Eagle-eyed Peter Woit spotted an interesting volume on AMS Books Online: a collection of survey articles called Mathematics into the Twenty-First Century. The papers were originally presented at a conference in 1988 held in honor of the AMS’ 100th anniversary. Collectively they provide a picture of the frontier of mathematical research.
AMS Books Online has also added a three-volume history of American mathematics to its General interest section.
Frequent commenter Kenny Easwaran (who also has a weblog, Antimeta, devoted to philosophy of math) has written several interesting essays on the interpretation of conditional probability:
The question is practically and philosophically interesting in the case that the event you are conditioning on occurs with probability zero.
I wanted to give the philosphers in our audience a chance to patronize me for my ignorance. I had no idea until the past few days that Saul Kripke is an important and widely influential philosopher. I knew him from his work in modal logic, but I imagined that he was a logician who worked on a technical subject on the margins of philosophy. (At least I’m better informed than a guy I know who assumed that Kripke must be a category theorist, because there’s something called Kripke-Joyal semantics, which is a translation of Kripke’s work into the language of topos theory.)
Many authors put up drafts of their books as they write them, only to take them down once the book is published. I understand the reasons, but it’s nice when such a book stays available in electronic format. David MacKay’s book Information Theory, Inference, and Learning Algorithms is still available, even though it’s been published by Cambridge University Press.
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 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…”
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.
Sanjeev Arora is writing a new book on computational complexity, and he’s posting the draft chapters online. Since the standard textbook on the subject, Papadimitriou’s Computational Complexity, is from 1993, a more-recent take on the subject is much appreciated.
I’m a total amateur in the subject, but my impression looking at the notes is that the big difference between 1993 and now is that in 1993 researchers were already beginning to suspect that the best idea in years for settling P versus NP (circuit complexity) wasn’t going to work, and now they’re pretty sure it won’t work.
In the introduction, Arora provides a more succinct summary of the post-1993 work than I possibly could:
The list of surprising and fundamental results proved in the 1990s alone could fill a book: these include new probabilistic definitions of classical complexity classes ( IP = PSPACE and the PCP Theorems) and their implications for the field of approximation algorithms; Shor’s algorithm to factor integers using a quantum computer; an understanding of why current approaches to the famous P versus NP will not be successful; a theory of derandomization and pseudorandomness based upon computational hardness; and beautiful constructions of pseudorandom objects such as extractors and expanders.
We’ve previously linked to some expository material on expanders.
Via Scott Aaronson.
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.
It’s been a while since we’ve had an open thread. Feel free to use this opportunity to boast about your own personal math-related website.