Ars Mathematica

Via It’s equal but it’s different, I’ve found a nice introduction to harmonic analysis: Adventures in harmonic analysis.

A few months ago I claimed that there was a “new“ Lothaire book, Algebraic Combinatorics on Words. This was a brazen lie on my part — the book, published in 2002, has already reached the advanced age of 3. I came across the web page of Jean Berstel, which has a link to an an actual new Lothaire book, Applied Combinatorics on Words. This one really is published in 2005.

Berstel also has a link to the text of Theory of Codes, a book he cowrote with Dominique Perrin.

As probably most of you have heard by this point, Serge Lang has passed away. He is most famous for his very many math textbooks. My impression was that most people were not fans of his books, but the discussion at Not Even Wrong was much more positive. What do you think? Have you read any of his books? Liked them? Hated them?

According to Wikipedia, the analogue of a random walk for a Lévy distribution is called a Lévy flight. I presume it’s called a “flight” because paths are no longer guaranteed to be continuous, but can have sudden jumps.

I also spotted a survey paper, More “Normal” than Normal: Scaling Distributions and Complex Systems, which argues that in physical applications, heavy-tailed distributions such as the Lévy distribution are more natural than the normal distribution. This seems to be emerging conventional wisdom in some circles, but I don’t know how true it is.

As a mathematician, something that I always envied physicists is the uninhibited way they use mathematics.
The classic example is the Dirac delta function, which is a function that’s zero everywhere except the origin, but has area one. The fact that no such function exists is only a minor inconvenience. Delta functions can be made rigorous as a distribution, but the concept well predates its formal definition. For example, Green’s functions, which are defined in terms of the delta function, date from 1828.

A more dramatic example is the concept of a dipole. A dipole is the limit of two electric charges of opposite charge as the distance between them goes to zero. It can also be thought of as the limit of the difference of two Dirac delta functions, or even the derivative of a delta function. Dipoles are used to approximate the effect of magnets from a long distance. In terms of distributions, a dipole is the derivative operator on the space of smooth functions, but this is far from the physical intuition.

Most people who learn measure theory secretly believe that all sets are “really” measurable. The one example of a nonmeasurable set anyone sees is that of Vitali, which requires the axiom of choice, so it’s tempting to believe that without the axiom of choice, every set is measurable. This belief is only reenforced by a result of Solovay’s that the axioms of set theory, removing the axiom of choice, and adding the axiom that every set is measurable remain consistent.

Pace the Solovay result, this belief is not quite right. Within set theory it is possible to construct sets without using the axiom of choice which are not necessarily measurable. The sets are not exactly nonmeasurable, but their measurability is independent of ZFC. (Solovay proved that their measurability can be added as a new axiom.)

These sets can be constructed as follows. The image of a Borel set under a continuous function is known as an
analytic set. Analytic sets are measurable. The complement of an analytic set (a coanalytic set) is also measurable, but the image of a coanalytic set under a continuous function in general has undecidable measurability.

This raises the metamethematical question, should they be measurable? My (vague) intuition is no: whether or not we assert the measurability of these sets, we have no way to actually assign a measure to them. Woodin, in his survey article about the continuum hypothesis that we linked to earlier argues yes.

I just spotted an interesting paper on Arxiv: Computing over the Reals: Foundations for Scientific Computing, which suggests a new model of computation with real numbers.

David Applebaum has a nice survey article on Lévy processes. As we’ve mentioned before, a persistent modelling problem in finance is that the variance of changes in financial time series, such as stock prices, seems to be infinite. This shows up as large jumps in price, larger than can be explained by Brownian motion. Lévy processes, a broad class of stochastic processes that generalize both Brownian motion and Poisson processes, are one candidate to model prices.

There’s a new paper available on ArXiv, The Universe from Scratch, which provides a lay introduction to a new approach to quantum gravity, causal dynamic triangulations.

The October Notices of the AMS is already out. It features an interview with Heisuke Hironaka. Hironaka is most famous for his proof of the existence of a resolution of singularities for an algebraic variety: every algebraic variety is birationally equivalent to a smooth variety, and the birational equivalence can be realized as a sequence of blow ups. The proof involves a famously fiendish sextuple induction. For a nice introduction, take a look at Hauser’s article, Hironaka Theorem on Resolution of Singularities.

Hironaka’s proof only works in characteristic zero, so a major research problem has been the situation in characteristic p. Abhyankar has proven it in the case of surfaces, but as far as I know, the question is still open in higher dimensions. Interestingly, people have been able to prove weaker results but by going in a radically different direction. The review of the book Alterations and resolution of singularities from the Bulletin provides some details.

Peter Woit spotted the new issue of the Notices a couple of days ago, and has some comments on the contents. He also passes along the interesting fact that Hironaka is celebrity in Japan, a big enough one that he appears on billboards.