poincare conjecture understanding

Question

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

Answer 1

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.