Did you know that according to MathWorld you can square the circle using the compass and straight-edge in the hyperbolic plane? How long have you known this? Why have you been keeping it from me?

MathForge links to an article on Hilbert’s “24th” problem. Hilbert, in his famous 1900 speech, proposed twenty-three open problems in mathematics, but apparently there was a twenty-fourth that he dropped from the list, to formalize the notion of simplicity of proofs, and prove that theorems have a unique simplest proof. Like many of Hilbert’s twenty-three, this is less a problem and more an open-ending research program.

Herbert Wilf has put several of his books online for download. I particularly recommend generatingfunctionology, which is an excellent account of the uses of generating functions in combinatorics. I regarded generating functions as a sleazy trick before I read that book. A=B, a book he co-wrote with Petrovsek and Zeilberger on combinatorial sums, is also very interesting. It turns out there is an eminently-implementable algorithm that will show, for a large class of formulas, when two such sums are equal.

J. S. Milne, author of Etale Cohomology, has a terrific set of lectures notes up at his website www.jmilne.org, which run the gamut from algebraic geometry to algebraic number theory (admittedly, not a very wide gamut). His notes on class field theory are particularly nice, but he also has extensive lecture notes on etale cohomology (I’m not sure how different these are from the book), modular forms, abelian varieties, and some more elementary topics such as elliptic curves.

In the comments to Programming Language and Logic Links, citylight asked for an example of what proof theory is good for. I wanted to sketch an example using Goodstein sequences.

Proving that Goodstein sequences always eventually go to zero requires transfinite induction up to the ordinal \epsilon_0. Gentzen proved, using transfinite induction up to \epsilon_0, that Peano arithmetic was consistent. By Godel’s second incompleteness theorem, not consistent system can prove its own consistency, so transfinite induction up to \epsilon_0 is not expressible in Peano arithmetic, and Goodstein’s theorem is independent of it.

More details can be found at Fast Growing Functions and Unprovable Theorems.

Back when I was a grad student, I would get asked weird questions about what I did all day. One day, after someone asked me if I worked on adding really big numbers together or something, I wrote a Math FAQ to answer questions about what mathematicians did. The link is to a copy of the FAQ that is being maintained by Sarah Marie Belcastro, a math professor at Xavier.

Here is an introduction to Symbolic Dynamics or, if you like books, there is Symbolic Dynamics and Coding.

One of the things that makes symbolic dynamics useful/interesting are Markov Partitions. A Markov Partition is a finite partition of the state space for a dynamical system. For a point x in the state space, look at it’s orbit and write down which element in the partition each point of the orbit belongs to. This infinite sequence of partition elements uniquely defines x. Doing this for all points in the state space yields a mapping from the dynamical system to a shift space. For more details, see the paper Symbolic Dynamics and Markov Partitions.

For an alternative to Euclid’s proof, there is Furstenberg’s topological proof of the infinitude of primes. The MAA‘s Mathematics Magazine published a variation of Furstenberg’s proof which does not use topology:

For each prime p, let S_p = pZ. Each S_p is periodic (it’s characteristic function is periodic). Let S be the union of all S_p. If S is the union of finitely many periodic sets, then S is also periodic. However, the complement of S is {-1, 1}, so S is not periodic. Hence, there must be infinitely many sets S_p, and infinitely many primes.