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.