**PROFESSOR SIR TIMOTHY GOWERS BA PHD FRS**

Royal Society Research Professor, University of Cambridge. Mathematician.

In 1911, over one hundred years ago, Georg Cantor was amongst the distinguished academics who visited St Andrews and was awarded an honorary degree as part of the University's 500th Anniversary celebrations. Cantor's work on set theory and the nature of infinity underlies modern mathematics although it encountered considerable opposition from many of his contemporaries. For example, his demonstration that infinity comes in many different sizes seemed revolutionary at the time but is now a standard part of our undergraduate courses.

Following Cantor's work, mathematicians embarked on the study of infinite dimensional spaces. For most practical purposes we live in three-dimensional space: thus the position of, say, an aeroplane can be specified by a list of three numbers or coordinates, perhaps latitude, longitude and height. To a mathematician, there is nothing special about 'three'. A list of four numbers represents four-dimensional space, five numbers five-dimensional space, and so on. But why stop here? An unending list leads to an infinite dimensional space where, just as in three dimensions, there are notions of size and shape, though with a much richer structure.

The Polish mathematician Stefan Banach led the investigation of such spaces in the early twentieth century, and Banach spaces remain fundamental mathematical structures today both because of their intrinsic interest and because of their applications across science. Over the years a great deal was discovered about these spaces but certain central aspects became regarded as inaccessible, and many questions and conjectures raised by Banach remained unanswered.

A completely new approach was required but it was not until the 1990s that Professor Timothy Gowers saw how methods from combinatorics, a seemingly very different area of mathematics, could be used to attack many of these fundamental questions. Using highly complicated constructions in a spectacular manner he was able to prove many of Banach's conjectures. On the other hand, he showed that several other conjectures were false by constructing a space which he described as the 'nastiest known Banach space' in that it had virtually no symmetry whatsoever. In the space of ten years Professor Gowers' work completely changed the landscape of infinite dimensional spaces and their geometry.

Professor Gowers' expertise in combinatorics also led to a new approach to central questions in number theory - that is the properties of the integers or whole numbers. For example, his techniques could show that within very large collections of numbers it is always possible to find certain regular patterns.

For his deep and highly original mathematics, Professor Gowers was awarded a Fields Medal at the International Congress of Mathematicians held in Berlin in 1998, the highest honour then attainable by a mathematician. In the same year he was elected to the prestigious Rouse Ball Chair at the University of Cambridge.

Despite the sophistication of his research, Professor Gowers places great importance on communicating the interest and excitement of mathematics widely and at all levels. His best-selling Mathematics: A Very Short Introduction gives an insight into the beauty of the subject and makes abstract ideas such as curved space and the square root of minus one accessible to those with minimal mathematical backgrounds. At another level, he recently edited the 1,000 page Princeton Companion to Mathematics, writing substantial parts himself. This brilliantly conceived and produced volume presents a wonderful overview of all areas of modern research mathematics making them accessible at the lowest feasible level. If an aspiring mathematician marooned on a desert island could salvage only one book, this would have to be it. From Cantor's set theory to recent developments in string theory, it is all made beautifully clear.

Professor Gowers has embraced the Internet era, and his blog, started in 2007, is widely followed by the mathematics community. The posts range across high school and university teaching, from research advances to new methodologies. In particular, he has used the blog to champion true open access to academic work, supporting new models for publishing research. His Cost of Knowledge blog, which highlighted the entrenched interests of certain academic publishers 'went viral' and has led to a widespread boycott of a major publishing firm.

Unusually for modern times, virtually all of Professor Gowers' research has been done alone rather than in collaboration with other mathematicians. Nevertheless, in 2009 he ventured to the other extreme, by posting a question on his blog 'Is massively collaborative mathematics possible?' He proposed a 'Polymath' project, where anyone who wished could contribute their ideas on a blog page aiming to reach the solution of a difficult research problem collectively. The first problem, involving colouring a high-dimensional cube, was posed in January 2009 and it was less than seven weeks before Professor Gowers blogged that the problem was 'probably solved'. The solution was written up and published in a top journal under the name of Polymath. Such projects, of which there have now been several, focus on the advancement of knowledge, rather than worrying about individual or indeed REF credit.

Some cynics have suggested that mathematicians work for the grudging approbation of a few friends. Professor Gowers is a splendid counter-example with his efforts appreciated by experts, by the mathematical community as a whole and by a wide public audience.

**Laureator:** Professor Kenneth Falconer, School of Mathematics and Statistics, University of St Andrews.