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Question about The Ricci Flow

  1. Mar 28, 2010 #1
    I started to read the book "The Poincare Conjecture In Search of The Shape of The Universe."
    It is a terrific book that explains the history of Poincare's conjecture in layman terms.

    The book changed the way that I visualize mathematics.
    I was wondering what is the mathematical background that is needed in order to be able to study and to understand The Ricci Flow?

    Appart from advanced calculus, real/complex analysis, PDE, Riemannian Geometry, and Topology. What else is needed?
     
    Last edited: Mar 28, 2010
  2. jcsd
  3. Mar 29, 2010 #2
    It's great that you want to learn more about mathematics. Many great mathematicians learned the art by themselves, not in a University setting.

    Riemannian geometry is just a branch of Differential Geometry, and many books on this subject cover Ricci flow too, so you'd be on the right track reading a basic differential geometry book. Unfortunately, I can't think of any popular ones right off the top of my head. The book we used was Elementary Differential Geometry by Barrett O'Neill, but I don't remember Ricci flow being covered. It was really just an introductory text.

    Contrary to what it's name might imply, you don't need much knowledge of differential equations (ODEs or PDEs) to be able to follow differential geometry.

    If you've already been exposed to things like advanced calculus and basic topology, and you are willing to follow long, complicated, sometimes counter-intuitive logical discussions, differential geometry should be a swallow-able pill for you. But if you haven't, you should definitely NOT pick up a differential geometry book as it will seem too arcane and will put you off entirely.
     
  4. Mar 29, 2010 #3
    Im teaching myself mathematics, the problem is that in my country there is no math research and the math major takes you just to analysis and probability. They wouldnt include topology and differential geomery.

    Im going to attempt taking a math and a physics major since the math is pretty straightforward.

    For differential geometry, I have been told that "Introduction to Smooth Manifolds" by John Lee is good.
    What about "Lectures on Differential Geometry" by Shing Tung Yau ?

    Appart from this what further mathematical background is needed to fully understand Perelman's result of the Poincare Conjecture?
     
  5. Mar 29, 2010 #4
    Most universities at least offer optional courses in these areas. If they don't, that's really strange, because I've been to universities in the hinterlands of third-world countries that offered at least basic diff geom.

    What kind of math have you done (in an academic setting or independently) ?
     
  6. Mar 30, 2010 #5
    Independently in my spare time im doing Apostol's book, later im looking forward to do Advanced Calculus by Loomis Sternberg and Principles of Mathematical Analisis by Rudin.
    Do you know any good book for complex analysis?

    Im almost sure that they dont offer differential geoemtry I will try to ask to the college soon.
     
  7. Mar 30, 2010 #6
    About complex analysis, you're probably asking the wrong guy.

    If you someday understand Perelman's proof, be sure to post your thoughts here on this forum so lazy people like me can be enlightened!
     
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