For the past several years, my good friend and fellow Medalist Timothy Gowers has been devoting an enormous amount of effort towards editing (with the help of June Barrow-Green and Imre Leader) a forthcoming book, the Princeton Companion to Mathematics. This immense project is somewhat difficult to explain succinctly; a zeroth order approximation would be that it is an “Encyclopedia of mathematics”, but the aim is not to be a comprehensive technical reference, nor is it a repository of key mathematical definitions and theorems; it is neither Scholarpedia nor Wikipedia. Instead, the idea is to give a flavour of a subject or mathematical concept by means of motivating examples, questions, and so forth, somewhat analogous to the “What is a …?” series in the Notices of the AMS. Ideally, any interested reader with a basic mathematics undergraduate education could use this book to get a rough handle on what (say) Ricci flow is and why it is useful, or what questions mathematicians are trying to answer about (say) harmonic analysis, without getting into the technical details (which are abundantly available elsewhere). There are contributions from many, many mathematicians on a wide range of topics, from symplectic geometry to modular forms to the history and influence of mathematics; I myself have contributed or been otherwise involved in about a dozen articles.

If all goes well, the Companion should be finalised later this year and available around March 2008. As a sort of “advertising campaign” for this project (and with Tim’s approval), I plan to gradually release to this blog (at the rate of one or two a month) the various articles I contributed for the project over the last few years. [As part of this advertising, I might add that the Companion can already be pre-ordered on Amazon.]

I’ll inaugurate this series with my article on “wave maps”. This describes, in general terms, what a wave map is (it is a mathematical model for a vibrating membrane in a manifold), and its relationship with other concepts such as harmonic maps and general relativity. As it turns out, for editing reasons various articles solicited for the Companion had to be removed from the print edition (but possibly may survive in the on-line version of the Companion, though details are unclear at this point); Tim and I agreed that as wave maps were not the most crucial geometric PDE that needed to be covered for the Companion (there are other articles on the Einstein equations and Ricci flow, for instance), that this particular article would end up being one of the “deleted scenes”. As such, it seems like a logical choice for the first article to release here.