Remembering Paul Cohen (1934–2007) Peter Sarnak, Coordinating Editor
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Remembering Paul Cohen (1934–2007) Peter Sarnak, Coordinating Editor Paul Joseph Cohen, one of the stars some classic texts. There he got his first exposure of twentieth-century mathematics, to modern mathematics, and it molded him as a passed away in March 2007 at the mathematician. He tried working with André Weil age of seventy-two. Blessed with a in number theory, but that didn’t pan out, and unique mathematical gift for solving instead he studied with Antoni Zygmund, writing difficult and central problems, he a thesis in Fourier series on the topic of sets of made fundamental breakthroughs uniqueness. In Chicago he formed many long- in a number of fields, the most spec- lasting friendships with some of his fellow stu- tacular being his resolution of Hil- dents (for example, John Thompson, who re- bert’s first problem—the continuum mained a lifelong close friend). hypothesis. The period after he graduated with a Ph.D. Like many of the mathematical gi- was very productive, and he enjoyed a series of Paul Cohen ants of the past, Paul did not restrict successes in his research. He solved a problem his attention to any one specialty. of Walter Rudin in group algebras, and soon To him mathematics was a unified subject that after that he obtained his first breakthrough on one could master broadly. He had a deep under- what was considered to be a very difficult prob- standing of most areas, and he taught advanced lem—the Littlewood conjecture. He gave the first courses in logic, analysis, differential equations, nontrivial lower bound for the L1 norm of trigo- algebra, topology, Lie theory, and number theory nometric polynomials on the circle whose Fourier on a regular basis. He felt that good mathematics coefficients are either 0 or 1. The British number should be easy to understand and that it is always theorist-analyst Harold Davenport wrote to Paul, based on simple ideas once you got to the bottom saying that if Paul’s proof held up, he would have of the issue. This attitude extended to a strong bettered a generation of British analysts who had belief that the well-recognized unsolved problems worked hard on this problem. Paul’s proof did hold in mathematics are the heart of the subject and up; in fact, Davenport was the first to improve on have clear and transparent solutions once the right Paul’s result. This was followed by work of a num- new ideas and viewpoints are found. This belief ber of people, with the complete solution of the gave him courage to work on notoriously difficult Littlewood conjecture being achieved separately problems throughout his career. by Konyagin and McGehee-Pigno-Smith in 1981. Paul’s mathematical life began early. As a child In the same paper, Paul also resolved completely and a teenager in New York City, he was recog- the idempotent problem for measures on a locally nized as a mathematical prodigy. He excelled in compact abelian group. Both of the topics from mathematics competitions, impressing everyone this paper continue to be very actively researched around him with his rare talent. After finishing today, especially in connection with additive com- high school at a young age and spending two years binatorics. at Brooklyn College, he went to graduate school As an instructor at the Massachusetts Institute in mathematics at the University of Chicago. of Technology (1958–59), Paul was introduced He arrived there in 1953 with a keen interest in to the problem of uniqueness for the Cauchy number theory, which he had learned by reading problem in linear partial differential equations. Alberto Calderón and others had obtained unique- Peter Sarnak is professor of mathematics at the Institute ness results under some hypotheses, but it was for Advanced Study in Princeton and at Princeton Univer- unclear whether the various assumptions were sity. His email address is [email protected]. essential. Paul clarified the understanding of 824 NOTICES OF THE AMS VOLUME 57, NUMBER 7 this much-studied problem by by the works of Gauss, Dirich- constructing examples in which let, and Riemann, but studying uniqueness failed in the context with them was not an option. of smooth functions, showing in However, Paul, whose work on particular that the various as- the continuum problem is re- sumptions that were being made corded in any good introductory were in fact necessary. He never text on mathematics, was appar- published this work (other than ently alive and well and living in putting out an Office of Naval California. Moreover, his repu- Research technical report), but tation and stories of his genius Lars Hörmander incorporated had reached all corners of the it into his 1963 book on linear mathematical world. Kathy partial differential operators, Driver (then Kathy Owen and so that it became well known. today dean of science at the This was one of many instances University of Cape Town) had showing that Paul’s impact on just returned from Stanford with mathematics went far beyond the following kind of description his published papers. He con- of Paul, which she emailed me tinued to have a keen interest in recently. linear PDEs and taught graduate “Paul was an astonishing man. courses and seminars on Fourier Impatient, restless, competitive, integral operators during the provocative and brilliant. He was 1970s, 1980s, and 1990s. a regular at coffee hour for the After spending two years in Paul Cohen as a young man graduate students and the fac- Princeton at the Institute for attending Stuyvesant High ulty. He loved the cut-and-thrust Advanced Study, Paul moved School. of debate and argument on any to Stanford in 1961; there he topic and was relentless if he remained for the rest of his life. He said that get- found a logical weakness in an opposing point of ting away from the lively but hectic mathematical view. There was simply nowhere to hide! He stood atmosphere on the East Coast allowed him to sit out with his razor-sharp intellect, his fascination back and think freely about other fundamental for the big questions, his strange interest in ‘per- problems. He has described in a number of places fect pitch’ (he brought a tuning fork to coffee hour (see, for example, [Yandell])1 his turning to work and tested everyone) and his mild irritation for the on problems in the foundations of mathematics. few who do have perfect pitch. He was a remark- By 1963 he had produced his proof of the indepen- able man, a dear friend who had a big impact on dence of the continuum hypothesis, as well as of my life. A light with the full spectrum of colours.” the axiom of choice, from the axioms of set theory. I set my goal to study with Paul in the founda- His basic technique to do so, that of “forcing”, tions of mathematics and was lucky enough to revolutionized set theory as well as related areas. get this opportunity. Paul lived up to all that I To quote Hugh Woodin in a recent lecture, “It will expected. Soon after I met him, he told me that remain with us for as long as humans continue to his interests had shifted to number theory and, in think about mathematics and truth.” A few years particular, the Riemann hypothesis, and so in an later, Paul combined his interest in logic and num- instant my interests changed and moved in that ber theory, discovering a new decision procedure direction, too. for polynomial equations over the p-adics and the Given his stature and challenging style, he was reals. His very direct and effective solution of the naturally intimidating to students (and faculty!). decision problem has proved to be central in many This bothered some and is probably the reason recent developments in the subject [MacIntyre]2. he had few graduate students over the years. I After 1970 Paul published little, but he con- have always felt that this was a pity, because one tinued to tackle the hardest problems, to learn could learn so much from him (as I did), and he and to teach mathematics, and to inspire many was eager to pass on the wealth of understand- generations of mathematicians. I was a member ing that he had acquired. Once one got talking to of the new generation who was lucky enough to him, he was always very open, and he welcomed be around Paul. with enthusiasm and appreciation others’ insights I first heard of Paul when I was still an under- when they were keener than his (which I must graduate in Johannesburg in the 1970s. I was taken confess didn’t happen frequently). As a student, I could learn from him results in any mathematical 1B. H. Yandell, The Honors Class, A. K. Peters, 2002, area. Even if he didn’t know a particular result, 59–84. he would eagerly go read the original paper (or, 2A. MacIntyre, In “Records of proceedings at meetings of I should say, skim the paper, often filling in his the London Math. Soc.”, June 22, 2007. AUGUST 2010 NOTICES OF THE AMS 825 Photo by C. J. Mozzochi. Group photograph from the 72nd birthday conference for Paul Cohen, taken on October 5, 2006. Cohen is in the front row, third from the left. own improvised proofs of key lemmas and theo- because every visit to our home was highlighted by rems) and rush back to explain it. His ideas on the one of his enthusiastic displays of tricks. Riemann hypothesis (about which I plan to give The 2006 meeting in Stanford, celebrating a brief description in his forthcoming collected Paul’s mathematics and also his seventy-second works) led him to study much of the work of Atle birthday, brought together a mix of world-leading Selberg and especially the “trace formula”.