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I have knowledge of basic school math, and in colleges I have read calculus (mostly forgotten now). I need to understand poincare conjecture, and hence I need to study a lot of things. I need to know from you what I should learn step by step to understand it fully if I have basic math knowledge. Thanks

T J. Kim
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debnath
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    Do you want to understand the basic idea of it or the proof by Perelman? – qmd May 23 '16 at 16:58
  • Proof by Perelman that actually comes from Ricci Flow – debnath May 23 '16 at 17:04
  • Then I won't be able to help you. I am not qualified to talk about Perelman's proof so another user will have to help you. Good luck! – qmd May 23 '16 at 17:10
  • Thanks. But I did not ask to explain it. I will understand it myself. I wanted to know which topics are prerequisites to understand it? :) – debnath May 23 '16 at 17:13
  • If I don't understand it, how would I be able to tell you about the prerequisites? – qmd May 23 '16 at 17:14
  • :) ok but thanks anyway – debnath May 23 '16 at 17:14
  • There is an overview of the Poincaré Conjecture here that will give you an idea of where to start. Basically algebraic topology and homology theory which will require basic knowledge of the theory of groups. – John Wayland Bales May 23 '16 at 17:19
  • Grt help! So I just looked into algebraic Topology and came to know about a book called "Basic Algebraic Topology". May I know what are the prerequisites to understand "Algebraic Topology". Here is a link from Amazon of that book-- https://www.amazon.in/Basic-Algebraic-Topology-Anant-Shastri-ebook/dp/B00GBC3INM?ie=UTF8&ascsubtag=c68a8a7a-fc41-43d0-8a6b-c6419a2d78dd&tag=googinhydr18418-21&tag=googinkenshoo-21 – debnath May 23 '16 at 17:44
  • I am checking Topology and Geometry book by Bredon. May I start with it? – debnath May 23 '16 at 18:03

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Terence Tao held a course on Perelman's proof. There are also slides and a paper somewhere. That material is probably the most pedagogic exposition to Perleman's proof you have. And, as you can see, it's very technical throughout.

My personal guess would be that, even with a pure math degree, you would have to professionally work on going through the proof for years in order to understand it on a satisfactory level (it took many years for several independent research teams to verify the proof). Perelman is super strong and well-read, and it took him $10$ years to complete it, so you shouldn't expect less.

If you really would like go through it, you would basically have to study an amount equivalent to a PhD in a very specific area of differential geometry before you could even start.

Matias Heikkilä
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