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.
This is a follow-up to this post.
Nilpotent infinitesimals allow you to define objects like the “double point”, which is the solution set of x2 = 0 on the line. Intuitively, the double point is the point x = 0, plus another point infinitesimally close to it. We can mimic this in nonstandard analysis by considering the solution set to x(x-h) = 0, where h is an infinitesimal. This has the property that if you evaluate any differentiable f at h, you get
f(h) = f(0) + f'(h) + o(h),
which also holds for nilpotent infinitesimals if you drop the o(h) term. (Here, a nonstandard number k is o(h) if k/h is infinitesimal.)
We can follow the same route to define the intersection of two double lines in the plane as the solution set to x(x-h) = 0 and y(y-k) = 0 where h and k are infinitesimals. In this case, we get a subtle difference from the nilpotent infinitesimal definition as the solution set for x2 = 0 and y2 = 0. In nonstandard analysis, xy is necessarily o(h) and o(k). If we neglect lower-order terms, we get the additional equation xy = 0 which is not satisfied by the intersection of two double lines defined using nilpotent infinitesimals. So seemingly we can’t simulate the intersection of two double lines using nonstandard analysis.
(Reasonably, you might think that maybe the nilpotent infinitesimal definition is just wrong, and that the intersection of two double lines really does morally satisfy the equation xy = 0. Just counting points, the intersection of two double lines should be some sort of quadruple point. For an ordinary set of n points, the ring of all differentiable functions from a set of four points to the real line is a vector space of dimension n. Nilpotent infinitesimals preserve this property: the ring of differentiable functions on the intersection of two double lines is a vector space of dimension 4, spanned by 1, x, y, xy. The nonstandard simulation by neglecting lower-order terms gives you a vector space of only dimension 3, spanned by 1, x, y.)
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.
Jos Leys emailed me to let me know about Dimensions, a computer-generated movie illustrating dimensions 2, 3, and 4. The trailer is here. The whole movie can be downloaded for free at dimensions-math.org. (It is also available on DVD.)
The movie was rendered using POV-Ray, and is joint work of Jos, Etienne Ghys, and Aurélien Alvarez.
Update. John Armstrong has downloaded and watched the whole movie, and has posted a lengthy review.
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.
If you’re trying to look up the term universal family, my recommendation is to not to use the search string “universal family” in Google.
Update. In the comments, Matt Heath points out that this post is already in the top ten for “universal family”, which means that I can be the change I want to see in the world. A universal family is basically a fine moduli space. A common example of a universal family is a classifying space in algebraic topology. A description of universal families in the context of elliptic curves can be found in Dan Edidin’s What Is… A Stack?
Lieven LeBruyn has a series of posts about the “field with one element”, here, here, and here. The field with one element does not exist, of course, but Tits pointed out a long time ago that you can think of the symmetric groups as Lie groups over a field with one element. I thought that there was all there was to the idea, but apparently there’s much more you can do with the idea.
This video explaining the basics of chemical reactions may be the definitive educational document of the YouTube Era.
Victor Shoup has written an introductory (abstract) algebra textbook that emphasizes the algorithmic aspects. The best part? You can download it here.
Via God Plays Dice.