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.
Monthly Archives: December 2007
DARPA Challenge Problems
DARPA has put out a list of 23 challenge problems in pure and applied mathematics. Some of them are specific, such as number 19:
SETTLE THE RIEMANN HYPOTHESIS
The Holy Grail of number theory.
Some are vague, such as number 3:
CAPTURE AND HARNESS STOCHASTICITY IN NATURE
Address Mumford’s call for new mathematics for the 21st century.
Via Peter Woit.
Artin-Zorn theorem
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.
RSS Reader
Does anyone have recommendations for an RSS reader for Firefox? On my old laptop I used Sage, which was okay, but since I’m reinstalling everything I’m willing to experiment.
Evince Project
Now that I have this new laptop with Ubuntu, I’m using the software it came with it to read PDFs and PS files, Evince. I had never heard of the software before, but I’m fairly impressed. It does a good job of rendering pages, and (unlike Acrobat Reader) does not paralyze my browser once a day.
Gröbner bases as sparse matrices
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.
Salamander Lemma
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.
Next they’ll train monkeys to remember where you put your car keys
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.
Filling in the Blank
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.