An Epsilon of Room, II: Pages from Year Three of a Mathematical Blog

by Terence Tao

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and nonrigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such "folklore mathematics". But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog. In 2007 Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to nontechnical puzzles and expository articles. The first two years of the blog have already been published by the American Mathematical Society. The posts from the third year are being published in two volumes. This second volume contains a broad selection of mathematical expositions and self-contained technical notes in many areas of mathematics, such as logic, mathematical physics, combinatorics, number theory, statistics, theoretical computer science, and group theory. Tao has an extraordinary ability to explain deep results to his audience, which has made his blog quite popular. Some examples of this facility in the present book are the tale of two students and a multiple-choice exam being used to explain the $P = NP$ conjecture and a discussion of "no self-defeating object" arguments that starts from a schoolyard number game and ends with results in logic, game theory, and theoretical physics. The first volume consists of a second course in real analysis, together with related material from the blog, and it can be read independently.

Genres: Mathematics

248 pages, Paperback

First published January 16, 2011

About the author

Terence "Terry" Tao FAA FRS (simplified Chinese: 陶哲轩; traditional Chinese: 陶哲軒; pinyin: Táo Zhéxuān) is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing and analytic number theory. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. Tao was a co-recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics.

Reviews

[Kevin K. Gillette · Rating 5 out of 5]

Before I write this brief review, I need to make two things perfectly clear:

1 - Terry Tao is a mathematical virtuoso. He is a truly world-class mathematician. I believe this is an undisputable claim. After all, he is the winner of the Fields Medal, which is the equivalent of a Nobel Prize in mathematics. The breadth and depth of his interests and expertise in mathematics are hard to match.

2 - I have not read the entire book, as it is a collection of expository articles and technical notes. I have sampled several of these, which I'll mention below.

Having said the foregoing, I must also add that Terry Tao is an outstanding expositor of mathematics. His ability to take the reader into the mind of a great mathematician is unmatched, in my experience. If you have read Richard Feynman, you will get an idea of where Terry is coming from. He makes the math he explores not just fun - he makes it magical.

I've read about a half-dozen of the articles in this tome - the first article on an explicitly solvable nonlinear wave equation is especially nice, as is the article a bit further on, dealing with Benford's Law, Zipf's Law, and the Pareto distribution, and yet another article further down on the AKS primality test. This latter article is a great introduction to the actual paper published by Messrs. Agrawal, Kayal and Saxena, assuring the reader that nearly all of the arguments in the article are elementary in nature. [NOTE: "elementary" in this context can be a bit misleading, as it means "elementary" to a first-year graduate student in mathematics.] Outstanding technical articles include the two on Szemeredi's regularity lemma, as well as the one on the Cohen-Lenstra distribution.

But is it "all there," as one might say? Well, the best way to find out is to read it for yourself. See what it's like to rattle around in the mind of a true genius!