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.
While thin on details, I found the article A new efficient algorithm for computing Gröbner basis intriguing. While a naive implementation of Gröbner bases is easy to come by, as a practical algorithm it is highly sensitive to the order in which you add new polynomials to the basis. This has given rise to a whole literature on strategies to add new polynomials. In the paper, Faugere seems to suggest that the whole question can be bypassed, and that polynomials can be added to the basis in bulk by using sparse linear algebra techniques.
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.
I had a pet theory that a large part of what made some people good at mathematics was simply memory. A large part of mathematical practice is remembering bits of trivia: standard counterexamples, definitions, little technical tricks. BBC News is reporting on a discovery that makes that seem less plausible: chimpanzees may have better memories than people do. Scientists compared the performance of chimpanzees and their closest relatives, university students, when tested on their ability to remember a random pattern of numbers on a screen. The BBC story comes with two videos of the chimps in action that are worth watching.
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.
A reader emailed me about a new online mathematics journal, Rejecta Mathematica. As far as I can tell from reading the FAQ, they only publish articles that have been rejected by peer reviewed journals. The twist is that with your submission, you must include a letter that describes the reasons the paper was rejected as well as any other flaws the paper might have. Submissions are not peer reviewed, and results are not required to be either correct or new, but submissions can be rejected, if they are deemed either incomprehensible or not mathematical.
This comment by klein4g helped me clarify for myself exactly what my objection to the examples then definition style of teaching. It’s that the author or speaker is pretending that we’re collectively coming up with the common definition as an act of creativity, when in reality there’s a right answer and the author knows it. It’s the pretense that annoys me.
Tim Gowers offers what he considered an uncontroversial pedagogical principle, examples first. He discovered, though, that on the internet there are no uncontroversial topics.
When I started reading Tim’s post, I expected to agree completely with his point. I think I understand subjects almost entirely through examples. I expected him to advocate putting examples before theorems, but he’s actually advocating putting examples before definitions. I dislike that style fairly strongly; it’s like being led down a road by someone who already knows the destination, but they won’t tell me where we’re going until we actually get there.