Dipoles

Posted on by

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.

The myth of measurable sets

Posted on by

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.

Lévy processes

Posted on by

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.

Hironaka’s resolution of singularities

Posted on by

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.

Most Disturbing Math Theorem Ever

Posted on by

I have identified the most disturbing math theorem ever. What makes it the most disturbing is that it does not involve the Axiom of Choice in any way. I’ve seen the theorem many times before, but I never really noticed how disturbing it was until a couple of days ago.

The theorem is this: for any positive constant c, there is an open set U that contains every rational point, but has measure less than c. Think about what that means, for a minute. The rationals are dense in the reals. Here’s a set that contains an open interval around every rational. Naively I would have believed that the set would have to be the whole real line (except with maybe a finite or countable number of exceptions). At the worst, I would have at least expected the set itself to have infinite measure, and the set’s complement to be measure zero. Instead, not only can we construct such a U with finite measure, we can make that measure be arbitrarily close to zero.

The proof of this is pretty easy, and is a standard result in any real analysis book that covers Lebesgue measure. Not only have I seen it before, I’m pretty sure I’ve seen it pointed out before that the result is surprising. Somehow I never took in how strange it is until just this week.

Div, Grad, Curl and All That

Posted on by

Cosmic Variance had three interesting &ldquot;greatest&rdquot; discussion threads:

In the comments to the Greatest Physics Textbook, Clifford (the original poster) joked that no self-respecting mathematician ever read Schey’s Div, Grad, Curl and All That. I don’t know about anyone else, but that’s the book I learned the subject from. The book gives incredibly hand-wavy proofs, and if I remember right it trumpets its lack of rigor, but it does a good job of giving the intuition behind the Green, Gauss, and Stokes theorems. Reading it made reading something like Spivak’s Calculus on Manifolds much easier.

September Notices

Posted on by

The September issue of the Notices of the American Mathematical Society is out, and the highlight is Robert Gompf’s 3 page introduction, WHAT IS… a Lefschetz Pencil?.

The WHAT IS… series is a terrific addition to the magazine, which began back in September 2002 with WHAT IS… an Amoeba?. Since then, it’s provided quick introductions to various terms that float around mathematics, such as the Monster group, motives, flips, and even operads.