Alain Connes, who we’ve talked about before, has added something very exciting to his website: the text of his famous book Noncommutative Geometry as a pdf. There is nothing quite like Connes’ book. It’s part popularization, part research monograph, and part manifesto. I highly recommend taking a look.
Via Not Even Wrong.
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.
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.
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.
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.)
Week 228 of John Baez’s This Week in Mathematical Physics is up. He’s still obsessed with sand dunes.
More logic can be found in John Baez’s latest This Week’s Finds in Mathematical Physics (no. 227), along with pictures of Martian sand-dunes.
One of the stranger developments of recent years is the influence of physics on algebraic geometry. A dramatic recent example is Gromov-Witten theory, which draws its inspiration from quantum field theory, but can be used to study the moduli space of complex algebraic curves.
A moduli space is a space that parameterizes all objects of a certain type. The classic example is the projective line, which classifies lines in the plane: each line in the plane corresponds to one and only one point in the plane. The moduli space of curves classifies complex-algebraic curves. The space itself is a geometric object, but its structure turns out to be very complicated, and recent progress has relied on these ideas from physics.
Ravi Vakil has posted a survey article on the subject, The moduli space of curves and Gromov-Witten theory, to Arxiv. He also has an older, more elementary article from the June/July 2003 Notices of the AMS, The moduli space of curves and its tautological ring.
Week 226 of This Week’s Finds in Mathematical Physics is up. This week is pretty far from his usual topics, as John Baez talks about cryptography, pseudorandom number generators, and P versus NP.
I hope I don’t end up feeling stupid for not writing that book.