## Math reading list

These are all expositions of topics that I am interested in. I'll attach a note if I have actually read the thing (sometimes even if I haven't.) Suggestions and reviews are welcome .

Rational Tangles are one of Conway's many fascinating discoveries (a basic introduction can be found here.) This paper by Kauffman proves the main theorem in a self-contained manner.

The Banach-Tarski Paradox A truly bizarre consequence of the axiom of choice is that you can take a solid ball, cut it up into a finite number of pieces and then reassemble those to get two balls of the same volume. This is a very nice and detailed exposition written for the Harvard minor thesis. It is readily accessible to advanced undergraduates.

A special case of Sharkovskii's theorem Let f be a continuous map from the unit interval to itself. A point is said to have period *p* if *f^p(x)=x* and *p* is the least integer for which this happens. This paper proves a special case of Sharkovskii's theorem - if *f* has any periodic point that is not fixed, then there exists a point of period 2. The paper is dense at first reading but the method is quite interesting.

The 'opposite' special case is that if *f* has a point of period 3 then it has also a point of period *n* for *arbitrary n*. That is the famous "Period Three Implies Chaos" theorem.

#### Lists of papers :

The Lester Ford award is given to upto five outstanding expository papers every year. You can find the awardees here. Big names on that list include Kac, Halmos and Milnor.

**E-mail** me with more suggestions at *abhishek at ocf dot berkeley dot edu*

*Last modified on 20th April 2003*

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