0708-1300/Class notes for Tuesday, October 30

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r:D^{n+1}\to S^n} .

Corollary. (The Brouwer Fixed Point Theorem) Every smooth Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^n} is not smoothly contractible.

Challenge. Remove the word "smooth" everywhere above.

Smooth Approximation

Theorem. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} be a closed subset of a smooth manifold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:M\to{\mathbb R}} be a continuous function whose restriction $f|_{A}$ to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is smooth, and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} be your favourite small number. Then there exists a smooth Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g:M\to{\mathbb R}} so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 SIAM Journal on Numerical Analysis, Vol. 13, No. 4. (Sep., 1976), pp. 473-483.

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.