STEPS
  • Study the Hamilton-Perelman theory of Ricci flow
  • Explore differential geometry applications in modern mathematics
  • Read the complete proof by Cao and Zhu for in-depth understanding
  • Investigate the historical contributions of Richard Hamilton to geometric analysis
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Mathematicians, geometric analysts, and students of topology seeking to deepen their understanding of the Poincaré Conjecture and its proof.

selfAdjoint
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Announced in http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf" . Differential Geometry meets Geometric Surgery on three-manifolds; Perelman clarified and (perhaps) corrected.


A COMPLETE PROOF OF THE POINCAR´E AND GEOMETRIZATION CONJECTURES – APPLICATION OF THE HAMILTON-PERELMAN THEORY OF THE RICCI FLOW

HUAI-DONG CAO† AND XI-PING ZHU‡

Abstract. "In this paper, we give a complete proof of the Poincar´e and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow. "

The first sections give a clear history of the recent approaches to the Poincare Conjecture and Thurman's Geometric Conjecture, which are joined at the hip. The guy who I feel sorry for is Hamilton, who did fantastic things to lay almost all of the groundwork for the solution but, like Moses, was not able to enter the promised land.
 
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neutrino said:
I don't understand an iota of all this, but here's some related discussion at NEW
http://www.math.columbia.edu/~woit/wordpress/?p=434


Right. I should have made it clear that's where I got the link to the Cao and Zhu paper from.

While I fully expect the actual proof to be over my head, and I can't hope to maike it through all 200 + pages, the historical account and the general idea of what they're doing is pretty clear to me.
 
selfAdjoint said:
the historical account and the general idea of what they're doing is pretty clear to me.
Even that's waaay over my head. :biggrin:
 
Why does it start at page 165? What are on the previous pages?
 
That's because it's been taken from a journal.
 
Last two sentences.

"Hence in case (2), M is diffeomorphic to a flat manifold and then it is also geometrizable.

Therefore we completed the proof of the theorem. "


That's a million dollar conclusion, heh.
 
  • #10
what said:
That's a million dollar conclusion, heh.

Welcome to modern mathematics.
 

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