Difference between revisions of "0708-1300/Class notes for Tuesday, October 30"

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(With Brouwer's fixed point theorem you can prove amazing things)
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3) You can through three potatoes in the air and with just one swing cut all of them in half.
 
3) You can through three potatoes in the air and with just one swing cut all of them in half.
  
4) Every non-bold person has a swirl of hair or some oder problem ordering their hair...
+
4) Every non-bold person has a swirl of hair or some other problem ordering their hair...
  
 
5) If you have a car with a loose antenna and you always go in your car in a trip exactly the same way every day then there is an initial position of the antenna such that it wont fall during your trip.
 
5) If you have a car with a loose antenna and you always go in your car in a trip exactly the same way every day then there is an initial position of the antenna such that it wont fall during your trip.

Revision as of 09:53, 30 October 2007

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Contents

Today's Agenda

Debts

A bit more about proper functions on locally compact spaces.

Smooth Retracts and Smooth Brouwer

Theorem. There does not exist a smooth retract r:D^{n+1}\to S^n.

Corollary. (The Brouwer Fixed Point Theorem) Every smooth f:D^n\to D^n has a fixed point.

Suggestion for a good deed. Tell Dror if he likes the Brouwer fixed point theorem, for he is honestly unsure. But first hear some drorpaganda on what he likes and what he doesn't quite.

Corollary. The sphere S^n is not smoothly contractible.

Challenge. Remove the word "smooth" everywhere above.

Smooth Approximation

Theorem. Let A be a closed subset of a smooth manifold M, let f:M\to{\mathbb R} be a continuous function whose restriction f|_A to A is smooth, and let \epsilon be your favourite small number. Then there exists a smooth g:M\to{\mathbb R} so that f|_A=g|_A and ||f-g||<\epsilon. Furthermore, f and g are homotopic via an \epsilon-small homotopy.

Theorem. The same, with the target space replaced by an arbitrary compact metrized manifold N.

Tubular Neighborhoods

Theorem. Every compact smooth submanifold M^m of {\mathbb R}^n has a "tubular neighborhood".

Entertainment

A student told me about this clip on YouTube (lyrics). Enjoy!

There is this one too but it is in Spanish. Romance of the Derivative and the Arctangent

Further Notes

With Brouwer's fixed point theorem you can prove amazing things

1) There are to antipodal points in the equator with the same temperature.

2) There are two antipodal points with the same temperature and the same pressure.

3) You can through three potatoes in the air and with just one swing cut all of them in half.

4) Every non-bold person has a swirl of hair or some other problem ordering their hair...

5) If you have a car with a loose antenna and you always go in your car in a trip exactly the same way every day then there is an initial position of the antenna such that it wont fall during your trip.