Ars Mathematica

Continuing the architectural theme, Isabel at God Plays Dice has a post on the ultimate fate of the real world Königsberg bridge problem. Königsberg had seven bridges, and in 1736 Euler proved it was impossible to find a path that allowed you to cross each bridge exactly once.

In World War II, several of the bridges were bombed, and later some were replaced. In present-day Königsberg, now Kaliningrad, there are now only five bridges, and you can now find a path that allows you to cross each bridge exactly once.

Now that this is primarily an architecture blog, here’s Falkirk Wheel, a science-fiction-looking rotating boat elevator in Scotland.

Actual math content soon.

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.)