Ars Mathematica

David Speyer gives a nice introduction to the representations of GL(n) at the Secret Blogging Seminar.

Todd at Topological Musings has posted an elementary proof of the Hairy Ball Theorem: the theorem that all vector fields on a even-dimensional sphere must vanish somewhere. The elementary proof is by Milnor.

The intersection of two interstate highways, I-95 and I-695 near Baltimore, is topologically non-trivial; it features a non-trivial braiding. Unfortunately, the interchange is scheduled to be redesigned.

Via Low-Dimensional Topology.

The Clay Mathematics Institute has placed their library of publications online. Their most high-profile publication (other than the Millennium Problems) is Morgan and Tian’s write-up of the proof of the Poincare Conjecture.

They have an interesting article by Bernd Stermfels, Can Biology Lead to New Theorems? You can guess his answer from the fact that the article exists at all.

Via Not Even Wrong.

A group of UK universities have put together a database of links to online resources in various academic subjects, called Intute. Their mathematics section is particularly impressive. (They’ve already linked to almost every online math book I can think of.)

Rigorous Trivialities is hosting the 36th Carnival of Mathematics.

The blog also has a long running series expounding the basics of algebraic geometry. The latest post covers blowing up.

I’m pretty sure that a certain theorem about cocomplete categories must be true, and I’m even pretty sure that I know how to write down a proof. (Famous last words, I know.) But I have the feeling that the result is already known, and I just haven’t seen it. I thought I would state the result here (in somewhat vague terms), and hopefully someone can point me to the result, if it already exists.

Every cocomplete category that is co-well-powered and has a set of generators can be constructed explicitly as follows. Each object X can be represented as:

1. A family of sets, X_i. This family is always a set. Each set represents a different sort, in the sense of multisorted algebras.
2. A family of relations, R_j defined on the X_i. The relations can be of arbitrary arity and signature (so you can have relations on X_1 x X_2, etc.) Infinite arities are allowed. The number of relations of a fixed arity and signature is a set, but the family of all relations can be a proper class.
3. A family of partially-defined operations. Each operation has as its domain all tuples that satisfy a certain relation.
4. The relations are required to satisfy a collection of specified Horn clauses. The left-hand side of the Horn clauses can contain infinite conjunctions.

The arrows of this category are all families of functions X_i -> X’_i that preserve the R_j and the partial operations.

An easy example of this is the category of small categories. Here X_1 is the set of objects, X_2 is the set of arrows. It has four operations: the id operation that sends an object to its identity element, the dom operation that sends an arrow to its domain, the cod operation that sends an arrow to its codomain, and the partial operation of composition, which is defined for all f and g such that cod f = dom g. The Horn clause it satisfies is the requirement that the identity arrow is an identity under composition. (This example is unusual in that the relation is an equality between two operations; the relations can be arbitrary in general.)

Yesterday, everyone was all atwitter over a new preprint by Xian-Jin Li containing a purported proof the Riemann Hypothesis. The optics of it looked good (Li is clearly not a crank), but Terry Tao has identified an apparent error.

More at Not Even Wrong.

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.)

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.