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Heini Halberstam, 1926-2014

2017; Wiley; Volume: 49; Issue: 6 Linguagem: Inglês

10.1112/blms.12115

1469-2120

Harold G. Diamond,

Mathematics and Applications

by Harold Diamond and Eira Scourfield Heini Halberstam was born in Brux, Czechoslovakia (today Most, Czech Republic), on 11 September 1926, the only child of Michael and Judita Halberstam. Heini's father had moved to Most from Vienna in the 1920s to become the town's Orthodox Rabbi. When Heini was ten years old, his father died suddenly from a heart attack, and soon after, he and his mother moved to Prague. Following the German invasion of Czechoslovakia, Judita arranged for Heini to study English and, in April 1939, to leave home for England on a Kindertransport train. Heini arrived a week later in London, never to see his mother again. In 1942, she, along with most of Prague's Jews, was deported to a Nazi work camp where she soon died of typhoid. After several placements in England, Heini had the good fortune to come in the care of Anne Welsford who recognized his ability and encouraged and supported him through his university studies. Heini began studying mathematics at University College, London. After completing his degree in two years, graduating about 1947, he began working for a PhD at UCL. He wrote his thesis on analytic number theory under the supervision of Theodor Estermann, and he was awarded his PhD degree in 1952. At that time Klaus Roth was a fellow research student who worked with Estermann and Professor Harold Davenport. Around 1948, Heini was appointed to a lecturing position at the University College of the South West in Exeter. The mathematics department then was small with about eight staff who taught the full syllabus for the External Degree of the University of London; in 1955 the College became the independent University of Exeter. A few months after arriving in Exeter, Heini married his first wife, Heather Peacock. He was subsequently appointed Warden of Crossmead Hall of Residence for men students, a position he held in addition to his lectureship. He and his colleagues Walter Hayman and Paddy Kennedy ran a mini research seminar with the encouragement of the Head of Department, Professor T. Arnold Brown. It was at Exeter that Heini's first paper 1 was published in 1949. Heini spent the academic year 1955–1956 in the United States at Brown University. One of his adventures there was getting a traffic ticket. In later years, Heini was amused to recount the conclusion of the court proceeding, at which the judge pronounced his fine with, ‘Rule Britannia, $5.00 please’. When Heini returned to Exeter in 1956 he undertook the supervision of his first research student, namely the second named author of this section. Like others subsequently, she found him to be an inspiring, challenging, and encouraging supervisor. In 1957 Heini moved to Royal Holloway College, University of London, where he was appointed Reader in Mathematics, and he arranged for Eira to transfer there for the second half of her Master's course and to write her thesis. She benefitted from and much appreciated his strong support throughout her university career and his maintenance of regular academic and personal contact by letter, at conferences and during sabbaticals for the rest of his life. While at Royal Holloway College, Heini regularly attended number theory seminars at UCL, and during this time he began his long involvement in the work of the London Mathematical Society (LMS). In 1962 he was appointed Erasmus Smith's Professor of Mathematics at Trinity College, University of Dublin. Two years later Heini moved to the University of Nottingham, where he served at various times as Head of Department and Dean of the Faculty. Heini and Heather had four children, two of whom live in the United States and two in Britain; Heather was tragically killed in a road accident in 1971. Heini subsequently married Doreen Bramley who has two children, both residing in Britain. They have eight grandchildren. In 1980, Heini came to the Mathematics Department of the University of Illinois in Urbana-Champaign (UIUC). He served as Department Head 1980–1988 and retired as Emeritus Professor in 1996. Heini was held in much esteem, and to mark his retirement, the department held an international conference on number theory in his honor. In spring 2014, another such conference was sponsored in memory of Heini and of Paul and Felice Bateman. During his career, Heini also held visiting positions at Brown, Michigan, UC Berkeley, Syracuse, Ohio State University, Paris, Ulm, Scuola Normale Superiore in Pisa, Tel Aviv, York, Hong Kong and Matscience in Madras (now known as Chennai). Heini was a major figure in number theory whose research ranged over several areas. He first studied probabilistic methods, and his later — and most important — work centered on sieves. Other interests of his were mean value theorems, Waring's problem and combinatorial number theory. Some of his research collaborators were Harold Davenport, Harold Diamond, Peter Elliott, Hans-Egon Richert and Klaus Roth. His conjecture with Elliott on the distribution of primes in arithmetic progressions remains one of the outstanding problems in analytic number theory. Sir William Rowan Hamilton (volume 3) 21 Harold Davenport (four volumes) 43 J. E. Littlewood (volume 2) 49 Loo Keng Hua 50 Recent progress in analytic number theory, Durham, 1979 (proceedings) 48 Analytic number theory, Allerton Park, 1990 (proceedings) 65. One of Heini's particular passions, perhaps remembering how he himself had been aided and encouraged as a child, was promoting talented young people. Heini was an inspiring (if demanding) teacher and mentor. He supervised fourteen PhD and four Masters' theses, and in addition, many others who came in contact with him as students or postdocs also received enormous help and support. Several of his charges went on to distinguished careers, among them Jean-Marc Deshouillers, Michael Filaseta, Kevin Ford, Richard R. Hall and Robert Vaughan. An example of the lengths to which Heini would go to help a young person was reported by Atul Dixit, a recent PhD at UIUC. Atul wanted to read a sixteen-page paper of the eminent nineteenth century Czech mathematician Mathias Lerch. This was a classic good-news/bad-news situation: Dixit was able to find the paper, but it was in Czech. However, here was Heini who had lived as a boy in Czechoslovakia. On the other hand, Heini's rusty Czech vocabulary was that of a twelve-year-old. Finally, good news: with a Czech–English dictionary, Heini wrote out by hand a translation of the whole paper. Heini also had a life-long passion to improve mathematical instruction. At Nottingham, he helped start the Shell Centre for Mathematical Education, was a director of the center, and was a member-at-large of the International Commission on Mathematics Education from 1979 to 1982. He continued work in mathematical education after coming to the United States and published several articles on this subject. Heini was a member of the LMS for 59 years, and he served as a Vice President of the LMS and as secretary of its Journal. Also, he was a member of the American Mathematical Society (AMS) for 57 years and wrote over 150 reports for Mathematical Reviews. In addition, he served on the editorial boards of several journals, including Acta Arithmetica and the Journal of Number Theory. Heini's accomplishments led to many awards and honors. He was elected to the Royal Irish Academy in 1963 (resigned 1966) and was a Fellow of University College, London, from 1967 onward. Heini gave an invited 1-hour lecture at an AMS Annual Meeting in 1980, and was named a Fellow of the AMS in 2012. Over the years, Halberstam held research grants from the U.S. Army, NATO, and the National Science Foundation. A gifted writer, Heini produced precise and elegant prose with seemingly little effort. He was frequently called upon for expository articles, book reviews and obituaries, as well as to provide insight and editorial assistance for the projects of others. An example of Heini's talent is seen in his review in the Notices of the AMS of The Indian Clerk, a fictionalized account ofthe interaction between the celebrated mathematician Ramanujan and his patron, G. H. Hardy. In the words of his wife, Doreen, Heini was ‘a voracious reader’. This enthusiasm was present from his youth in England, when a friend, the local waste collector, lent him many books. When he married Doreen and they were combining their households, his main request was that she bring all her bookcases. He found most everything interesting, including mysteries, biographies and works, particularly on the Holocaust. Heini always hoped to learn more about his mother's fate. After retiring, he visited Prague and followed his mother's path of deportation. More recently, Heini participated in a reunion organized by the Kindertransport Association, and he gave talks in Champaign and elsewhere on the Holocaust and his personal experiences in the Kindertransport. One of Heini's presentations can be seen at https://www.youtube.com/watch?v=3oyAhpax8w0. An interesting and insightful account of Heini's life is given in an obituary prepared by his daughter Jude at http://www.jackhalberstam.com/obituary-for-heini-halberstam-1252014/. For this obituary, she has written about Heini's recollections of leaving Czechoslovakia in 1939. Heini died at home in Champaign, Illinois, on January 25, 2014 at the age of 87. He had a mathematical career extending over 60 years and had been active until the last months of his life. Heini was an internationally known figure in number theory, particularly for his work in sieve theory. In addition to his scholarship, Heini was treasured for his encouraging and optimistic manner, beautiful writing, energy and his interest in people. by Michael Filaseta When I came to the University of Illinois in Champaign-Urbana as a graduate student in August, 1980, Heini had just arrived as head of the Mathematics Department. This additional strength of the U of I number theory group increased my enthusiasm for being a student there. On my first day, as I eagerly walked around campus, I recall hearing among other things the bell tower chimes in Altgeld Hall. The theme song from ‘The Flintstones’ was playing; someone had unusual taste, and maybe I did, too, for recognizing the piece. It says something about my opinion of the U of I number theory group that this was the only graduate school that I had applied to. In addition to Heini, the faculty included Paul Bateman, Bruce Berndt, Harold Diamond, Walter Philipp, Bruce Reznick and Ken Stolarsky, as well as several algebraic number theorists. Postdocs then included Brian Conrey and David Richman. Adolf Hildebrand was also to hold a postdoc position toward the end of my four years at the U of I. Many of these people I had already met while attending the West Coast Number Theory Conference as an undergraduate. Recent breakthroughs in number theory around that time included Roger Apéry's proof that ζ ( 3 ) is irrational and, a little earlier, Chen Jingrun's proof that every sufficiently large even integer could be written as a sum of two primes or as a prime plus a product of two primes. In addition to Paul Erdős' numerous uses of sieve methods, Chen's theorem helped stoke my eagerness to learn more about sieve techniques. Knowing Heini was an expert in this topic made me think, even before arriving on campus, that I would want him to direct my doctoral dissertation. As an undergraduate student, I had lived at home to keep down costs for my parents who had six children. My dad was in the Army, and we had a rather strict family household to help make life with six kids somewhat orderly. Now, more-or-less on my own for the first time (‘less’ because my brother was also at the U of I, pursuing a doctoral degree in physics), I wanted to express some independence. This urge probably showed itself in ways that I am not even aware of, but the hair that covered my face during much of my four years at the U of I was certainly part of it. By good luck, Heini offered a course in sieve methods during my first year at Illinois. This was Heini's first-year teaching here, and I wondered how the course might be taught. Heini was a great lecturer: he wrote and spoke clearly, was very organized, and faced us as he discussed the material. While he frequently asked whether there were questions, I could not help but feel that he did not really want any. But I asked … . I recall believing that I understood very well what he was saying in the lectures, but I was not so comfortable with the bigger picture of how everything tied together. I blamed myself and not him for the situation, but this feeling and my good grasp of the lectures as they were presented led me not to ask as many questions later in the semester. I recall at some point, when Heini had finished an explanation and I had raised no question, he looked at me and asked, ‘What does the skeptic think about that’? Perhaps asking too many questions during Heini's first-year teaching at the U of I was not such a good idea. I was determined early on to impress Heini. The questions during his class may not have helped, but something else unexpectedly did. I attended the number theory seminars regularly and even presented a couple of talks, but these did not necessarily help either. One day, however, as I entered the seminar room, Heini said to me, ‘You are looking more and more impressive every day’. This was a reference to my facial hair; by then I had a rather thick beard. Having finally impressed Heini, I was ready to ask him to be my advisor. He agreed. At that time, I was finishing a combinatorics paper on the classical ballot problem. This arose in a transcendental number theory course of Ken Stolarsky, where a homework problem was related to Padé approximants for e x . The problem was to show that an n × n determinant with the entry 1 / ( i + j − 1 ) ! in the ith row and jth column is non-zero. Another student in the course had shown me a nice combinatorial method of evaluating the determinant that proved clearly that it was non-zero, and I stubbornly stayed up late the night before the assignment was due, determined to find a different solution. I ended up showing that a multiple of the determinant counted the number of lattice paths of certain types; since these lattice paths existed, the determinant was non-zero. The paper was based on extending this idea further. I recall Heini taking my paper and then returning it the next day with very detailed corrections and comments. This was perhaps my earliest realization of the amount of time he was to devote to me as a student. As department head, one might expect him to have little time for students, but this was not the case. His door was always open to me, and he gladly spent time with me, suggesting material to read, discussing research, and meticulously going through drafts of papers and my dissertation. At some point while I was a student, Heini wanted me to have the experience of refereeing a paper, so he handed me one to referee along with a detailed chart of notation to use for marking it up. When I returned the paper with a report, I inquired about an item from the chart, a circle with a dot in the center. Next to it was the description, ‘full stop’. I asked him what this meant. He looked at me, with no understanding of my confusion and said, ‘You know — a dot at the end of the sentence’. Only then did I realize he had given me British descriptions of the editorial markings. Heini wanted students not to spend too much time working toward a doctoral degree. I was thinking of finishing in five years, but he told me at the beginning of my fourth year that I was ready to move on, so I adjusted my thinking. After sending out many job applications, offers came from three places, one with a strong undergraduate program, one in industry, and one from the University of South Carolina. Heini was supportive of whatever decision I would make, and I recall him expressing points for taking the job in industry as well as the one that I did take at the University of South Carolina. The former had offered me three-halves the salary of the latter, but the university position would allow me to pursue my passions for both teaching and research. As an added bonus, David Richman was going to South Carolina, and we had already discussed mathematics extensively while he was a postdoc at the U of I. Over the years, Heini helped me greatly in my career, undoubtedly also in ways that I am unaware of. On a number of occasions when I returned to Illinois to give a talk, whether Heini was involved in the arrangements or not, he made a point to see that my expenses were covered, to meet with me and discuss mathematical as well as non-mathematical topics, to offer career suggestions, to take me out to eat and visit his home, to meet his lovely wife Doreen, and to share time with my wife and family. He corresponded with me on a regular basis, often on a personal, non-mathematical level. Early on, I saw in Heini a serious stern personality marked by a commendable sense of conviction. As time passed, his more gentle and kind nature became evident in ways that I could not help but admire. Heini Halberstam was my doctoral advisor, but he was at times more of a father figure to me and always a dear friend. by Kevin Ford My first memories of Heini Halberstam are from the spring of 1991 when I took his course on divisor theory. I was a first-year graduate student, very interested in number theory, but unsure of the direction I wanted to pursue for research. By the end of the term, however, I had fallen in love with the subject, thanks in large measure to Heini's presentation. His enthusiasm for the subject was infectious, and his lectures were a model of clarity. I do not think I had ever learned so much in a single mathematics course as I did that term, not just about primes and divisors but the whole ‘Erdős method’ of thinking about problems. The text for the course, a terrific little Cambridge tract of Richard Hall and Gérald Tenenbaum 〈〈12〉〉, was densely written and I greatly appreciated having an expert field guide to help navigate it. One of the main theorems treated in the course has a simple statement but a long, difficult proof, and I began to appreciate Heini's viewpoint, rarely stated explicitly, about what he considered to be worthwhile mathematics. That same term, Paul Bateman gave a fascinating seminar on Carmichael's conjecture, an old open problem about Euler's totient function. I latched onto it and began reading many papers about Euler's function, including a 1935 paper by Paul Erdős titled ‘On the normal number of prime factors of p − 1 and some related problems concerning Euler's ϕ-function’ 〈〈8〉〉. I recognized in this paper many of the same methods which I was learning in Heini's course, but at one place I was baffled, where Erdős invoked ‘Brun's method’. Well, it is sometimes said that half the battle in research lies in asking the right question of the right person. When I went to Heini's office to ask him what this was all about, I did not know that the modern term for ‘Brun's method’ is sieve methods nor that Heini had written, with H.-E. Richert, the definitive book on the subject 39. Needless to say, very soon afterward I was learning about sieve methods, another subject that would play a large role in my future work. Despite my early interest in divisors, Euler's function and sieve methods, my PhD dissertation was written in a completely different area, and it was not until my postdoctoral years that I returned to these subjects. The story of the thesis began, I believe, in my second year of graduate studies, when I invented what I thought was a new problem, that of representing positive integers as the sum of a perfect square, perfect cube, fourth power, etc. using a minimal number of summands. I turned to Paul Bateman to inquire about the problem, chiefly because he was an expert on related problems of representing integers as sums of squares, and also because of his encyclopedic knowledge of the area. Paul said that Heini might know about this, and phoned Heini to confirm. I then walked next door to Heini's office to discuss the problem with him. My problem turned out not to be new, and in fact first appeared in the PhD thesis of Heini's fellow graduate student Klaus Roth in 1948. Heini himself had studied very similar problems in his own PhD thesis (both he and Roth were supervised by Theodor Estermann) before moving on to other topics later in his career. This was the start of my regular meetings with Heini, at first going through Bob Vaughan's Cambridge Tract on the Hardy–Littlewood circle method 〈〈31〉〉, which would have been much more difficult without Heini's guidance, and later reading the literature related to Roth's problem. Heini treated each session as a learning opportunity; a sticky point in a proof would lead to a lecture about the methods being employed, and perhaps reading some additional papers. Our conversations were mainly on mathematics, although sometimes Heini could not help but make some witty comment on the state of education in the United States or Britain. He once referred to himself as a ‘Thatcher refugee’. He never spoke about his personal life. It was many years later that I first learned that he had been one of the children saved by the Kindertransport. Eventually I proved my first significant theorem, reducing the known upper bound for the number of summands required to represent sufficiently large integers, and was quite proud of myself. I had no idea how to publicize my result, these being the days before the World Wide Web, the math arXiv, and such, but Heini knew what to do — he advertised my result to his friends and colleagues, including Vaughan (who, as another student of Estermann, also had worked on this problem for his PhD thesis). I am sure Heini also helped get my paper published in the Journal of the LMS; however, he was not totally satisfied and pushed me to do even better — he had uniformly high standards of excellence, which he applied to his students as well as to himself, and not only for mathematics. Heini shared with me a letter in which it was mentioned that someone else was working to improve my result. Well, I could not let that happen! Thus spurred on, I redoubled my efforts, and about two months before I was to defend my thesis I improved my previous theorem by reducing the number of required summands for large integers from 15 to 14. I am certain that Heini's publicity efforts helped me tremendously in finding a job after graduation. This was very useful, since the Russian border had just opened and there was much competition for jobs. Just three weeks after my breakthrough, I received an offer of a postdoctoral position. Heini was definitely ‘old school’ in many ways; he dressed conservatively, gave very careful, meticulous lectures, and was not up to speed on the latest technological advances, although he did use email. Of course he welcomed the newest mathematical advances with gusto. I, on the other hand, had long hair, wore blue jeans and ratty T-shirts, and grew up using computers. When I was still a student, I was playing around with a problem about gaps between numbers coprime to a given integer and asked Heini if he knew about this. He thought Erdős might know and suggested that I write to him. I asked Heini for an email address, not knowing that Erdős was an even greater technophobe. ‘No’, Heini said, ‘you'll have to write him a letter’. Heini knew where Erdős was at that moment (which was remarkable, since he was always on the move), and I handwrote my letter. A few weeks later I received a nice reply; it turned out my question was related to ones about large gaps between consecutive primes, one of Erdős' favorite problems, and one to which I would return twenty years later. Heini and I wrote only one joint paper, titled ‘The Brun–Hooley sieve’ 80, which appeared in 2000. Our paper had its genesis in a seminar talk Heini gave when I was still a graduate student, on a variant of Brun's original sieve method recently discovered by Christopher Hooley. I thought this was really cool. Afterward I had some ideas about how to optimize various parameters, worked at it for a few days, showed it to Heini, and then returned to my thesis problem. I had almost completely forgotten this topic, until Heini wrote to me in 1998 about possibly writing a paper based on my calculations along with some ideas of his for streamlining Hooley's treatment. Once the mathematics was finalized, Heini wrote and organized the paper as only he could have done. To say that the writing was immaculate is an understatement! In 2001, shortly after we wrote our paper, I returned to Urbana, now as an assistant professor. Heini was by then retired, and although we were now formally colleagues, it was difficult for me to view our relationship in these terms. Heini remained very active mathematically for many years, in particular completing his large project on combinatorial sieves with Harold Diamond and Hans-Egon Richert. He continued to attend our number theory seminar regularly until 2013, when he was unable to continue on account of declining health. Heini was a fantastic mentor, colleague and friend, and I miss him dearly. by Harold Diamond My family and I met Heini, Doreen and their family in the fall of 1973, while on sabbatical at the University of Nottingham. Heini was just returning from leave at that time and had a full plate of activities, but he was very generous of his time and helpful to me and my family. I soon found that Heini was a wonderful source for sound advice, well expressed. Shortly after we arrived in Nottingham, I went to Heini with a family problem: what to do about our son who was completely miserable at school. Andy was five years old, had never been to school before and suddenly found himself in a classroom with 50 (!) children he did not know. He came home each day exhausted from concentrating in this new environment. Also, he spoke funny and got ribbed for it. And the food that was served in the school cafeteria offended him beyond measure. I always treasured Heini's response to this lamentation. He replied in his calming, optimistic manner that Andy, like all other children, would get used to school; he soon would acquire the worst Nottingham accent we ever had heard; and he would be demanding the same food at home as was served at school. Of course, this is just what happened. During the winter we were in Britain, there were many crippling strikes, including the great coal strike that brought Mrs Thatcher to power. Buildings were heated even less than usual, the city was dark because the street lights had been turned off, and it was quite grim. The Halberstams did not let such things get them down. When we visited their home, we gathered in one room, they closed all the doors and turned on every electric heater they owned. It was warm and cheerful being with them. Heini had another side, one that saw the world the way it was, and he had the wit and tongue to comment on it pithily. Once, following some major natural catastrophe in America, perhaps a hurricane or earthquake, I asked him whether analogous disasters happened in Britain. ‘No’, he replied with a wry smile, ‘no, ours are all man-made’. My mathematical connection with Heini arose from a common interest in sieve theory. He and Hans-Egon Richert, of the University of Ulm, took me on as an equal partner in their sieve research. During the course of this work, which went on for many years, Richert died, and Heini and I finished the project, with much computer help from several of my PhD students. Our result extended a classical sieve theorem, and some of the applications improved on earlier results. We wrote up a form of this work with my former student Will Galway in a book published by Cambridge University Press 85. When I began working on these projects with Heini, I was apprehensive about meetings when I had no progress to report, particularly since Heini could blow his top when he sensed laziness or indifference. But Heini felt that an effort was being made, and he was always encouraging, so I came to feel comfortable working with him. I was fascinated to watch Heini write his long complicated sentences. About when I started wondering whether a sentence could conclude both sensibly and grammatically, it would wind up with a triumphant flourish. Readers of our joint papers can easily decide which of us wrote a particular part. He was for five years an active member of the Philosophy Club at the University of Illinois, a group that meets weekly during the term for dinner, reading and conversation. He was distressed when his hearing and concentration prevented him from participating in its discussions. My wife and I continued our close association with the Halberstams in the years after his retirement. We had many pleasant get-togethers, often when one of their children visited. We will miss Heini. by Robert C. Vaughan I first met Heini in 1969. Of course, I knew his early work in which he applied the Hardy–Littlewood method, since it related to part of my thesis. I had also acquired a copy of the superb book Sequences 18 which he had written with Klaus Roth, and had already read large parts of it avidly. He gave a seminar in the spring of 1969, probably sometime in March, at University College London where I was a postgraduate student. I seem to recall that it was on some aspect of sieve theory. I was just finishing my PhD with Theodore Estermann, and was on the job market 6. Although I was present in the seminar I did not meet him then. However, Estermann must have had a word with him because soon afterwards I was invited to attend a job interview at Nottingham. I knew that he was a graduate of UCL, so I wore my UCL colors tie. He spotted it at once, and as