I found an elementary introduction to the Catalan numbers at Tom Davis’ web site. The Catalan numbers arise in several counting problems, such as counting the number of ways of parenthesizing an expression and the number of ways to cut up a polygon into triangles.
Author Archives: Walt
Computing Chern classes
Zach Teitler has written some notes on how to explicitly compute Chern classes in algebraic geometry.
Woodin on the Continuum Hypothesis
Hugh Woodin has two survey articles on recent work on the Continuum Hypothesis: I and II. Most mathematicians consider the continuum hypothesis as a settled question: since it is independent of ZFC, its truth is unknowable.
Set theorists, on the other hand, sometimes hold out the hope that new, intuitive axioms will be found that will provide a definite answer. Woodin thinks that we are close to finding such an axiom, and it seems to indicate that the cardinality of the reals is aleph two. (The continuum hypothesis states that it’s aleph one.)
There is a simpler example of an intuitive result that implies that the continuum hypothesis is false. Details can be found here and here.
E. Lee Lady
E. Lee Lady, a mathematician at the University of Hawaii, has a terrific collection of lecture notes in algebra. He also has posted a draft manuscript of a book on torsion-free modules over Dedekind rings, which years of graduate school brainwashing will convince you are the natural generalization of the ring of integers.
Euler on ArXiv
Leonhard Euler is one of the most prolific mathematicians in history. So prolific, in fact, that he has posted 14 articles to ArXiv, despite being dead for 222 years.
The Euler Archive has many more papers of Euler’s, both in the original (either Latin or French) and in translation.
The (Mis)Behavior of Markets
Michael of comment board fame had lent me Benoit Mandelbrot and Richard Hudson’s The (Mis)Behavior of Markets a while ago, and I finally had a chance to read it. The verdict? Still not sure.
Mandelbrot offers an eloquent critique of contemporary financial theory, and speculates on some alternatives. The limitations of the financial theory presented in textbooks is well known: rare events happen more often than predicted by a normal distribution (so-called “fat tails”), and changes in the volatility of financial time series tend to persist, so this part of Mandelbrot’s book is not original, while he does a good job of explaining it.
The part that is new is a series of alternative proposals for financial models. Unfortunately, since the book is written for a general audience, it’s thin on technical details, so I’m not really sure if they’re a good idea or not. I tracked down some links which I’ll work through as I get the chance:
- SmartQuant has a bunch of links to papers
- Levy distribution
- MIT has an online course on Power laws
- Cosma Shalizi has a weblog post criticizing the tendency to see power law distributions everywhere.
Daniel’s ArXiv highlights
Daniel Doro Ferrante has been picking out weekly highlights from ArXiv, with a particular emphasis on cosmology, and the mathematics related to it.
Among the papers he spotted this week is The world problem: on the computability of the topology of 4-manifolds by James van Meter. For some reason, I was thinking about this topic a couple of days ago. Markov proved that every possible finitely-presented group occurred as the fundamental group of a 4-manifold. Post proved that it is undecidable whether two finitely-presented groups are isomorphic. Ergo, deciding when two 4-manifolds are homeomorphic is undecidable. Van Meter sketches both the Markov and Post results.
Keeping Secrets about Squared Circles
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?
Hilbert’s 24th Problem
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
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.