Problem from A Beautiful Mind

[Dragonfall]

Problem from "A Beautiful Mind"

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

$$V=\{F:\mathbb{R}^3\backslash X\rightarrow \mathbb{R}^3:\nabla\times F=0\}$$

$$W=\{F=\nabla g\}$$

$$\dim (V/W)=?$$

What's X?

 

[Fabius2]

V/W is the first deRham cohomology group of the space R^3\X, where X is presumably some subset of R^3.

 

[Freddy_Turnip]

Just finished reading the biog. A good read. Can't help with the above math though.

 

[Dragonfall]

Can't solve it without X.

 

[icystrike]

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

inverse cohomology problem to find a manifold M with a 8-dimensional fundamental group.

 

[AlephZero]

What's X?

$$\times$$ is the vector cross-product.

$$\nabla\times$$ is the vector calculus "curl" operator.

 

[disregardthat]

$$\times$$ is the vector cross-product.

In that case, what does $$\mathbb{R}^3 \slash \times$$ signify?

 

[Kevin_Axion]

Firstly the person is right in saying $$\times$$ is the x but that is not what the person is referring to, they are asking what the $$X$$ means in $$X\rightarrow \mathbb{R}^{3}$$. But I believe the person is right in saying it is the first deRham cohomology group.

 

[Edgardo]

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" .

3) http://www.math.harvard.edu/~knill/teaching/math21a/nash.pdf" , 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" , R. Jason Parsley