J. L. Bell’s A Primer of Infinitesimal Analysis (an intro to synthetic differential geometry) begins with a series of quotes to motivate why we should think of the reals as containing infinitesimals. The quotes all involve the idea that philosophically speaking the continuum is an indivisible whole. This quote from Hermann Weyl is a typical example:
A true continuum is simply something connected in itself and cannot be split into separate pieces; that contradicts its true nature.
I find this line of reasoning completely unconvincing as motivation for allowing the reals to have nilpotent infinitesimals. I can grant, for the sake of argument, that maybe its unnatural that our model of the line can be split cleanly into two or more parts, but to me this is an argument for constructivism, not infinitesimals.
I deny the fact that I just went three weeks without posting. Nothing on the Internet can be trusted, even timestamps on blog posts.
The math librarian at Drexel University, Peggy Dominy, has a blog. Most of the posts are about acquisitions by Drexel’s library, but some are about new public math resources. For example, from this post I learned that the Hiroshima Mathematical Journal is now open access.
When I first took abstract algebra, I loved theorems classifying all of the groups of a certain order. Here is a paper I would have loved, The Groups of Order Sixteen Made Easy. Normally, the classification of groups of order 16 is described in terms of group extensions and the theory of p groups. The author bypasses all that to give a more elementary derivation.
Via God Plays Dice.
This is a test post to see what’s involved in uploading images.
This is of course the Sierpinski carpet. What’s interesting to me is that many objects that, in a previous age dominated by a picture of the physical world as a continuum, seemed deeply pathological, have straightforward computer-language descriptions. For example, you can check whether or not a point in the plane is on the Sierpinksi plane by looking at the ternary expansion of its coordinates, which is a couple of lines of computer code. From the point of view of the computer, the Sierpinksi carpet is not much more complicated than a parabola. I suspect that the popularity of fractals marks a change in the popular imagination of the dominant metaphor for mathematics, from mathematics as mechanics to mathematics as computer program.
The 2008 Abel Prize has been announced. This year’s winners are John Thompson and Jacques Tits.
Thompson is most famous for his work on the Feit-Thompson theorem, that every group of odd order is solvable. Solvable groups resemble upper triangular matrices: a solvable group is constructed in layers out of abelian groups.
Tits invented the notion of BN pairs and buildings. The opposite of a solvable group is a simple group, which cannot be split up into layers. Simple groups tend to resemble the set of all invertible n-by-n matrices over a field (which itself is not simple, but is pretty close to it). Tits identified the key property that makes the resemblence work: the existence of special subgroups B and N. For the group of invertible matrices, B is the set of upper-triangular matrices, while N is the set of permutation matrices. Buildings are a geometric explanation of BN pairs.
What makes a well-written proof? Who writes proofs well? Discussions of mathematical exposition usually revolve around larger-scale questions, such as how to organize the material, what kinds of examples to use, or how much background is necessary; by and large, around the question of what to put between the proofs. There is the art of choosing a proof, which is a subjective measure of taste. But once you’ve chosen the proof, what’s the best way to lay it out?
The style of my own proofs tends towards alternating sentences that begin with “Let” and sentences that begin with “Thus”.
A reader sent me two news articles (here and here) announcing a generalization of the Schwarz-Christoffel mapping in complex analysis. The paper itself is not freely available, but I found this summary from SIAM news that fills out many of the details.
The Schwartz-Christoffel mapping an explicit mapping from the inside of a polygon to the unit disk that is conformal: it preserves angles (it does usually preserve straight lines). The recent work extends this to give conformal mappings from polygonal regions with polygonal holes to circular regions with circular holes. It was known before this that you couldn’t necessarily map any polygon region with holes to any circular region with holes while preserving angles. The two regions must share the same moduli, which are a sets of numbers you can associate with a region. (These moduli are related to the moduli that arise in the theory of Riemann surfaces.)
The breakthrough is not showing that a conformal map exists when the moduli agree, but giving an explicit means of calculating it. The result is not as explicit as the original Schwartz-Christoffel result, but can be calculated numerically.
While I was driving in my car today, I thought of a proof of the Cauchy-Schwartz theorem. I’m sure that it is completely unoriginal, but it has the advantages of both being longer and requiring more background than the usual proof (which you can find on the Wikipedia page).
Let (,) be an inner product. From the definition, we know that for two vectors x, y and two scalars a, b that
(ax+by, ax+by) = a2 (x,x) + 2ab (x,y) + b2 (y,y) ≥ 0
This is a positive-definite quadratic form in a, b, which means that its associated matrix has positive determinant:
(x,x) (y,y) – (x,y) (x,y) ≥ 0,
which is the result.
The real advantage of the proof, I suppose, is that if you already have the linear algebra background there’s no trick involved. It also means that using the same determinant argument there are analogues of the inequality that involve n vectors instead of two.
Peter Woit quotes from a reminiscence by Peter Goddard from a physics conference in 1971:
With great technical mastery, he was covering the board with special functions, doing manipulations that I knew from my studies with Alan White (who was also at the School) could be handled efficiently and elegantly using harmonic analysis on noncompact groups. Just as I was wondering whether it might be too impertinent to make a remark to this effect, the lecturer turned to the audience and said, â€œThey tell me that you can do this all more easily if you use group theory, but I tell you that, if you are strong, you do not need group theory.â€
Count me among the weak.
I’m intrigued by the beginning of a new series of posts at the Everything Seminar about harmonic analysis. This particular post talks about the relationship of singular integral operators and Carleson’s Theorem. Carleson’s Theorem (that Fourier series of functions in Lp for p > 1 converge pointwise almost everywhere) is a famously difficult result; the post gives some idea of where the difficulty lies.
For the ambitious, a complete proof is available in a preprint by Michael Lacey.