Ars Mathematica

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.

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

• Greatest Physics Textbook
• Greatest Popular Science Book
• Greatest Physics Paper

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.

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.

This is kinda off-topic for the site, but the Spirit rover on Mars has just crested a hill. NASA has put together a beautiful panoramic picture of the view from the top of the hill.

John Baez has a new This Weeks Finds in Mathematical Physics up. He links to a reasonably elementary survey article about loop quantum gravity by Abhay Ashtekar, which I’m reading right now.

The bulk of his post is about operads, which aren’t something I know much about. They originally arose in algebraic topology, but have turned out to have applications in abstract algebra. Somehow there is a connection between a particular operad from algebraic topology (the little disks operad) and the deformation theory of associative algebras, but I’m murky on the details.

Last week’s claim to be back online turned out to be greatly exaggerated. I think I really am back online this time, though.