A review of The Princeton Companion to Mathematics

Ronald Graham in American Scientist:

ScreenHunter_04 Dec. 20 12.25 This impressive book represents an extremely ambitious and, I might add, highly successful attempt by Timothy Gowers and his coeditors, June Barrow-Green and Imre Leader, to give a current account of the subject of mathematics. It has something for nearly everyone, from beginning students of mathematics who would like to get some sense of what the subject is all about, all the way to professional mathematicians who would like to get a better idea of what their colleagues are doing.

The 75-page introduction, which was written by Gowers, gives a very readable account of the basic branches of mathematics (algebra, geometry, analysis) and how these overlap and relate to one another, how they have developed and are continuing to do so, and how they are driven in large part by the types of questions mathematicians ask. This section should be mandatory reading for any prospective mathematics student.

Most of the articles that make up the rest of the book were written by leading experts. For example, Carl Pomerance has contributed a stimulating essay on computational number theory, Cliff Taubes provides a wonderful overview of differential topology, and Jordan Ellenberg gives a thoughtful summary of arithmetic geometry.

More here.