**Aurel Wintner**'s father, Eduard Wintner, was a businessman who had lived in Rotterdam and has emigrated to Budapest while his mother, Charlotte Eugenie Hirshfeld, was from Vienna. Aurel attended school in Budapest, completing his school education there in 1920.

Wintner's mathematical ability was recognised by one of his school teachers and, realising that he was capable of progressing much faster than the other pupils, arrangements were made to enable Wintner to use the mathematics library at the University of Budapest. This was a major influence in allowing him to discover his love of the subject, for at school it was not clear that he would continue to study science. The reason was that Wintner was extremely musical and showed great talent for the violin.

As he came towards the end of his school career in 1920 he realised that he could not pursue both music and science. Many might have decided to aim for a scientific career and still keep the violin as a major hobby - in fact many of the mathematicians in this archive played the violin to a very high standard. Not so Wintner, for once he had made the decision for a scientific career he never played the violin again.

Although being allowed to use the mathematics library at the University of Budapest was a major influence in directing his interests in that direction, there was another influence which pushed him towards astronomy. This came from his uncle, S Oppenheim, who was professor of mathematics at the University of Vienna. As a schoolboy Wintner had spent holidays in Vienna and had used his uncle's astronomy library.

After leaving school Wintner entered the University of Budapest in 1920. However, although he came from a fairly well-off family, this was a period of hyper-inflation and it became increasingly difficult for him to continue with his education. In 1924 he withdrew from university by this time he had reached a level where he was undertaking research. During the years 1924 to 1927 he published about 20 papers on astronomy and mathematics. The quality of the work that he was producing made a marked impression on the world of mathematics and Leon Lichtenstein, who taught at the University of Leipzig began to correspond with him on mathematical topics. Lichtenstein began to encourage Wintner to consider studying for his doctorate.

Wintner entered the University of Leipzig to study for his doctorate in 1927. During the following two years he served as assistant to Lichtenstein as editor of *Mathematische Zeitschrift* and also *Jahrbuchs über die Fortschritte der Mathematik* Ⓣ. Wintner always felt a great debt of gratitude towards Lichtenstein for supporting his studies and throughout his life Wintner kept a picture of Lichtenstein on the wall in his office.

George W Hill had published an account of his lunar theory in 1878. Hill's methods used infinite matrices and series expansions which he assumed were convergent but gave no proof. Wintner wrote a number of papers putting Hill's theory on a rigorous mathematical foundation.

He published the first proofs of the basic facts about Hilbert spaces in 1929, the year in which he was awarded his doctorate by the University of Leipzig. In fact at this time the development of Hilbert spaces had become particularly important for the study of quantum theory since this mathematics underlay the theory. This should have made Wintner's work of particularly topical interest but unfortunately for him mathematicians were developing their theories in terms of operators, following the approach of von Neumann. This should not have meant that Wintner's approach via infinite matrices was not of great value but it had the effect that his contributions were not appreciated as they should have been. This in turn led to Wintner feeling bitter at his lack of recognition for his contribution.

After completing his doctorate in 1929 Wintner won an International Board Fellowship to enable him to study abroad. He made two important visits during 1929-30, one to Rome where he worked with Levi-Civita, and the other to the observatory in Copenhagen where he worked with E Strömgren.

While studying at Leipzig Wintner had been taught by Hölder. This became more than just a relationship between teacher and pupil for in 1930 Wintner married Irmgard Hölder, Otto Hölder's daughter. In the same year he accepted a post at Johns Hopkins University in the United States where he continued to be employed until his death. He spent the year 1937-38 at the Institute for Advanced Study at Princeton, also visiting Harvard during that period to work with G D Birkhoff. Wintner was awarded a Guggenheim Fellowship in 1941 which enabled him to visit Cambridge, Massachusetts. He had planned to write a book with Norbert Wiener during this time but Wiener was involved with war work which meant their plans could not be carried through as they had intended. In 1946 Wintner was appointed to a full professorship at Johns Hopkins University, a position which Wintner should have been offered many years before.

In 1944 he became editor of the *American Journal of Mathematics* having already been associate editor from 1936. Clearly his editorial duties as Lichtenstein's assistant in 1927-29 stood him in good stead, and editorial duties were clearly something which gave Wintner great satisfaction. He played an important role in developing the editorial policy of the *American Journal of Mathematics* and, working with André Weil, he made major changes to the editorial policy from 1953 until his death, which resulted in significantly increasing the standard of the journal.

Wintner published on analysis, number theory, differential equations and probability (with several joint papers with Norbert Wiener). These interests appear to be unconnected but this is in fact far from true. Although it is true that Wintner studied certain areas of mathematics for their own sake, he was led to these areas through his work in celestial mechanics.

Along with Poincaré and George Birkhoff, he placed celestial mechanics on a more sound mathematical basis. These innovators were more concerned with the underlying theory, less concerned with quantitatively accurate prediction of celestial body motion. A study of certain astronomical equations led Wintner to consider almost periodic functions. Hartman, Wintner's student and then colleague, writes in [3] that an interest in perturbations of planets and other related work:-

Wintner published 437 papers during his career and 9 monographs. Of these monographs six were published after he emigrated to the United States. These were... led Wintner to various interests: first, his interest in almost periodic functions as such; second, analytic number theory and summability; third, the asymptotic distributions of almost periodic functions; and finally, the theory of distribution functions as such. In this connection, Wintner used to like to point out the debt of analytic number theory to dynamics, noting that in a certain sense the oldest Tauberian theorems date back to the dynamical work of Sundman and Hadamard.

*Lectures on asymptotic distributions and infinite convolutions*(1938),

*Analytical foundations of celestial mechanics*(1941),

*Eratosthenian averages*(1943),

*Theory of measure in arithmetical semigroups*(1944),

*The Fourier transforms of probability distributions*(1947), and

*An arithmetical approach to ordinary Fourier series*(1945). Of these monographs, those of 1943, 1944 and 1945 were published privately. Wintner [3]:-

As to Wintner's character Sternberg writes in [1]:-... took rather a dim view of American publishing houses which would not publish specialised monographs with limited sales. He felt that he proved his point by not losing money in this capacity as entrepreneur.

Hartman, in [3], writes of:-Wintner, a man of high moral principles, opposed direct government support of scholarly research, for fear of interference. He not only accepted considerable financial hardship by personally refusing such support but also was willing to forgo fruitful scientific collaboration in order to maintain his ideals.

As to his personal relations with students, he [3]:-... the intensity and the great energy which[Wintner]brought to every task, mathematical and non-mathematical. ... in the last few years of his life he seemed to prefer working closely with one or two students to lecturing, but earlier in his career he gave inspiring lecture courses in which he transmitted his love and enthusiasm for mathematics.

His interests outside mathematics included hiking in the mountains but, in a similar way to never playing the violin past the age of 17, he strongly believed that one should not have non-essential interests. As a result he had many friends among mathematicians, but few others. He died suddenly from a heart attack while still at the height of his mathematical productivity.... sought out the promising young students and offered them friendship, help and suggestions. On the other hand, he could be very abrupt in dropping these students if he decided that they did not work as hard as he thought they should.

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

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