I have a question about projective determinancy that I was hoping someone could answer. Is projective determinancy provably consistent with ZFC , or does its consistency require large cardinals to prove?

I’ve been trying to learn something about Banach spaces, the kinds of things you wouldn’t see in an introductory functional analysis course. I found this dense survey paper, Basic concepts in the geometry of Banach spaces by Johnson and Lindenstrauss, on William Johnson’s website.

Alon Levy is organizing a Carnival of Mathematics.

I’ve run across these on biology weblogs, but I have to admit I’m not 100% sure how weblog carnivals work. I know that this makes me sound like an old fogey who doesn’t know how to work his VCR and bitches about kids these days, but as far as I know I may already be an old fogey who doesn’t know how to work his VCR and bitches about kids these days. In my day, we had to do integrals in our head while walking to school in five feet of snow. Not only was it uphill both ways, but we weren’t allowed to use elementary functions, only power series. And not power series the way you use them today, where you can just use the first few coefficients, or a closed-form formula. No, we had to write out the whole power series, term by term. Back then, we were tough.

Commenter tdstephens3 in this thread at Mathematics Under the Microscope:

Mathematicians aren’t born from school math competitions in the same way that poets do not grow out of spelling bees.

At the rate I’ve been writing up this post, I’m surprised I finished it before March. The January Notices of the AMS have been out for a while. The feature article, Homological Sensor Networks, describes an application of homology to network design. I predict computational homology will be a major growth area for applied mathematics in the future.

What is… a projective structure introduces manifolds that are locally modelled after projective space. There’s also a review of Fearless Symmetry, which is a popularization of advanced number theory (going so far as to talk about the relationship between number theory and representations of Galois groups, apparently). My library has this book, so I plan taking a look to see how the authors do.

(The February Notices are already online, but I’ll save a post for that when I finally get a chance to look at it. If the March Notices are already online, I don’t want to know about it.)

Sometimes I think I have a handle on Banach spaces. Then I contemplate the example of Tsirelson space, which is a Banach space that does not contain as a subspace any classical sequence space (c₀ or l_p).

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.