A couple of weeks ago, Peter Woit linked to some slides by David Vogan on the orbit method in representation theory of Lie groups. The slides give some of the flavor of subject, but in PDF form are very repetitive, for reasons that are completely clear if you’ve ever attended a Powerpoint presentation with one of those remote controls that allow you to use visual effects. Much better is Vogan’s review of Kirillov’s Lectures on the Orbit Method, a book that I have taken out of the library without reading more times than I care to admit to.

Wedderburn’s theorem (proof here) states that any finite division ring is a field. Interestingly, apparently this generalizes to nonassociative division rings that are alternative. This is known as the Artin-Zorn theorem. The best online reference I could find was here.

Back in November Anton Geraschenko had an interesting post at the Secret Blogging Seminar based on a preprint by George Bergman. Homological algebra is full of diagram chasing arguments that lead to scary-looking theorems like the Snake Lemma. Bergman claims that these are all special cases of an even-scarier looking but more obvious result he christens the Salamander Lemma. I’ll definitely be looking more closely at this when I get a chance.

There’s a quote that I have rattling around in my head that I can’t quite remember. The quote was of the form “___ is more interesting as a source of questions than a source of answers.” I don’t remember what went in the blank (the Brauer group, maybe?) or who said it, so if anyone happens to know I’d appreciate it.

I was browsing through Wikipedia today when I came across the definition of pretopological space. The notion seemed very exotic until I thought of a family of examples, which I’m christening ennui spaces.

A pretopological space prescribes for each point the set of (not-necessarily open) neighborhoods of that point. The set of neighborhoods of a given point are required to satisfy some natural axioms, but neighborhoods of one point can be completely unrelated to neighborhoods of another point. A sequence in a pretopological space converges if for any neighborhood, the sequence eventually enters that neighborhood and never leaves it again. A topological space can be turned into a pretopological space by taking as the set of neighborhoods of a point to be all sets that contain an open set that contain that point. You can try to reverse the process by borrowing the characterization of the closure of a set in terms of sequences (or nets), but usually the topological space you construct will have a coarser notion of convergence than the pretopological space.

An ennui space has the same underlying set as a metric space. A neighborhood of a point is any set that contains the unit ball around that point. A sequence in an ennui space converges to a point if is guaranteed to be eventually within one unit of the point. The mental image I have is that the sequence gets close to its destination, but then gets bored. If you try to construct a topology out of this space, you get the indiscrete topology, where all sequences converge to all points. Essentially, all information about the convergence properties of the ennui space are lost.

A practical example of an ennui space would be your computer whenever it simulates a convergent sequence of operations, such as numerical integration or Newton’s method. The computer gets within machine precision of the correct answer, and then stops to light up a Gauloise and discuss L’Être et le néant in a cafe.