I was glad to see that Wikipedia’s page for the Baker-Campbell-Hausdorff formula actually explicitly states the formula. When I was first learning the subject of Lie groups and algebras, the authors would only show the first few terms, and then an ellipsis. It always left me with the impression that the actual formula was so hideous that no one ever mentioned it explicitly, but only passed over it in discreet silence.

I just ran across a quote by Hilbert from 1930:

The real reason for Comte’s failure to find an unsolvable problem is, in my opinion, that an unsolvable problem does not, altogether, exist.

I assume that Comte is Auguste Comte, the sociologist, but I don’t know what remark Hilbert is alluding to. Gödel published his incompleteness theorem in 1931.

Last year, the entire editorial board of the journal Topology resigned in protest of the high subscription fees charged by the publisher, Elsevier.

Peter Woit notes that many of the same people have now announced a new journal, the Journal of Topology. The annual subscription price is $570. (The price for Topology was $1665.)

We have readers of all backgrounds here at Ars Math, so I thought I would experiment with a more expository post than usual. Commenter Wendell Dryden is teaching himself the Chinese remainder theorem.

The result has been considerably generalized (as the Wikipedia entry makes clear), and one variant is easier to understand (I think): polynomial interpolation. Given n numbers x_i on the x-axis, and n numbers, y_i, on the y-axis, you can always find a polynomial p such that p(x_i) = y_i. The steps in solving this problem and the integer congruence problem are similar, and both problems can be solved by using the extended Euclidean algorithm.

This analogy has been taken much further in algebraic geometry. From that point of view, an integer is no longer just a number, but actually (like a polynomial) a function. The integer, in its function guise, sends prime numbers to that integer modulo that prime. So the function 67 sends 2 to 1, 3 to 1, 5 to 2, 7 to 4, 11 to 1, etcetera. (The value eventually stabilizes, in this instance at 67, which always seemed to me must be a fact of some significance, but I’ve never seen it used for anything.)

So now you know: to an algebraic geometer, integers are functions. Mathematics is like drugs, but cheaper.

Here’s an analogy that I try to complete from time to time.

Integers:Reals::Free Group on Two Generators:?

(Under addition, the integers are the free group on one generator.) I’m not precisely sure what properties of the construction of the reals I’m trying to generalize to the right-hand side of the analogy, beyond the fact that the answer should be a group that is a path-connected topological space.

Two candidates I’ve considered are: 1) certain sets of paths on the plane (which is naturally a groupoid, but you can bully it into being a group) or 2) the Lie group corresponding to the free Lie algebra on two generators (I don’t know in what sense, if any, such an object exists).

Finally, somebody (Rudbeckia Hirta) willing to tell the truth about the harmonic series:

I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.

Via Let’s Play Math.