0708-1300/Class notes for Tuesday, October 30

0708-1300/Navigation Panel   Add your name / see who's in! # Week of... Links Fall Semester 1 Sep 10 About, Tue, Thu 2 Sep 17 Tue, HW1, Thu 3 Sep 24 Tue, Photo, Thu 4 Oct 1 Questionnaire, Tue, HW2, Thu 5 Oct 8 Thanksgiving, Tue, Thu 6 Oct 15 Tue, HW3, Thu 7 Oct 22 Tue, Thu 8 Oct 29 Tue, HW4, Thu, Hilbert sphere 9 Nov 5 Tue,Thu, TE1 10 Nov 12 Tue, Thu 11 Nov 19 Tue, Thu, HW5 12 Nov 26 Tue, Thu 13 Dec 3 Tue, Thu, HW6 Spring Semester 14 Jan 7 Tue, Thu, HW7 15 Jan 14 Tue, Thu 16 Jan 21 Tue, Thu, HW8 17 Jan 28 Tue, Thu 18 Feb 4 Tue 19 Feb 11 TE2, Tue, HW9, Thu, Feb 17: last chance to drop class R Feb 18 Reading week 20 Feb 25 Tue, Thu, HW10 21 Mar 3 Tue, Thu 22 Mar 10 Tue, Thu, HW11 23 Mar 17 Tue, Thu 24 Mar 24 Tue, HW12, Thu 25 Mar 31 Referendum,Tue, Thu 26 Apr 7 Tue, Thu R Apr 14 Office hours R Apr 21 Office hours F Apr 28 Office hours, Final (Fri, May 2) Register of Good Deeds Errata to Bredon's Book

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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.

6) It doesn't matter how much you stir your coffee at least one point will be in the same position.

Constructive proof of Brouwer's Fixed-Point Theorem

Several proves of Brouwer's theorem had been given. See

A Constructive Proof of the Brouwer Fixed-Point Theorem and Computational, R. B. Kellogg; T. Y. Li; J. Yorke

Most of them motivated by the fact that the first proof of the fixed-point theorem was a non-constructive indirect proof i.e. using reductio ad absurdum and hence using the excluded middle axiom. This axiom is rejected by Brouwer's itself paradigm of Foundations of Mathematics and the intuitionist school of which Brouwer is one of the founders.