A common view of mathematics is that it consists of a collection of tried and trusted computational techniques of various levels of difficulty, that can be hauled out when needed to solve a particular problem. This view is accurate as far as it goes: but it does not go very far.

Mathematics is an evolving and creative science. It is growing and developing in response to theoretical advances within itself, to the availability of new technology, and to the cogency of applications. These influences combine to shape the progress of the subject, and in turn the creation of new mathematical ideas generates original solutions to previously unsolved problems, establishes theoretical frameworks, stimulates technological change, and reveals further opportunities for the applications of mathematics.

The evolution of mathematics takes place in the minds of research mathematicians, where intuition and imagination coexist with ingrained precision and logical rigour. Fusing with a deep understanding of theory and technique, these facets of thought intermingle and coallesce into new mathematical ideas. It is a mysterious process. It is activated and experienced in individual ways by different mathematicians, searching for some far-reaching insight or for the next small piece in a complex array of reasoning. Eventually these new ideas are distilled into a theoretical result, or into an algorithm, or into a mathematical model.

To preserve the validity and reliability of mathematical ideas, the subject makes austere demands for assurance: for absolute certainty of the logical rigour of a theory, for the predictive power of a mathematical model, for elegance in an explanation. The outcome is the most secure body of knowledge assembled in human history.

The mathematician works against a cultural background of hundreds or even thousands of years of the development of mathematics, and also in response to the urgent demands for new or improved solutions to pressing, practical problems.

Who are the mathematicians? What sort of people are they? *Faces of
Mathematics* offers a glimpse into the lives and personalities of some of
the most successful and influential mathematical researchers currently
working in the UK and shows mathematical research as a creative, human
endeavour. We see engaging, passionate and enthusiastic people, with
a shared commitment to the development of mathematical ideas and with
a shared secret. They have looked long and hard at some of the most
challenging problems in modern mathematics, statistics, and mathematical
physics; and they share the secret of success, because they have solved
these problems and made their own extraordinary contribution to the
subject.

Marc Atkins has captured these individuals in his distinctive style, whilst they are engaged in talking about mathematics. Nothing could be more characteristic, for communication is the essence of modern mathematical research. Mathematicians talk about their subject all the time: informally to colleagues over coffee, or in detail and at a highly technical level to co-workers; face-to-face and by email; at one remove by sending out preprints and preliminary versions of research; and finally in the formal conversation represented by a published article in a research journal. Our Faces of Mathematics are seen talking about mathematics, and this brings us close to mathematics research in action.

This project depended for its success on the cooperation and enthusiasm of the mathematicians whose images we sought. Marc and I wanted to meet them in a usual working environment - typically this was in a departmental office, or in a college room, but was occasionally at home. We travelled up and down the country during 2000 and early 2001, exploring not only the different intellectual aspects of mathematical research, but also sampling the different atmospheres and architectural styles in which it is carried on. At times we were ensconced within walls that had witnessed hundreds of years of the cultural background to modern science: at other times we wandered through a University building constructed in the 1960's in response to, and in anticipation of, technological change.

We were always made very welcome, though the welcome was at times a little guarded. Mathematicians are not often in the public eye and so few are used to being the sole focus of the attention of a professional photographer, and Marc Atkins' height lends him a dominating presence in an average-size room. Yet once the conversation began to flow - and once I had abandoned my scripted questions - mathematics took over, and we settled into the style of a coffee-break exchange between two mathematicians about the general direction and of progress research.

Our aim was to generate or, given my own limited expertise, to simulate this sort of discussion, so that Marc could track down the image he wanted when mathematics research was in the foreground. Despite the regular punctuation of the smooth snap of Marc's SLR shutter, the act of photography was allowed to drift into the background, and the filming of a video segment could be done quite unobtrusively. The conversations proceeded in a fairly technical style, so that Marc heard the rhythms of the flow of ideas and the innate poetry of the precisely crafted vocabulary of mathematics. Taking his cue from the animation behind the words, Marc found the image and the essence of the meeting.

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