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Structure and Randomness: Pages from Year One of a Mathematical Blog

Terence Tao
AMERICAN MATHEMATICAL SOCIETY 2009, 298 PAGES
PRICE (PAPERBACK) £27.50 ISBN 978-0-8218-4695-7

Terence Tao is a distinguished mathematician, perhaps best known for his work in combinatorics and number theory, linked especially to the theory of arithmetic progressions of prime numbers. In 2007, he turned his homepage into a weblog, and this book collects some of his online writings which first appeared there. In the book’s collection of some of these blogs, it sketches out unusual proofs for classical theorems, the texts of three of his invited lectures, a selection of discussions of open problems, and a few number curiosities.

The book follows themes. One of these is given by the title, and treated most fully in the chapter called "structure and randomness". The idea is to divide mathematics into three parts: the highly structured or regular objects, the effectively random objects, and the "hybrid" objects built from both structured and random components. Tao then goes on to describe how mathematics tries to deal with these different cases using algebra, geometry, analysis and probability, going into detail on the problems faced when working, in his area, with prime numbers.

Prime numbers he gives as an example of a hybrid set and explains how you need a large variety of mathematical tools to attempt to work in this area, such as algebra, geometry, analysis, probability, decompositions, algorithms and evolution equations. The style of the book gives sketches of mathematical ideas with extensive bibliography and at times I would have liked more mathematical detail to allow my journey to become continuous through the book rather than in discrete steps.

Another interesting theme running through the book is tricks and patterns of establishing results which can be used across many different types of problems. Parts of most chapters in this book are put aside to this theme. For example in Chapter 1 Tao gives a family of tricks for improving inequalities, a technique he calls amplification.

These collections of blogs, are for mathematicians with a good understanding of abstract algebra, algebraic geometry, functional analysis, graph theory, harmonic analysis, Lie algebras, mathematical logic, measure theory, number theory, partial differential equations and real analysis. My knowledge in some of these areas is patchy, forgotten and sometimes a little vague, but the book still gave me a wonderful insight into the world of a mathematician working at the edge of our understanding. The book will be of interest to anyone who is, or who has studied mathematics, and also those interested in mathematics coming from a physics, statistics, economics and computer science background.

Steve Humble FIMA NCETM 
Mathematics Today December 2010

Structure and Randomness: Pages from Year One of a Mathematical Blog can be purchased at Amazon.co.uk