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.
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:
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.
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.