Author Archives: Walt

Brouwer Fixed Point Theorem

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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.

Universal Family

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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?

Field With One Element

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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.

Spam Must Go On. It Will Go On.

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I happened to check the spam filter, and discovered that our spam is getting depressed, but has determined to persevere. Today’s spam included this comment:

I have become my own version of an optimist. If I can’t make it through one door, I’ll go through another door – or I’ll make a door. Something terrific will come no matter how dark the present….

The Internet attributes the sentiment to Rabindranath Tagore.

Nilpotent Infinitesimals I

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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.

Perspective

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I was looking at the stats for Ars Math, when I saw that we’ve had 525 posts. I thought “Wow, that’s a lot of posts.” Then my eye happened to glance at the count of the number of comments the Akismet plug-in has deleted as spam: 304,061.

Selling Infinitesimals

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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.