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Kakeya problem, number theory, Apollonian circle, Zaremba’s conjecture, Pythagorean triples.

The Belgian mathematician Jean Bourgain was born in Ostendein 1954. After a whirlwind career during which hesolved many deep problems and transformed several areas ofmathematics, he passed away in Bonheiden on 22nd December2018 (the birth anniversary of Ramanujan). Bourgainwas the modern-day equivalent of Leonhard Euler, makingprolific contributions to a wide variety of problems in mathematicsand physics. The Mathematical Reviews cite 511 publicationsunder his name. Such a wide spectrum of work cannotbe described in one article with any justice even if theauthors were to possess expertise in a number of these areas.The breadth and depth of Bourgain’s work can be fathomedfrom the following phrase used in a review by a renownedmathematician Ben Joseph Green; he said, “It is beyond thecapability of the reviewer to give anything like a meaningfuldescription of the argument here, save to repeat the authors’comments....” The review was of a paper that completelysolved Vinogradov’s mean value conjecture in analyticnumber theory. We choose a small assortment of the topics towhich he contributed so deeply and, describe some technicaldetails in rough terms. In the end, we make a brief mention ofhis results in a wide variety of areas (see [3–5], [7–14] for technicaldetails). The topics we dwell on in this write-up includethe so-called Kakeya problem and some striking applicationsto number theory.

Sury B¹ Kaushal Verma²

1. Stat-Math Unit Indian Statistical Institute 8th Mile Mysore Road Bangalore 560 059 India.
2. Department of Mathematics Indian Institute of Science Bengaluru 560 012, India.

Published

3 May 2019