Ars Mathematica

A. J. Berrick has an interesting paper explaining how a topologist thinks about group theory. Topology and group theory are connected throught the fundamental group. For every group, topologists can construct a space with that group as its fundamental group. Some of these can be very complicated, even for comparatively uncomplicated groups. For example, perfect groups lead to very scary-looking constructions.

The paper is A topologist’s view of perfect and acyclic groups.

I have just learned from multiple Google searches that a) apparently I think Hilbert is spelled Hibert (with no l), but b) Google is smart enough to correct me. Next year I’m letting Google do my taxes.

There is a new site, outofprintmath, that is collecting information on which out-of-print books the mathematical reading public would like to see brought back in print. At the moment there are 67 books listed, many of them are surprisingly well-known. I had no idea so many classic texts were out of print. For example, Curtis and Reiner’s Methods of Representation Theory, Kuratowski’s Topology, and Riordan’s Combinatorial Identities are all out of print.

Timothy Chow tells the story behind the site’s creation here.

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.)