The Princeton Companion to Mathematics is the kind of book that comes along only rarely—a vast compendium of mathematical information in the form of essays by experts on a wide variety of fields, in most cases bringing the reader up to date on developments in recent decades in a way that nonexperts can understand and appreciate. The Committee recommending this award realizes that it is the work of many: Professor Gowers and his two associate editors (June Barrow-Green and Imre Leader) as well as 133 distinguished contributors, a list that includes George Andrews, Sir Michael Atiyah, Béla Bollobás, Alain Connes, Ingrid Daubechies, Persi Diaconis, Jordan Ellenberg, Andrew Granville, János Kollár, Peter D. Lax, Barry Mazur, Dusa McDuff, Karen Hunger Parshall, Carl Pomerance, Peter Sarnak, and Terence Tao, to name but a few. Gowers singles out two of the many contributors for special recognition for their especially valuable help both in writing text and in editing work by others: Jordan Ellenberg and Terence Tao. The Committee, in recommending the award, singled out Professor Gowers because of his extraordinary achievement in putting this whole volume together (over 1000 pages of text) and also for writing a beautiful 76-page introduction as well as 68 of the 288 individual entries. The organization is thematic, with sections on the origins of modern mathematics, mathematical concepts, the various branches of the subject, the big problems, biographical essays, and, though the subjects are mainly confi ned to what we call “pure” mathematics, a section on the influence of mathematics on other fields.Tim Gowers' Response
That the level of exposition in this volume is so impressive will come as no surprise to anyone familiar with Professor Gowers’ superb but diminutive volume (a sharp contrast in length at roughly 150 pages), Mathematics: A Very Short Introduction (Oxford University Press, 2002). Both books display an exceptional talent for mathematical exposition.
The Companion has something for everyone who has any interest in mathematics. Many sections can be read with great benefit and considerable pleasure by mathematical amateurs and students. Overwhelmed as we are in the twenty-first century by the enormous size of mathematics, the professional mathematician can benefit from finding out what colleagues are doing in branches of mathematics that did not exist when many of us were in school. Anyone who wonders about “mirror symmetry,” “quantum groups, “vertex operator algebras,” “automorphic forms,” or “Ricci flow,” topics referred to in the current mathematical literature or even in the newspapers, can find help here.
Gowers in his preface points out that in deciding what to include he “simply aims to present for the reader a large and representative sample of the ideas that mathematicians are grappling with at the beginning of the twenty-first century, and to do so in as attractive and accessible a way as possible.” The book, he is quick to add, is not an encyclopedia and “does not have a serious online competitor: rather than competing with the existing Web sites, it complements them.”
For a long time I have felt that there was a gap in the market for mathematics books that are much less formal than textbooks and monographs, but aimed at an audience that already knows a substantial amount of mathematics. The Princeton Companion to Mathematics was an attempt to do something about this. I am honoured and delighted that the effort that went into the book has been rewarded with the 2011 Euler Book Award, and in a more general way I am also very pleased that the committee has chosen to recognise a book that demands more of
the reader than most popular mathematics books.
The Princeton Companion to Mathematics was very much a collective undertaking. It could not have been finished without the hard work of June Barrow-Green and Imre Leader, my associate editors, and without the work of the large number of contributors, who were willing not just to send us their contributions, but also to cooperate in a long editing process. The editors also received huge support from Sam Clark of T&T Productions, who converted the authors’ files into a unified format and did a large amount of copyediting. We also received just the right balance of pressure and encouragement from Anne Saverese, the reference editor at Princeton University Press.
The response to the book has been very positive, which suggests to me that the gap in the market that I thought I had identifi ed was real. I hope that people are not just buying the book but also reading it, and that one of its central aims, to improve communication amongst mathematicians by helping them to understand what other mathematicians are doing, is to some extent being fulfilled.