I just recently got my hands on a copy of the new Princeton Companion to Mathematics, and I fear that this is likely to seriously impact my ability to get things done for a while, as I devote too much time to happily reading many of its more than 1000 pages.
The book is an amazing document (and physically, a beautiful, if weighty object), unlike anything else I know of. Its coverage of mathematics and mathematical culture is very wide and sometimes deep, but it makes no attempt to be comprehensive. Thus, the accurate title “Companion to” rather than “Encyclopedia of”. The most remarkable aspect of the book is the extremely high quality of the contributions from a large number of different authors. It includes many wonderful long expository articles, mostly at a level that a good undergraduate math student could hope to appreciate, with much of the book accessible to an even wider audience. The articles are often written by some of the best researchers and expositors around. For example, one can find Barry Mazur writing on Algebraic Numbers, Janos Kollar on Algebraic Geometry, Cliff Taubes on Differential Topology, Ingrid Daubechies on Wavelets, Persi Diaconis on Mathematical Statistics, and many, many others of similar quality. The table of contents is available here.
The book also includes extensive articles on historical topics in mathematics and short biographies of a large number of mathematicians, as well as coverage of applications and a section largely devoted to describing the art of problem-solving and how mathematics really gets created. This section includes a beautiful set of five essays called “Advice to a Young Mathematician”, which give five different equally fascinating perspectives from some of the best in the subject about how they achieved what they did, as well as what they have learned from years of helping students become researchers. The authors of these pieces are Michael Atiyah, Bela Bollobas, Alain Connes, Dusa McDuff, and Peter Sarnak. Luckily for all young (and old) mathematicians, this chapter is freely available here.
The person most responsible for this is clearly the editor (and author of some of the pieces), Fields Medalist Timothy Gowers, who had help from many others, including fellow Fields Medalist Terry Tao. Gowers has a weblog, and he has written about the book in these entries (and there’s a podcast interviewing him on the book web-site at PUP). Terry Tao has a posting about the book here.
If you’re looking for a gift for someone with a serious interest in mathematics, no matter what their background, you won’t do any better than this.