Charles Wells, author (with Michael Barr) of Toposes, Triples, and Theories, now has a blog, gyre and gimble, devoted to how mathematicians use language.
He notes that the idea of completed infinity, which mathematicians take for granted, is still not well-liked outside of mathematics. The tone of the Wikipedia page on the subject (which consists mainly of quotes) tends towards the negative, for example.
Doug Ravenal has made the latest version of his book Complex Cobordism and the Stable Homotopy of Spheres available online.
I see, via Yet Another Sheep, that nonstandard analysis has spread to mathematical economics. Robert Anderson has a book manuscript available, Infinitesimal Methods in Mathematical Economics which explains how to apply nonstandard analysis to approximate economies with large numbers of agents. The main technique is Loeb measures, which is something that I plan on writing a post on, once I actually know anything about them.
One idiosyncratic interest of mine is mathematical economics. I was looking through Volume 2 of the Handbook of Mathematical Economics when I spotted a paper by Scarf called “The Computation of Equilibrium Prices: An Exposition”. The real subject of the paper is an incredibly clear exposition of how to find fixed points of maps of the unit n-cube to itself. The Brouwer Fixed Point Theorem promises that at least one fixed point exists. I knew that there was a combinatorial approach based on Sperner’s lemma, but it had always struck me as rather technical. Not so; Scarf gives a straightforward algorithm for finding the fixed point. Sperner’s lemma is just the result that dictates that the algorithm terminates.
The proof is stated for an n-simplex, which is the n-dimensional analogue of a triangle. The algorithm works by cutting up the simplex into smaller simplices, and identifying which of the smaller simplices contains a fixed point. It then repeats, trapping the fixed point in smaller and smaller simplices until it eventually converges. What’s interesting is that the test for whether a particular simplex contains a fixed point is fantastically crude; it amounts to just checking a simple condition on the map at the vertices. (The condition is not satisfied for every simplex containing a fixed point, and in fact the algorithm will miss some fixed points. At least one fixed point will satisfy the condition, though.)
The article does everything from scratch. Brouwer’s theorem is derived as a consequence of the algorithm. It is simple enough that it could easily be included in an undergraduate analysis textbook. The whole article is so simple that it makes me wonder if there is an elementary combinatorial subject lurking under the intimidating algebra of modern-day homology theory. An interesting test case is if a constructive version of the Lefschetz fixed point theorem. (Lefschetz’ original proof was apparently combinatorial, but extremely difficult to follow. I doubt it was constructive.)
Here is two artists’ take on the Brouwer fixed point theorem.
Update. Commenter Mio spotted this elementary introduction to the topic on Herb Scarf’s web page. Poking around some more, I found the original article I mentioned above here.
I’ve been writing a post explaining the practical difference between the synthetic and nonstandard notions of infinitesimals. It was getting a bit long, so I’m splitting it into two posts, of which this is the first.
Synthetic differential geometry (or smooth infinitesimal analysis) is a way to add infinitesimals to the reals; one that is an alternative to the nonstandard analysis approach. In SDC, infinitesimals can be nilpotent: their square or some higher power can be zero. This allows you to formalize arguments such as “this quantity is so small, we can treat its square as if it is zero”. You can also formalize these arguments in nonstandard analysis, but with more care (you can’t actually set the square to be zero, but you can treat it as an even smaller infinitesimal). Nonstandard analysis cannot have nilpotent infinitesimals directly, because it is required to preserve first-order theorems about the reals (which includes the theorem that the only nilpotent element of the reals is zero).
You can formalize infinitesimal arguments at the level of calculus equally well using either approach, so why would you ever want nilpotent infinitesimals? Here are some examples with a differential geometric flavor. Consider two points on the real line, and move them together so that they coalsce into one point. In ordinary differential geometry, that’s all they are — one point. In the synthetic approach, you can treat this as a double point, with defining equations x2 = 0. You can imitate this in nonstandard analysis; I’ll explain how in the next post.
Here’s an example that’s considerably harder to simulate in nonstandard analysis. Consider four lines in the plane, two horizontal, and two vertical. Collectively they intersect in four points. If we let the two horizontal lines move towards each other, they become a double line, which intersects each of the two vertical lines in a double point. If we now let the two vertical lines degenerate into a double line, we have two double lines intersecting in a quadruple point. But not just any quadruple point (there’s more than one kind), they intersect in the quadruple point with defining equations x2 = 0 and y2 = 0.
Topological Musings has a post up about a cute little result, the Stolz-Cesaro theorem. The result is a discrete analogue of l’Hôpital’s rule.
Via God Plays Dice.
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.