Polytopes with Non-Rational Coordinates

Now that I have external evidence that someone is still hoping for new posts, I thought I better write one.

Here’s a result that not only would have I not have guessed, but I would have assumed the opposite is obviously true. There are convex polytopes that cannot be presented in Rn as a polytope all of with rational coordinates. I would have assumed that this is wrong because you can always take the vertices, and perturb them slightly so that they become rational. This argument doesn’t work in general, but you can prove using other techniques that in 3 dimensions that every convex polytope can be written with rational coordinates. There already exist counterexamples in dimension 4.

The survey paper Non-rational configurations, polytopes, and surfaces, by Günter M. Ziegler, gives an explicit construction in 13 dimensions. The paper provides a decent overview of many other, related, topics.

Model Categories

Sorry for the dearth of posts recently. I’ve been spending time I would ordinarily spend on blogging instead obsessively learning the theory of model categories and related ideas such as (∞,1)-categories. I found the subject completely impenetrable until I read Greg Friedman’s expository article on simplicial sets. (I wrote an earlier post about it here.)

Functor of Points Versus Locally Ringed Spaces

A year ago, the Secret Blogging Seminar had a long thread on how to teach algebraic geometry, one that I never managed to read in its entirety before now. The thread was very interesting. The original post started with the premise that the best way to introduce the idea of an affine scheme is to use the set of maximal ideals, rather than the set of prime ideals as is standard. This is sufficient for classical algebraic geometry.

The thread then wanders off into a different question, of whether the best way to define schemes is as locally ringed space, or rather in terms of a scheme’s functor of points. In the functor of points view, you think of a scheme S as being given by the functor Hom(Spec A, S), as A varies of all rings. You can think of “Spec A” as representing a generalized point — true points correspond to fields. The idea of a generalized point never appealed to me, but still the functor of points view seems more natural to me. What do we know about a general scheme? We know how to map affine schemes into it, and how these pieces are glued together. I remember when I first tried to read Hartshorne, and I learned about sheaves, and then ringed spaces, and then when I got to the requirement that the induced map between stalks had to be local homomorphisms, I got frustrated that after all that machinery, you still needed a weird extra condition to get the right definition. With the functor of points, we start with affine schemes as the dual of the category of rings, and then say what we mean by “gluing” affine pieces together. The Secret Blogging Seminar thread, though, has several practicing algebraic geometers arguing that some notions, such as proper morphism, are much clearer from the locally-ringed space point of view.

Tai’s Method

I just ran across a slightly odd story. In an old post, the (now-defunct) physics blog Flip Tomato points to an abstract for a paper in a diabetes journal that makes a curious claim. The author claims that she has found a new method (named after herself) for calculating the area under a curve by subdividing it into rectangles and triangles and adding it up. This sounds almost exactly like the trapezoid rule from first-year calculus, reported as a new discovery in 1974.

If you’re curious, you can see the abstract of the paper here.

nLab link

The nLab — a math wiki devoted to n-categories and related topics — is having temporary DNS troubles, so if you go to the usual address you’ll get one of those landing pages that domain squatters love so much. There’s a secret alternative URL, but since I love you guys so much, I’m going let you in on the secret.

Axiom of Replacement in Category Theory

I had a slightly ironic experience on Math Overflow. A couple of months ago, I started wondering to what extent you could develop category theory “below a cardinal”. When you consider the category of groups (for example), you’re probably not literally interested in groups of arbitrarily large sizes — you just want enough space so that you can perform any operation you need to. I started writing this post here arguing that for concrete categories, sets smaller than a limit cardinal were big enough. Limit cardinals are not usually large cardinals in the sense of set theory, but they’re pretty big — the category of sets smaller than a limit cardinal is closed under the power set operation, for example.

Before I finished the post, I thought I should check the claim and look over some proofs in a category theory book. I realized that, under the usual definition of a diagram in the literature, my proposed restriction would make the category of sets fail to be either complete or cocomplete — even countable diagrams could have to have limits or colimits. You could finesse the issue by changing the definition of diagram, but I thought “No one will stand for that”. Under the standard definition, the construction of limits or colimits requires the Axiom of Replacement, which means that the right condition is inaccessibility, or equivalently you need Grothendieck universes.

So now I thought I understood the big picture. Completeness required replacement, which leads naturally to Groethendieck universes, which explains why the main competitor in textbooks to either Goedel-Bernays or Morse-Kelley set theory is to postulate one or more Grothendieck universes. The only thing that puzzled me was that while people using category theory seemingly made casual use of replacement, people would also argue that replacement is never used in ordinary mathematics. I thought that maybe I was confused on some issue, so I asked on Math Overflow.

It turns out that at least some people really don’t want to use replacement. They would rather change the definition of what it means to be a small diagram so as to be able to avoid the axiom. Avoiding replacement has lots of little consequences. For example, you have to require that the image of a small diagram is a set. Even with the corrected definition, the General Adjoint Functor Theorem becomes false as stated, and you have to strengthen the solution set condition. It means lots of fiddly little details have to be changed. You also no longer have as clean of a distinction between large and small. (You can have categories that are locally small, and have only countably many objects, and yet are not small categories, for example.)

But I could have stuck with my original idea for this post.

Universal Differential Equation

Here’s something I didn’t know. There exists nonlinear (but algebraic) ordinary differential equations such that solutions to that differential equation are dense in the space of continuous functions. These are known as universal differential equations. An explicit construction of one is given in this preprint by Keith Briggs. If I understand the construction correctly, the trick seems to be that the nonlinearity gives you branch points where you have a choice for the direction in the solution. This allows you to paste together solutions in enough ways that you can achieve density.