On Some of Jean Bourgain’s Work

Abstract

The Belgian mathematician Jean Bourgain was born in Ostende in 1954. After a whirlwind career during which he solved many deep problems and transformed several areas of mathematics, he passed away in Bonheiden on 22nd December 2018 (the birth anniversary of Ramanujan). Bourgain was the modern-day equivalent of Leonhard Euler, making prolific contributions to a wide variety of problems in mathematics and physics. The Mathematical Reviews cite 511 publications under his name. Such a wide spectrum of work cannot be described in one article with any justice even if the authors were to possess expertise in a number of these areas. The breadth and depth of Bourgain’s work can be fathomed from the following phrase used in a review by a renowned mathematician Ben Joseph Green; he said, “It is beyond the capability of the reviewer to give anything like a meaningful description of the argument here, save to repeat the authors’ comments.…” The review was of a paper that completely solved Vinogradov’s mean value conjecture in analytic number theory. We choose a small assortment of the topics to which he contributed so deeply and, describe some technical details in rough terms. In the end, we make a brief mention of his results in a wide variety of areas (see [3–5], [7–14] for technical details). The topics we dwell on in this write-up include the so-called Kakeya problem and some striking applications to number theory