As mentioned in the title, I want to understand the proof of Poincare Conjecture by Perelman, what prerequisites do I need?

1$\begingroup$ I think there's 7 or more books on the topic now. There's MorganTian. There's Topping. There's Chow (2 different books). There's CaoXiPing. And there's Kleiner. Simon Brendle. Zhang. On and on. I've got a vague memory of a few others. I think different approaches demand different backgrounds. Some are more traditionally 3manifoldsish, some are more DG/PDEish in flavour. $\endgroup$– Ryan BudneyFeb 28, 2012 at 9:28

2$\begingroup$ What prior background do you have in this or related areas? $\endgroup$– Yemon ChoiFeb 28, 2012 at 9:29

1$\begingroup$ Also, there's a rather slender set of John Morgan's lecture notes from a lecture series he gave at Stanford. Authors are Morgan and Frederick Fong. IMO as far as pencilsketch "warm up" type notes go, they seem to be some of the friendliest reading. This is in the University Lecture Series, Volume 53. $\endgroup$– Ryan BudneyFeb 28, 2012 at 9:31

12$\begingroup$ Why not start reading and look up unfamiliar stuff when you come across it? I find that's a good way to read anything. Say they use Hamilton's maximum principle for tensors and you look up Hamilton's paper and are completely lost so you go back to some basic PDE book and learn about the maximum principle for scalars under parabolic flows. You learn something and eventually you get the feel for what's going on and can move on to the next perplexing point. This kind of reverseengineering is a good way to decide what you need to learn. $\endgroup$– Jonny EvansFeb 28, 2012 at 9:57

1$\begingroup$ This should be retagged. $\endgroup$– Sean TilsonFeb 28, 2012 at 14:29
2 Answers
If I were going there I wouldn't start from here.
If you're new to 3manifolds, it might better to familiarise yourself with them intimately before starting on Perelman's work. In fact, learning some knot theory (in particular Dehn surgery) would be a good first step. I don't remember where I first learned this stuff, but I do remember sitting on the floor in the library in front of the lowdimensional topology section and looking at lots of books (perhaps a better search mechanism than Google when you're not quite sure what you're looking for). One good such book is Rolfsen's "Knots and Links". I remember being very happy when I worked out why $S^1\times S^2$ is the result of doing 0surgery on $S^3$ (there's a nice picture).
Maybe using the Wirtinger presentation and van Kampen's theorem to compute the fundamental group of the Poincaré sphere would be a good exercise to convince yourself you understand what's going on with Dehn surgery.
The basic observation in all of this is that the 3sphere is the union of two solid tori (or indeed of two handlebodies of arbitrary genus).
If that grabs your imagination then a good step would be to convince yourself that every 3manifold can be presented as (a) a Heegaard splitting, (b) a sequence of Dehn surgeries on the 3sphere. This uses the Lickorish theorem (that the mapping class group of a surface is generated by Dehn twists) and that will lead you into studying 2manifolds (see Farb and Margalit's book on mapping classes for an excellent presentation).
When you have convinced yourself that the classification of 3manifolds is an interesting and worthwhile subject then there are Hatcher's survey, Allen Hatcher's notes on 3manifolds and Hempel's book (amongst other places). You could have a look at Stalling's "How not to prove the Poincaré conjecture" (available on his website) and maybe at the proof of the Poincaré conjecture in high dimensions (either Smale's original paper or Milnor's wonderful hcobordism theorem book) to get an idea of what you're missing by living in three dimensions.
Perelman's approach comes from a completely different world to any of this: the world of Thurston's geometrisation conjecture. Thurston's book introduces some of these ideas (with an emphasis on the hyperbolic) and his papers are full of beautiful insights. Once you have at least some familiarity with this stuff you could reasonably crack open a book on Ricci flow and start learning about that, but be warned that it won't necessarily bear much resemblance to anything else you've read about 3manifolds.
Of course you don't need all this background to understand Ricci flow, but at least you'll know what a 3manifold is.
I also stand by my comment that the best way to learn something is to pick up a difficult book containing something you would like to understand and then look stuff up as and when you need it. Google and Wikipedia are wonderful for quick reference but they are not an easy place to learn a subject thoroughly for the first time.
Edit: As Deane Yang points out below, if you're more interested in Ricci flow itself, there may be better learning approaches. For instance, Chow and Knopf have a nice book in which they introduce Ricci flow and use it to prove the uniformisation theorem in two dimensions. They also cover Hamilton's theorem that a positively curved 3manifold admits a metric of constant positive sectional curvature. These are both strictly easier than Perelman, while still involving hard differential geometry. Of course, you need to learn some differential geometry but there are plenty of good books about that.

$\begingroup$ As you say, Perelman's proof is completely different from Thurstonstyle topology. I am under the impression that you don't need to know much of the latter to understand Perelman's proof, which is all either analytic estimates or a careful geometric analysis of regions of the 3manifold where the flow breaks down. It is useful to know at least a little bit of 3manifold topology. In addition to Thurston's book, I recall a nice Bulletin of the AMS survey by Peter Scott. $\endgroup$ Feb 29, 2012 at 9:04

2$\begingroup$ I also endorse Jonny's endorsement of the books written by Ben Chow with others (disclaimer: Ben and I have been friends since graduate school). I think Ben and his coauthors have gone to great pains to present the foundations of the Ricci flow very carefully and in great detail. $\endgroup$ Feb 29, 2012 at 9:59

2$\begingroup$ The book by Chow and Knopf is in my opinion the best available place to learn about Ricci Flow for itself, and its 3 sequels entitled "Ricci and its applications" are excellent references for any one working in the field. I should add two more references for Ricci flow for itself, there is the book by Chow, Lu and Ni "Hamilton's Ricci flow" which in my opinion is a bit harder than Chow and Knopf but covers a bit more material. There are also notes by P. Topping, available on his website, which goes up to the beginning of Perelman's work (F and W functionals) in a nice selfcontained way. $\endgroup$ Feb 29, 2012 at 12:26

2$\begingroup$ Your answer is quite encouraging!!!..I will start my learning journey!.. $\endgroup$ Mar 1, 2012 at 9:00
My humble advice for learning about Ricci flow generally, after obtaining some background in Riemannian geometry, would be to start with a book which gets you to important results quickly. An excellent book is the one by Peter Topping. (The only typo I observed there is the one regarding backwards uniqueness, which is now due to Brett Kotschwar.) After that, there are excellent books on the differentiable spherical space form theorem by Brendle and AndrewsHopper; see also the original papers of BoehmWilking and BrendleSchoen.
What is irreplaceable is to read and to reread the original works by the masters, i.e., Hamilton and Perelman. A collection of Ricci flow papers, mostly by Hamilton, is edited by H.D. Cao, etal.; this is a convenient place to get Hamilton's papers in one place. Perelman's papers are on arXiv. There are a number of excellent expositions of their work (focused on Perelman's work), which actually go beyond expositions and include various degrees of original work, namely in alphabetical order: BessieresBessonBoileauMaillotPorti, CaoZhu, KleinerLott, and MorganTian.
The above remarks only pertain to Riemannian Ricci flow.