Understanding Pereleman's Proof: Essential Fields of Interest in Topology

[MathematicalPhysicist]

What fields of interest in topology (or else) should I learn to fully understand his proof?
Is there a comprehensible list?

Thanks in advance.

 

[yyat]

"Ricci Flow and the Poincaré Conjecture" by John Morgan and Gang Tian http://claymath.org/library/" requires only minimal background (basics from differential geometry, homotopy theory and PDEs).

 

[MathematicalPhysicist]

This is why I ask is there a comprehensible list, but thanks nontheless.

 

[sundried]

My guess is you are not going to FULLY understand his proof unless you spend the next ten years at it, and even then perhaps not.

 

[zhentil]

You need to understand the basics of course (differential geometry, topology), as well as advanced Riemannian geometry (comparison theorems, etc.) as well as advanced PDE (specifically hyperbolic equations). Furthermore, it wouldn't hurt to have some background in the calculus of variations, since those ideas form the basic framework for techniques like the Ricci flow.

 

[Geometrick]

Perelman uses a lot of advanced techniques in differential geometry and geometric analysis. I would recommend you read Peter Petersen's text on Riemannian Geometry, L.C. Evans text on Partial Differential Equations, Peter Li's notes on Geometric Analysis and then read Richard Hamilton's seminal papers on Ricci Flow and Geometrization from the 80s. Finally, read a full proof of Perelman's argument; there is John Morgan and Gang Tian's proof (which is available on arXiv), then there is also one done by some of Yau's students and I believe Bruce Kleiner and John Lott also wrote a proof while they were at Michigan.

You need a background in graduate Riemannian Geometry, a workable background in PDEs. There are some good "intro" books, like Knopf and Chow then there is Chow and Ni (both from San Diego). But these are generally 2nd or 3rd year grad differential geometry texts.

It should be noted that this is a geometric analysis argument. John Morgan recalls talking to ST Yau a few years ago saying that the Poincare Conjecture (which is a topological problem) will be solved only using topological methods while Yau said you need to implement geometry. Perelman is NOT a topologist by trade; he is a remarkable differential geometer and incredibly innovative. Perelman originally published his papers on arXiv and the papers were less than 40 pages a piece, while Morgan and Tian's proof is approximately 500+ pages. Perelman is from the Russian school of thought where some proofs are not totally explicit and leaves a lot to the reader.

Also, Perelman solved the bigger problem of Thurston Geometrization, of which the Poincare Conjecture is a "corollary." Quite an extraordinary corollary!

Sadly Perelman quit after the controversy involved with Yau and his students. I feel like I have been robbed that such a talented mathematician has quit; I will never able to read any new papers by Perelman. Please come out of retirement Grisha!

 

[zhentil]

There is one professor at my university who understands the proof, and this is only because his research area is comparison Riemannian geometry with heavy use of PDE. Unless you plan on going into research in Riemannian geometry, you probably won't ever fully understand the proof.

 

[MathematicalPhysicist]

Zhentil, I know it's quite hard, but I have patience, you need to have it if you are dealing with maths and physics, don't you? (-:

And a good memory, which I have because of lack of coffee drinking! (-:

 

[wofsy]

Zhentil, I know it's quite hard, but I have patience, you need to have it if you are dealing with maths and physics, don't you? (-:

And a good memory, which I have because of lack of coffee drinking! (-:

If you have the patience why not start out with Richard Hamilton's papers on Ricci flow? He invented this idea which underlies the proof of Thurston's Geometrization conjecture.

His collaborator Matt Grayson also has a famous paper in the Annals of Mathematics on curvature flow of closed curves.

This work is certainly important background to Perleman and is much simpler (though not simple).