**Norman Levinson**'s family were very poor Russian Jewish immigrants to the United States. Norman, born in Lynn, Massachusetts just before World War I, was the son of a shoe factory worker there, who earned three dollars a week and whose education consisted of attending a yeshiva for a few years. Norman's mother was illiterate but, despite the poverty, she managed to feed Norman and his sister Pauline. Zipporah (Fagi) Levinson, Norman's wife, wrote in [5]:-

Relatives and friends would give[Norman's mother]worn hand-me-down clothes. She would take them apart, and using the undersides which didn't show any wear, would make clothes for her children and husband.

Despite a childhood of desperate poverty, and an education that consisted of attending rundown vocation schools, Norman said of his childhood:-

We were very poor, but we didn't think of ourselves as poor.

Norman's father changed jobs, working for Forbes Lithograph, and the family bought a little house in Revere. It had no bathroom, and was heated by a single oil stove in the kitchen. Norman's mother did a remarkable job of providing for the family, planting fruit trees and vegetables behind the house. With little else for food, she would walk over a kilometre each day to buy stale bread for the family which was sold at half price.

Norman attended Revere High School but the family doctor said that as he suffered from rheumatic fever he should not take any physical exercise. These school years were not easy for Norman but he helped his fellow pupils with their homework which put him on good terms with them. These years had a lasting effect on Norman, however. They may have been the cause for him being very shy, and they certainly were the reason that he became a hypochondriac. To try to help the family finances, Norman worked in the evenings in a grocery store while studying at Revere High School.

Levinson entered the Massachusetts Institute of Technology in 1929 but he did not register for a mathematics degree, rather he studied for a degree in Electrical Engineering. In June 1934 he was awarded a Bachelor of Science degree and a Master of Science degree, both in electrical engineering. However he had by this time taken twenty graduate courses in mathematics at MIT, almost all the graduate courses the mathematics department offered. In fact by this time, while still an electrical engineering student, he had written a thesis with Norbert Wiener and, according to the Head of Mathematics, had:-

... results sufficient for a doctoral thesis of unusual excellence.

The turning point in Levinson's studies had come when he signed up for Wiener's graduate course on *Fourier series and integrals* in 1933-34. Levinson described what happened (see for example [6]):-

I became acquainted with Wiener in September1933, while still a student of electrical engineering, when I enrolled in his graduate course. It was at that time really a seminar course. At that level he was a most stimulating teacher. He would actually carry on his research at the blackboard. As soon as I displayed a slight comprehension of what he was doing, he handed me the manuscript of Paley-Wiener for revision. I found a gap in a proof and proved a lemma to set it right. Wiener thereupon sat down at his typewriter, typed my lemma, affixed my name and sent it off to a journal. A prominent professor does not often act as secretary for a young student. He convinced me to change my course from electrical engineering to mathematics.

After the award of his Master's degree in electrical engineering, Levinson applied to MIT to begin studying for his doctorate in mathematics. However the Mathematics Department were convinced that he had already done sufficient for the Ph.D. before starting the course! Instead Wiener together with Phillips, the Head of Mathematics, arranged for Levinson to receive an MIT Redfield Proctor Traveling Fellowship so that he could spend the year at Cambridge in England. He was assured that he would receive his doctorate on his return to MIT irrespective of any work he did in Cambridge. The attraction of Cambridge for Levinson was the fact the G H Hardy, one of the world's most respected mathematicians, taught there.

Arriving in Cambridge at the end of August he wrote back to MIT in January:-

I have been researching along here at quite a rate in sheets of paper consumed per day. However, usually the end product of my effort warms my soul only by blazing up in the fireplace. So far my stay here has produced two completed papers and several semi-completed ones.

This is rather a remarkable statement, if we look beyond the modesty and the joke; for a young student four months into his doctoral studies to have written two papers and have several more underway is truly remarkable.

Rota, in [9], makes a rather surprising statement about Levinson's year at Cambridge which we reproduce without comment:-

Norman told me that the first thing he did when he arrived in England was to buy a new wardrobe for himself. His year in Cambridge was uneventful: he never even met Hardy, as a matter of fact he never had a high opinion of Hardy, he thought Littlewood to be a stronger mathematician.

When Levinson returned to MIT in 1935 he was awarded a Doctor of Science degree for a thesis entitled *Non-vanishing of a function*. He then was awarded a National Research Council Fellowship which enabled him to spend two years at the Institute for Advanced Study at Princeton. At the Institute for Advanced Study, Levinson was attached to von Neumann who was to act as a supervisor, but Levinson was a fully independent research worker by this time and certainly did not need a supervisor.

The Great Depression began in 1929, the year Levinson entered MIT, and by 1932 one quarter of the workers in the United States were unemployed. As Levinson undertook his research at Princeton he felt that he had little prospects of gaining a university job, partly because of the high unemployment, but also because anti-Semitism in the United States at this time meant that Jewish mathematicians found it much harder than others to get posts. There was no shortage of high quality Jewish mathematicians, too, since by 1937 many such people were fleeing from Germany and surrounding countries and emigrating to the United States. Levinson formed a plan to train as an actuary and try for a job with an insurance company after his Fellowship at Princeton ended.

If Levinson really had not met Hardy during his year at Cambridge, it was certainly Hardy who fought for Levinson to get a permanent job when he visited the United States. In the autumn of 1936 Jesse Douglas, who was awarded one of the first Fields' Medals that year, became ill and could not teach his courses at MIT. Levinson was an obvious person for them to hire and was recommended to them by Wiener but anti-Semitism at MIT tried to prevent such a move. The university's provost, Vannevar Bush, turned down Wiener's recommendation that Levinson be offered a position as Instructor but Hardy, on a visit to MIT, went with Wiener to the provost's office to protest against the decision. Hardy is reported to have said:-

Tell me, Mr Bush, do you think you're running an engineering school or a theological seminary? Is this the Massachusetts Institute of Theology? If it isn't, why not hire Levinson.

Levinson was appointed as an Instructor at MIT in February 1937 having been released from his Fellowship by Princeton before its term was complete. He married Zipporah Wallman (known to all as Fagi) on 11 February 1938 and their two daughters, Sylvia and Joan (Zorza) were born in 1939 and 1941. Levinson was promoted to Assistant Professor in the year that his first daughter was born.

In 1940 Levinson published *Gap and density theorems* in the American Mathematical Society Colloquium Publication Series. It was a great tribute to the young mathematician that he had been invited to write a book in a series which was reserved for distinguished senior mathematicians. The book contains his early researches on Fourier transforms in the complex domain which had developed from his study of the Paley-Wiener book which began his research career. *Gap and density theorems* [2]:-

... subsumes much of Levinson's brilliant early research in harmonic and complex analysis.

Martin explains in [6] how Levinson changed the direction of his research after the publication of this treatise in 1940:-

Norman decided to shift his field from gap and density theorems to non-linear differential equations, both ordinary and partial. I recall our talking about this decision in1940, and how difficult is was to move into this new field, and how hard Norman worked over a period of two or three years before he felt that he had enough mastery to obtain substantial results in this field; but this mastery he did achieve, and his outstanding contributions to non-linear differential equations were recognised officially in1954when the American Mathematical Society awarded Norman the Bôcher Prize.

By 1954 Levinson had been a professor for five years. He had been promoted to Associate Professor in 1944, and to full Professor at MIT in 1949. However his career nearly came to a premature end in the McCarthy era.

Levinson believed in employment for all, fighting anti-Semitism, and fighting discrimination against blacks. He found that these were exactly the views of the American Communist Party which he therefore joined. When he learnt of the direction that Stalin had taken Communism in Russia, Levinson left the American Communist Party. However in 1953 he, and two other colleagues at MIT, were forced to testify to the House Un-American Activities Committee. The Committee demanded that Levinson name other members of the American Communist Party but, although he agreed to talk freely about what he did as a member of the Party, he refused to name others since he knew the consequences that would have. He had already worked out in his own mind what he would do if fired by MIT, but a skilful lawyer saved him from this fate. It did mean some difficult years for Levinson and his family, but they came through them.

Let us return to discuss further Levinson's mathematical contributions. In 1955 he published another text which quickly became a classic. This was *Theory of ordinary differential equations* (written jointly with Earl Coddington) which [2]:-

... has literally become the bible for students that helped train several generations of mathematicians, scientists and engineers since it was published in1955.

Levinson wrote only two papers on time series, but these had a large impact. They had implications for geophysical signal processing (and signal processing in general) and they contributed to improved methods of oil exploration, particularly in off-shore oil fields.

To gain a full appreciation of Levinson's mathematical contribution we quote at length from the Preface to [2]:-

The deep and original ideas of Norman Levinson have had a lasting impact on fields as diverse as differential and integral equations, harmonic, complex and stochastic analysis, and analytic number theory during more than half a century. Yet, the extent of his contributions has not always been fully recognized in the mathematics community. For example, the horseshoe mapping constructed by Stephen Smale in1960played a central role in the development of the modern theory of dynamical systems and chaos. The horseshoe map was directly stimulated by Levinson's research on forced periodic oscillations of the Van der Pol oscillator, and specifically by his seminal work initiated by Cartwright and Littlewood. In other topics, Levinson provided the foundation for a rigorous theory of singularly perturbed differential equations. He also made fundamental contributions to inverse scattering theory by showing the connection between scattering data and spectral data, thus relating the famous Gelfand-Levitan method to the inverse scattering problem for the Schrödinger equation. He was the first to analyze and make explicit use of wave functions, now widely known as the Jost functions. Near the end of his life, Levinson returned to research in analytic number theory and made profound progress on the resolution of the Riemann hypothesis.

For a paper on number theory Levinson received the 1971 Chauvenet Prize of the Mathematical Association of America. Shortly before his death he wrote a series of important papers on the Riemann hypothesis arising from this fundamental number theory paper.

E A Robinson, see [1], writes:-

Levinson ... was on the threshold of perhaps his greatest achievements in mathematics at the time of his death.

Although Levinson had feared all his life that he would die of a heart attack, and was extremely careful with his diet in an attempt to avoid this fate, it was a brain tumour which lead to his death. Rota writes:-

One day shortly after his paper on the Riemann zeta function appeared, he knocked at the door, came in, and sat down. He looked pale and ill. He complained of a strong headache. ... Shortly afterwards, he entered Massachusetts General Hospital. ... in the August of that summer[I]visited him ... His head was shaven, and red and black lines were drawn on it. ... I never saw him again.

After his death the MIT Faculty prepared a tribute to Levinson. It read:-

Norman Levinson was the heart of mathematics at MIT, a man who combined creative intellect of the highest order with human compassion and unremitting dedication to science and to excellence in its pursuit. Throughout the mathematical world the name of MIT and the name of Norman Levinson have been synonymous for many years. To those of us who were fortunate to have him as a friend and colleague, this is entirely fitting, because we are aware that, with extraordinary effectiveness and caring, he devoted forty six years of his life to mathematics and to this Institute.

**Article by:** *J J O'Connor* and *E F Robertson*

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