An appreciation of Jean Bourgain’s work

Jean Bourgain viewed himself as an “analyst”, and as the record shows he was uniquely gifted as such, and much more. Analytic, combinatorial, and probabilistic reasoning is at the heart of many central problems of modern mathematics and its applications, and these naturally attracted Jean’s attention. The combination of his brilliance, his thirst to solve long-standing problems, and his many fruitful collaborations led him to transformative contributions in a striking number of areas. Jean’s research and its impact remind one of the great Russian analyst Kolmogorov. It is said that Kolmogorov made major contributions to all fields except number theory. A list of areas to which Jean made decisive contributions include functional analysis, harmonic analysis, probability theory, ergodic theory, partial differential equations, mathematical physics, number theory, group theory, and theoretical computer science. It is impossible in a single volume, let alone an issue of the Bulletin of the American Mathematical Society, to give anything like a comprehensive account of Jean’s mathematical achievements. Gathering his over 500 (and counting) publications in a collected works would be physically impossible. Fortunately, Jean was very purposeful and proactive in preparing his papers for publication, and almost all of these are in print and are accessible. He made sure that anyone committed to understanding and using his work would have it available. The four articles in this issue give very clear and insightful accounts of some highlights of Jean’s work in functional analysis (Ball), harmonic analysis (Demeter), dispersive partial differential equations (Kenig), and a deconstruction of some of the techniques that Jean invented (Tao). The reports on some of Jean’s earlier works highlight how his breakthroughs impacted and reshaped each of these fields. In many cases his resolution of a long-standing problem introduced fundamental new tools, insights, and viewpoints allowing others to achieve further lofty goals. Many view Jean primarily as a problem solver, perhaps because that is the impression he liked to give. The first solution of a fundamental problem almost always comes with new tools, insights, and theory, so naturally Jean was both a problem solver and a theory builder of the highest calibre. The report on Jean’s more recent work on l decoupling (Demeter) explains a striking application resolving a long-standing problem in analytic number theory—the Vinogradov mean value conjecture. Jean’s expectation that the decoupling theory will yield much more has certainly materialized recently, and it will no doubt continue to do so. I was fortunate enough to be Jean’s colleague and collaborator and to witness a number of his breakthroughs first hand. I mention a few of these that I am particularly fond of and which are not discussed in the four reports. The first is around the sum-product phenomenon, as Jean liked to call it, and its applications. In [Bou03] Jean gave a proof of a local version of the Erdős–Volkmann conjecture.