William Timothy Gowers **1998 Fields Medalist William Timothy Gowers**

William Timothy Gowers has provided important contributions to functional analysis, making extensive use of methods from combinatorial theory. These two fields apparently have little to do with each other, and a significant achievement of Gowers has been to combine these fruitfully. Functional analysis and combinatorial analysis have in common that many of their problems are relatively easy to formulate, but extremely difficult to solve. Gowers has been able to utilize complicated mathematical constructions to prove some of the conjectures of the Polish mathematician Stefan Banach (1892-1945), including the problem of "unconditional bases." Banach was an eccentric, preferring to spend his time in the café rather than in his office in the University of Lvov. In the twenties and thirties he filled a notebook with problems of functional analysis while sitting in the "Scottish Café," so that this later became known as the Scottish Book. Gower has made significant contribution above all to the theory of Banach spaces. Banach spaces are sets whose members are not numbers but complicated mathematical objects such as functions or operators. However, in a Banach space it is possible to manipulate these objects like numbers. This finds applications, for example, in quantum physics. A key question for mathematicians and physicists concerns the inner structure of these spaces, and what symmetry they show. Gowers has been able to construct a Banach space which has almost no symmetry. This construction has since served as a suitable counterexample for many conjectures in functional analysis, including the hyperplane problem and the Schröder-Bernstein problem for Banach spaces. Gowers' contribution also opened the way to the solution of one of the most famous problems in functional analysis, the so-called "homogeneous space problem." A year ago, Gowers attracted attention in the field of combinatorial analysis when he delivered a new proof for a theorem of the mathematician Emre Szemeredi which is shorter and more elegant than the original line of argument. Such a feat requires extremely deep mathematical understanding.

**William Timothy Gowers** (born 20 November 1963) is lecturer at the Department of Pure Mathematics and Mathematical Statistics at Cambridge University and Fellow of Trinity College. From October 1998 he will be Rouse Professor of Mathematics. After studying through to doctorate level at Cambridge, Gowers went to University College London in 1991, staying until the end of 1995. In 1996 he received the Prize of the European Mathematical Society.

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