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Problem from A Beautiful Mind

  1. Aug 27, 2009 #1
    Problem from "A Beautiful Mind"

    In the movie when he first walks in to teach that ad cal class, this is the problem he wrote on the board (verbatim):

    [tex]V=\{F:\mathbb{R}^3\backslash X\rightarrow \mathbb{R}^3:\nabla\times F=0\}[/tex]

    [tex]W=\{F=\nabla g\}[/tex]

    [tex]\dim (V/W)=?[/tex]

    What's X?
     
  2. jcsd
  3. Aug 27, 2009 #2
    Re: Problem from "A Beautiful Mind"

    V/W is the first deRham cohomology group of the space R^3\X, where X is presumably some subset of R^3.
     
  4. Aug 27, 2009 #3
    Re: Problem from "A Beautiful Mind"

    Just finished reading the biog. A good read. Can't help with the above math though.
     
  5. Aug 27, 2009 #4
    Re: Problem from "A Beautiful Mind"

    Can't solve it without X.
     
  6. Dec 23, 2010 #5
    Re: Problem from "A Beautiful Mind"

    Dim(v/w) = 8 instead of question mark.

    inverse cohomology problem to find a manifold M with a 8-dimensional fundamental group.
     
  7. Dec 23, 2010 #6

    AlephZero

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    Re: Problem from "A Beautiful Mind"

    [tex]\times[/tex] is the vector cross-product.

    [tex]\nabla\times[/tex] is the vector calculus "curl" operator.
     
  8. Dec 23, 2010 #7

    disregardthat

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    Re: Problem from "A Beautiful Mind"

    In that case, what does [tex]\mathbb{R}^3 \slash \times[/tex] signify?
     
  9. Dec 23, 2010 #8
    Re: Problem from "A Beautiful Mind"

    Firstly the person is right in saying [tex]\times[/tex] is the x but that is not what the person is referring to, they are asking what the [tex]X[/tex] means in [tex]X\rightarrow \mathbb{R}^{3}[/tex]. But I believe the person is right in saying it is the first deRham cohomology group.
     
  10. Dec 24, 2010 #9
    Re: Problem from "A Beautiful Mind"

    Material on the Nash problem:

    1) http://answers.yahoo.com/question/index?qid=20080501140727AAb2Ys1", Yahoo Answers

    2) A beautiful mind http://www.math.harvard.edu/~huizenga/LECTURE35WS.PDF" [Broken].

    3) http://www.math.harvard.edu/~knill/teaching/math21a/nash.pdf" [Broken], Spring 2004

    4) "www.wfu.edu/~parslerj/math733/lecture%20notes%201-4.pdf"[/URL], Math 733: Vector Fields, Differential Forms, and Cohomology
    [PLAIN]http://www.wfu.edu/~parslerj/math733/note.html" [Broken], R. Jason Parsley
     
    Last edited by a moderator: May 5, 2017
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