... exceptional gifts of musicianship and technical command. Even more important is her innate sense of style: a priceless gift which lifts her playing to a level of exceptional maturity.

I had another inspirational teacher, Norman Routledge, who had been a fellow of King's. He did not allow himself to be limited to the syllabus but ranged far more widely. In my last two years at Eton, the mathematics specialists were given a weekly sheet of challenging problems which were only loosely based on the syllabus, if at all. Of course, boys being boys, we tended to do nothing for five days and then rush at them for two days, but even so it was a very valuable experience.

I became certain that I would like to be a professional mathematician some time when I was an undergraduate - though, even then, I had little idea of what this meant.

Looking back it is amusing to remember how little idea I had of what research in different areas would be like when I made such an important choice. But I was lucky and found myself in an area that suited me very well, and with an excellent supervisor.

I used to commute from Cambridge, and found the train a congenial place to work, making at least one genuine breakthrough on it.

Dr W T Gowers of University College, London, is awarded a Junior Whitehead Prize for his work in applying infinite combinatorics to resolve a series of longstanding questions in Banach space theory, some originating with Banach himself. Dr Gowers' achievements include the following: a solution to the notorious Banach hyperplane problem (to find a Banach space which is not isomorphic to any hyperplane), a counterexample to the Banach space Schröder-Bernstein theorem, a proof that if all closed infinite-dimensional subspaces of a Banach space are isomorphic then it is a Hilbert space, and an example of a Banach space such that every bounded operator is a Fredholm operator. Over the past five years, Gowers has made the geometry of Banach spaces look completely different. The techniques he uses are highly individual; in particular, he makes use of a Ramsey theory for linear spaces, stating a dichotomy for subspaces rather than subsequences. In this area, where there is initially little structure, imagination and technical strength of a high calibre are needed. The work demonstrates both characteristics, and the techniques seem likely to find further application in different fields in the future.

William Timothy Gowers' work has made the geometry of Banach spaces look completely different. To mention some of his spectacular results: he solved the notorious Banach hyperplane problem, to find a Banach space which is not isomorphic to any of its hyperplanes. He gave a counterexample to the Schröder-Bernstein theorem for Banach spaces. He proved a deep dichotomy principle for Banach spaces which, if combined with a result of Komorowski and Tomczak-Jaegermann, shows that if all closed infinite-dimensional subspaces of a Banach space are isomorphic to the space, then it is a Hilbert space. He gave (jointly with Maurey) an example of a Banach space such that every bounded operator from the space to itself is a Fredholm operator. His mathematics is both very original and technically very strong. The techniques he uses are highly individual; in particular, he makes very clever use of infinite Ramsey theory.

William Timothy Gowers has provided important contributions to functional analysis, making extensive use of methods from combination theory. These two fields apparently have little to do with each other, and a significant achievement of Gowers has been to combine these fruitfully.

A year ago, Gowers attracted attention in the field of combination analysis when he delivered a new proof for a theorem of the mathematician Emre Szemerédi which is shorter and more elegant than the original line of argument. Such a feat requires extremely deep mathematical understanding.

In this breakthrough paper the author proves his version of the Hypergraph Regularity Lemma and the associated Counting Lemma. As an application, he gives the first combinatorial proof of the multidimensional Szemerédi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound.

The Princeton companion to mathematics is a unique text, which does not fall neatly into any of the usual categories of mathematical writing. It is not quite a mathematical encyclopedia, it is not quite a collection of mathematical surveys, it is not quite a popular introduction to mathematics, and it is certainly not a mathematics textbook; and yet it is still an immensely rich and valuable reference work that covers almost all aspects of modern mathematics today (although there is certainly an emphasis on pure mathematics at the research level). An encyclopedia might focus primarily on definitions, a survey article might focus on history or on the latest research, and a popular introduction might focus on analogies, personalities or entertaining narrative; in contrast, this book is intended to answer (or at least address) basic questions about mathematics, such as "What is arithmetic geometry?", "Why do we care about function spaces?", "How is mathematics used today in biology?", "What is the significance of the Riemann hypothesis?", "Why are there so many number systems?'", or "Is mathematical research all about proving theorems rigorously?".

In summary, this unique book is an extraordinarily broad and surprisingly accessible reference work for a remarkably large fraction of modern (and historical) mathematics. While it does not substitute by any means for more traditional textbooks in mathematics, it complements these more detailed, precise, and technical texts nicely, and is one of the rare places where one can actually see all of mathematics as a unified subject, with coherent themes and goals.

... if a large group of mathematicians could connect their brains efficiently, they could perhaps solve problems very efficiently as well.

Using principles similar to those employed in open source programming projects, [Gowers] used blogs and a wiki to organize an open mathematical collaboration attempting to find a new proof of an important mathematical theorem known as the density Hales-Jewett theorem.

In very general terms I suppose if you divide mathematics into that which uses 'elementary' methods and mathematics that uses a lot of sophisticated theory and well-established techniques, then I'm drawn towards the former rather than the latter.

I just try and find certain problems that I might be able to think about profitably, and quite often I think about problems and get absolutely nowhere. ... with most problems I think about I get absolutely nowhere, but with Banach spaces and combinatorics it was just the case with both of them, there were problems that seemed reasonable to tackle, and I found them interesting.

I like to talk about completing the square, when you have result A that generalises in one direction to result B and in another direction to result C: then you want to find the generalisation of C that corresponds to how B generalises A.

I can work at home and in my office, and those are the two places I work most. Just anywhere where I've got a pad of paper and a biro ... But if you're sitting waiting in an airport lounge, which for many people would be a very boring experience, for a mathematician it isn't. Get out some paper and have a think about things.