| Citation | Tim Gowers is indisputably the finest young mathematician in the world working with primarily combinatorial methods. He has made several of the most important discoveries in functional analysis and probabilistic number theory of our times. Specifically, his work has revolutionized the theory of infinite dimensional Banach spaces, settling a whole series of distinct and longstanding problems by the invention of a highly original version in linear spaces of Ramsey theory in combinatorics. More recently, he has made deep progress on the problem in probabilistic number theory, which grew out of Van der Waerden's theorem that if one partitions the integers into a bounded number of subsets then at least one of these subsets contains arbitrarily long arithmetic progressions. Erdos and Turan conjectured that the same assertion should hold if one replaces the set of all integers by any subset of positive density. A complete proof of this conjecture was first given by Szemeredi using deep combinatorial methods, and then by Furstenberg using ideas from ergodic theory. Earlier, a partial result towards the conjecture had been obtained in 1953 by Roth by a beautiful argument involving exponential sums. In spectacular work, Gowers has now discovered the ideas in harmonic analysis needed to generalize Roth's proof to the general case, obtaining strong quantitative results from this method. |