Ars Mathematica

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 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.

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.

The three body problem in physics is the study of the trajectories of three bodies mutually attracted by (Newtonian) gravity. Unlike the two body problem, the three body problem cannot be solved in general, but some specific solutions are known. An article in the Notices of the AMS reports that a new (well, new circa 2001) solution has been found where all three bodies travel in the same orbit — a figure eight.

The figure eight orbit is stable, which means that it is robust under small perturbations. Since any actual physical system will be perturbed slightly by the gravitational pull of other bodies, stability is a prerequisite for a solution to actually be found in nature. So the figure eight orbit may someday be observed by astronomers.