Jean Bourgain (1954-2018) Prolific Unifier of Mathematics
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OBITUARY COMMENT Jean Bourgain (1954-2018) Prolific unifier of mathematics. ver the course of a wildly prolific needed to be totally free,” Kontorovich career, Belgian mathematician Jean explains. “He was going to connect so many Bourgain would repeatedly enter fields — so let him go do it, and people after Osome area, solve several of its outstand- him will come pick up the pieces.” ing problems and create an entirely new In the early 2000s, Bourgain settled a field of study in the process. His stomping question about addition and multiplication ground was analysis — a discipline that uses that is simple to state, but hard to prove. TANIA/CONTRASTO/EYEVINE estimates to capture the essence of messy From a given set of numbers, you can build mathematical quantities. But, unlike many two new sets by listing the sums and the mathematicians, he turned the powers of products of all the pairs. Each of these two his discipline outward, using the lens of lists generally contains some redundancies; analysis to bring far-flung areas such as for example, 8 equals both 3 + 5 and 1 + 7. number theory and computer science into But Bourgain and his collaborators showed, focus. for a variety of different number systems, Bourgain, who died on 22 December, was that the sums list and the products list born in 1954 in Ostend, Belgium. He earned usually cannot both have a high degree of a PhD in 1977 from the Vrije Universiteit redundancy — at least one will have signifi- Brussel, where he quickly started producing cantly more numbers than the original set. mathematics at the highest level. In 1985, he This elementary statement about numbers took up joint appointments at the Univer- has been used in computer science to show sity of Illinois at Urbana-Champaign and that a wide class of networks are ‘expanders’, the Institute of Advanced Scientific Studies valuable tools for generating random struc- (IHES) near Paris. because Bourgain would often pull some tures, such as a well-shuffled deck of cards. Some of his notable early work concerned novel technique out of his hat to solve a Bourgain was diagnosed with pancreatic the geometry of high-dimensional convex problem the other mathematician had been cancer in late 2014. Between blasts of shapes, such as how to find slices of the struggling with for months. (One math- chemotherapy, the pace of his research shape that are especially large or round. ematician ripped up a manuscript he had scarcely slackened. In late 2015, for instance, Many of these studies later found applica- planned to submit for publication, after Bourgain and his collaborators settled an tions in high-dimensional data analysis — Bourgain — off the cuff — came up with an 80-year-old conjecture in number theory. It for example, to create feature-extraction easy proof of the paper’s main result.) concerned an inequality that governs a host algorithms. Bourgain lived and breathed mathematics. of questions about the properties of whole In the early 1990s, Bourgain launched A night owl who travelled frequently back numbers, such as which can be expressed as the modern era for dispersive, nonlinear and forth between the IAS and California, a sum of four perfect squares or nine perfect partial differential equations. These model where his wife and son lived, he would cubes. how waves spread out from each other over often, when at Princeton, show up at the IAS Many mathematicians had expected time, like ripples when a pebble drops into shortly before afternoon tea and work with the solution to come from number theory. a lake. He developed a broad framework for collaborators until well past midnight. Then, Characteristically, Bourgain tackled the understanding when such equations have after the others had gone to bed, he would problem by means of harmonic analysis. well-behaved solutions, and how waves in continue into the night. His collaborators Lists of numbers such as perfect powers this setting interact (as when a child’s kicks would often wake up to an e-mail from can be converted into waves of those fre- amplify or dampen the motion of a swing). Bourgain sent at 4 a.m., containing the solu- quencies. Starting there, Bourgain and his In 1994, these advances won Bourgain tion to a problem that, the night before, had collaborators proved a much more general a Fields Medal, widely seen as the high- seemed as if it would take weeks to crack. result about such waves than was needed to est honour awarded in mathematics. But In the papers he co-authored, mathe- prove the conjecture. Bourgain — who, in that same year, began maticians say, it is easy to spot which The flow of ideas has ended, but the flow an appointment at the Institute for Advanced sections Bourgain wrote. As with his solo of papers probably has not. Mathematician Study (IAS) in Princeton, New Jersey, where papers, they are impenetrable thickets, Peter Sarnak, Bourgain’s colleague at the he spent the rest of his career — was just making giant conceptual leaps with little by IAS, predicts that as many as 50 more might getting started. way of explanation. One page of a Bourgain be published posthumously by Bourgain’s Mathematics poured out of him at a dizzy- paper might take weeks for the uninitiated collaborators. Together with his densely ing rate: he wrote more than 500 papers over to decipher. But the deep ideas and tech- written earlier papers, these will give math- his lifetime, easily 5–10 times what a typical niques within were ample reward for the ematicians treasures to mine for decades to research mathematician might produce. persevering reader. come. ■ Telling Bourgain about one’s research Asking Bourgain to make his papers more could be an intimidating prospect. This readable would have been a disservice to Erica Klarreich has been writing about was not because of his personality, which mathematics, says Alex Kontorovich, a fre- mathematics for more than 15 years. She colleagues describe as kind, optimistic quent collaborator of Bourgain’s at Rutgers lives in Berkeley, California. and intellectually generous. Rather, it was University in Piscataway, New Jersey. “He e-mail: [email protected] ©2019 Spri nger Nature Li mited. All ri ghts reserve14d. FEBRUARY 2019 | VOL 566 | NATURE | 183 .