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Abstract: Harmonic analysis and combinatorics: How much may they
contribute to each other?" asked Jean Bourgain in a 2000 review
article. Since then, there has been an explosion of interest in the
subject, with recent developments ranging from the work of Green and
Tao on the arithmetic progressions in primes to the hypergraph
regularity theorems of Gowers and Nagle, Rodl, Schacht and Skokan; and
yet it seems that we could go much further. In this talk, I will
attempt to survey some of these developments, their context, and the
emerging new directions in the field.
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