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.
Author Archives: Walt
SIGFPE’s Law
On his blog, Sigfpe has proposed SIGFPE’s Law, that unlike the exponential growth of ordinary computers, the number of qubits in a quantum computer will only grow linearly. So when the mafia hacks into your bank account a la Sneakers because of their quantum public-encryption-cracking supercomputer, blame Sigfpe for getting it so terribly wrong. Also, blame him for the lack of flying cars.
(One subtlety to keep in mind — which Sigfpe alludes to — a 2-qubit computer is not the same thing as 2 1-qubit computers. The the states of each qubits must allow quantum superposition with each other to really count.)
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.
Behavioral Economics
I spotted a survey article, Behavioral Economics: Past, Present, and Future, which gives a guide to this fairly-new field of economics. The subject was born from a mathematical failure. Economists had given precise axioms as to how people would take into account time and uncertainty when making decisions. The axioms allowed precise predictions that (unlike most economics) could be tested in small-scale experiments with a few test subjects. The result was almost-total failure: nearly every prediction turned out to be wrong. Instead of this being the last word on the subject, this has inspired large amounts of research into finding empirical regularities in the discrepancies between the predictions and the experimental results, and formulating a new theory that is both precise and correct. It’s interesting because the original failure could have led to a turn away from mathematical modeling altogether, but it instead has led to research in improved mathematical modeling.
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.
Two-Envelope Paradox
Have any of you ever heard of the two-envelope paradox? It’s a paradox so important that Wikipedia manages to have two articles on it: Two envelopes problem and Envelopes paradox. The only thing that puzzles me about it is that I’m having trouble seeing how it’s a paradox — unlike, say, the Monty Hall problem, the naive answer is the correct one.
April Notices
The April Notices of the AMS is up. The bulk of the issue is devoted to the centennial of Kurt Goedel’s birth. Sadly, the internet has managed the rather remarkable feat of making me sick of the Incompleteness Theorems, but the issue also has an article about one of my favorite mathematical topics, syzygies, in Roger Wiegand’s article, What is… a syzygy?.
The Only Flame War You’ll Ever Need
This post on Scott Aaronson’s weblog is every internet discussion thread in microcosm (except for ours, which are shockingly polite by internet standards). Scott thought he was explaining the notion of non-constructive proof to a dense but argumentative student. If you read the comment thread, though, you’ll discover that the “student” thought Scott was some sort of nut who imagined he’d invented a computer more powerful than a Turing machine.
Back from vacation
I’m back from vacation. Tragically, my laptop, one that I literally took with me around the world, has died. But my commitment to your mathematical pleasure is so high that I plan on breaking into my neighbors’ houses to use their computers to post to the site.