Blogging has created a whole new way of cultivating one’s obsessions. There are many mathematical bloggers, from enthusiastic high school students to professional journalists. Active researchers have also gotten into the game, producing blogs that focus on mathematical ideas and intuitions, open problems, and recent work. These allow public access to conversations that used to be limited to a small circle; phrases such as “what’s really going on” and “here’s the essential idea,” long banished from journals, have found a home online. Terry Tao’s mathematical research blog, entitled What’s New, is one of the best of these.
Much of what appears in the blogosphere is ephemeral, of course, and this is true of mathematical research blogs as well. But there is quite a bit that is worth preserving, citing, and re-reading; for those purposes, books are a better vehicle than the web. And there are old fogeys like me, for whom reading a book is far more comfortable than reading online. Hence Structure and Randomness, a collection of “pages from year one of a mathematical blog.” There are more volumes to come.
Tao has selected 32 of the 93 articles he posted during 2007 and polished them for publication. The articles are grouped into three sections, corresponding to different kinds of writing about mathematical research:
According to the preface, a possible fourth category, namely discussion of recent work by others, has been left out.
I was particularly impressed by the first group of articles, which range from playful to deeply serious but are always clear and full of good ideas. There is a fascinating discussion of “Soft analysis, hard analysis, and the finite convergence principle” that would be accessible to (good) undergraduate math majors. It wrings real insight out of a distinction that often just generates heated disagreement. The chapter on non-standard analysis is also very good, providing an incredibly lucid explanation of what is going on.
Tao often finds different ways to approach things. Rather than focusing on actress Danica McKellar’s book, Math Doesn’t Suck, he explains the Chayes-McKellar-Winn theorem in statistical mechanics — yes, the same McKellar. (For the curious: Journal of Physics A, 31 (1998), 45, pp. 9055–9063.)
In one article, Tao gives a fairly elementary proof of the Hilbert Nullstellensatz. “I was a little unsatisfied,” he says, “with the proofs I was able to locate—they were fairly abstract and used a certain amount of algebraic machinery, which I was terribly rusty on—so, as an exercise, I tried to find a more computational proof that avoided as much abstract machinery as possible.” Another article is a technical discussion of how to use “arbitrage” to strengthen inequalities.
My least favorite of the expository articles in this section is the one that seems to have started it all, an article illustrating the weirdness of the quantum world via the computer game Tomb Raider. Maybe that just reconfirms my old fogey status.
The articles in the second group, while still interesting, represent a more standard way of writing about mathematics: lecture notes. Tao has preserved some of the informality of his lectures, so the notes are pleasant to read and informative. They deal mostly with Tao’s own work, so there is (necessarily) some repetition. The dominant theme of these lectures, the tension between structure and randomness and techniques for taking advantage of each, provides the title for the whole book.
The third section contains discussions of unsolved problems, focusing, in most cases, on what makes these problems hard. Tao’s approach is to explain the problem, consider possible strategies for solution, and then to explain why no one has (so far) been able to get those strategies to work. As he puts it, this is the kind of conversation students have with their advisors and experts have among themselves. They should both spur new work on the problems and provide helpful guidance. (It is commonly said that in mathematics one cannot publish articles like “How I Tried and Failed to Prove the Poincaré Conjecture.” Tao shows us how it can be done.)
Structure and Randomness is fun to read and full of good ideas. Even though most of the articles dealt with mathematics outside my main interests, I enjoyed it immensely.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.