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Category Archives: Mathematics
Baez Week 230
Week 230 of John Baez’s This Week’s Finds in Mathematical Physics is out. He has returned to one of his favorite subjects (and really, one of everyone’s favorite subjects), Dynkin diagrams. We covered some of the same topics here and here.
Modular Forms
Modular forms have been thrust into mathematical prominence by Wiles’ proof of Fermat’s Last Theorem. Wiles in actuality proved a special case of the Shimura-Taniyama conjecture, which relates elliptic curves and modular forms.
Fred Calegari has written a nice introduction to the topic of modular forms in the guise of a book review of A first course in modular forms by Diamond and Shurman. (The review also features the best variant of the “kids today” sentiment I’ve seen recently: “With today’s Ipod generation more likely to study elliptic curves and modular forms before learning any class field theory…”
Peer-review and its discontents
The latest issue of the Post-Autistic Economics Review is now out, available here.  It has an interesting article by philosopher Donald Gillies arguing against the centrally-organized reviews of university research activities which British academics have had to endure these last 20 years, and which now look likely to be adopted in Australia, NZ and elsewhere. One argument he makes is that one’s peers are usually quite bad at judging the long-run impact and quality of one’s research, especially when the research is innovative, and Gillies gives the example of Frege’s Begriffsschrift, the first axiomatic treatment of propositional and predicate calculus. When this was published in 1879, it was slammed by Frege’s contemporaries, and it was only recognized for the seminal work it is two decades later. If Frege had been working in a British University a hundred years later, both he and his department may have faced termination by his university administration, given the hostility that his own peers felt towards his work; lots of departments have been closed, and academics made unemployed, as a result of the peer assessments of the British Research Assessment Exercise (RAE).
A longer version of Gillies’ paper is available on his web-site, here. Â
Bayesian Detente
I’ve been reading a bunch of papers on Bayesian statistical inference lately, somewhat to my regret. I have no particular objection to Bayesian statistics, but distressingly often, a Bayesian paper will include a gratuitous slam of all other types of statistics. D. V. Lindley’s papers (which are classics in the literature) are particularly noxious in this regard. It’s a strange pattern, and I’d be curious to know the history of the habit.
More pleasant is a paper by Brad Efron based on an address he gave at Phystat2003, Bayesians, Frequentists, and Physics, which offers a detente in the Bayesian-frequentist debate. He describes Stein’s paradox, which is a challenge from both the Bayesian and classical points of view, and discusses means of inference, such as empirical Bayes, which are (arguably) neither purely Bayesian nor purely frequentist.
Baez Week 229
Week 229 of John Baez’s This Week’s Finds in Mathematical Physics is up. And this time I’m even including the link right away, just to show there’s no hard feelings about the lack of flying cars.
May Notices
The May Notices of the AMS are up. The lead article is Cells in Coxeter Groups. This month’s What is… is What is… Percolation?.
It also includes obituaries for Serge Lang and Raoul Bott.
Bulletin of the AMS, Vol. 43, No. 2
The new Bulletin of the AMS is out. It has a review of Computational Homology, a book that I have not read, but was very tempted by at the bookstore. Sadly, my library doesn’t have it. Homology provides an interesting pedagogical challenge. If you just wanted to convey the idea of it, you would probably start with simplicial or cubical homology (I think this is the approach Rotman takes in his book), but if you wanted to train future researchers in the subject, you’d be tempted to skip that and go straight to singular or cellular homology. Most graduate courses probably opt for the latter, but perhaps we’ll begin to see applied courses that take the former route.
Baez Week 228
Week 228 of John Baez’s This Week in Mathematical Physics is up. He’s still obsessed with sand dunes.
Cosman on Sets of Probabilities
I’ve been doing some reading into alternatives to subjective probability, and one interesting alternative is to model an assignment of subjective probability by a convex set of probability distributions, rather than a single distribution. Convex sets encompass several natural situations where you have a vague sense of probabilities, but would be unwilling to specify an exact value. For example, a range of probabilities for an event can be expressed as a convex set, as well as the idea that one event is more likely than another (without expressing exact probabilities for each event). Convexity also has a natural probabilistic interpretation: if two distributions are in the set, then any mixture of the two is also in the set.
A nice introduction to the subject is Fabio Cozman’s online tutorial Introduction to the Theory of Sets of Probabilities. For some additional surveys on related approaches, see the homepage of the Imprecise Probabilities Project.