**Rolf Nevanlinna**'s parents were Otto Wilhelm Neovius and Margarete Romberg who spoke both Swedish and Finnish. Of course this immediately prompts the question of where the name Nevanlinna came from since his father's name was Neovius. In fact Rolf Nevanlinna was known as Rolf Neovius for the first ten years of his life, only becoming Nevanlinna in 1906 when his father Otto changed the family name. Otto, born in 1867, came from a mathematical family and studied mathematics, physics and astronomy at Helsinki University. After obtaining a doctorate in physics he studied with the astronomer Herman Romberg at Pulkova Observatory and married Romberg's daughter Margarete in 1892. They settled in Joensuu where Otto taught physics, and there their four children were born: Frithiof (born 1894), Rolf (the subject of this biography, born 1895), Anna (born 1896) and Erik (born 1901). Frithiof also became a mathematician and he collaborated with Rolf on a number of publications which we mention below.

In 1902, at the age of seven, Rolf began his schooling in Joensuu. Having already been taught to read and write at home, he went straight into the second grade but he still found the work boring and after a while he refused to attend the school. He was educated at home for about eighteen months but then, after the family moved to Helsinki when he father was appointed as a teacher, he began attending the school there where his father was now teaching mathematics and physics. At school he learnt French and German to add to the languages he already spoke. The family also had a love for music encouraged by their mother [12]:-

Margarete was an excellent pianist and Frithiof and Rolf would lie under the piano and listen to her playing. At13they went to orchestra school and became accomplished musicians - Frithiof on the 'cello and Rolf on the violin. Through free tickets from the orchestra school they got to know and love the music of the great composers, Bach, Beethoven, Brahms, Schubert, Schumann, Chopin and Liszt, as well as the early symphonies of Sibelius(1865-1957), conducted by the composer. Rolf first met Sibelius's music in1907, when he heard his Third Symphony. Although later he met Hilbert, Einstein, Thomas Mann and other famous people, Rolf said that none had had such a strong effect on him as Sibelius. The boys played trios with their mother and their love of music - in particular of chamber music - lasted all their lives.

At the Helsinki High School Nevanlinna's interests were firstly classics, secondly mathematics. He was taught by a number of excellent teachers, but perhaps the best of them all was his own father who taught him physics and mathematics in his final years at school. He graduated from the school in 1913 having performed very well although he was not the best student in his year. However he went beyond the school syllabus in the summer of 1913 when he read Lindelöf's* Introduction to Higher Analysis* before beginning his university studies. From that time on he became an enthusiastic analyst working on that topic for the whole of his life. (Ernst Lindelöf was a cousin of Nevanlinna's father and so part of this mathematical family.)

Nevanlinna entered Helsinki University in 1913. There he was inspired by Lindelöf's teaching but the beginning of World War I in the following year threatened to disrupt his studies. Finland had been under Russian domination throughout Nevanlinna's life but this domination had increased during the first years of the 20^{th} century resulting in a strong independence movement within the country. After the war began, the independence movement looked to Germany for support, and young volunteers went to Germany where they received military training and formed the Jägar Battalion. Nevanlinna was keen to go to Germany and join the Jägar Battalion but his parents persuaded him that he should continue with his studies. He avoided being conscripted into the Russian military since he was a rather frail young man and so was able to continue his studies throughout the war years. In fact Helsinki was the scene of a civil war after Finland declared independence from Russia in December 1917. His thesis *Über beschränkte Funktionen die in gegebenen Punkten vorgeschriebene Werte annehmen*, presented in 1919, investigates the existence of regular functions satisfying certain fixed conditions. The work exhibits the care and attention to detail which was to typify all his work. He was awarded his doctorate on 2 June 1919 and on the same day married his cousin Mary. They had been engaged to marry for two years but they had been friends since 1911 when Nevanlinna, together with his older brother Frithiof, had returned to Joensuu to remember their childhood there. Rolf and Mary had four children: Kai (born 1920), Harri (born 1922), Aarne (born 1925), and Sylvi (born 1930).

University posts were not available in Finland in 1919 so Nevanlinna became a school teacher. His brother Frithiof worked as a mathematician for the insurance company Salama and, in addition to full time school teaching, Rolf Nevanlinna also worked for Salama. In 1920 he received an invitation from Edmund Landau to go to Göttingen but he did not accept immediately. He became a lecturer at Helsinki University in 1922 but he did not give up school teaching until he was appointed professor at Helsinki in 1926. Remarkably, despite this heavy workload, it was during these years that he produced what today is called 'Nevanlinna theory'. His working habits may explain this very high output of work. These are described in [12] as follows:-

Rolf never needed much sleep. In the late1930's he used to rise at4, join his family at10till lunch time, then work again till7, and spend the evening with his family.

Olli Lehto, who was a student of Nevanlinna, describes the birth of Nevanlinna theory in [20]. Walter Hayman, in a review, gives a summary of Lehto's paper which:-

... gives a most interesting account of the beginnings of value distribution theory. He starts with the construction by Weierstrass of an entire function with a preassigned sequence of zeros, the introduction by Laguerre of the genus and the results by Poincaré and Hadamard on the link between the exponent of convergence of the zeros and the order of the function which had first been defined by Borel. Borel also proved the extension of Picard's theorem which states that the exponent of convergence of the a-points of any entire function is equal to its order except for at most one value of a. This was the situation before Nevanlinna introduced his characteristic which not only extended the theory to meromorphic functions but also obtained a much more precise formulation of the results even in the special case of entire functions by means of his first and second fundamental theorems and the deficiency relation. The development proceeded in the1920's ... through R Nevanlinna's successive papers culminating in the monograph 'Le Théorème de Picard-Borel et la théorie des fonctions méromorphes'(1929).

The main results of Nevanlinna theory appeared in the 100 page paper *Zur Theorie der meromorphen Funktionen* in 1925. This paper was described by Weyl as one of the few great mathematical events of the twentieth century. To illustrate Nevanlinna's remarkable output, we note that this 100 page paper was one of eight papers he published in 1925, with thirteen papers having been published in the preceding three years. It is even more amazing when we realise that this huge output was achieved during years when he was working both as a school teacher and as a university lecturer.

Nevanlinna remained on the staff at the University of Helsinki for the rest of his life except for extensive travels he made to many different countries. Perhaps his enthusiasm for meeting other mathematicians was fired by his attendance at the 5^{th} Scandinavian Mathematical Congress at Helsinki in 1922. There he met Mittag-Leffler for the first time - he had been professor in Helsinki for five years, long before Nevanlinna was born. He also met, among others, Harald Bohr, Niels Bohr, Torsten Carleman and Harald Cramér. His first visit abroad was to belatedly accept Edmund Landau's invitation to go to Göttingen which he did in 1924. In addition to Landau he met Hilbert, Courant and Emmy Noether in Göttingen and, while in Germany he also met Pavel Sergeevich Aleksandrov, Pavel Samuilovich Urysohn, and Constantin Carathéodory in Munich. Later visits included one to Paris in 1926 where he met Hadamard, Montel and he also visited André Bloch in a mental hospital. He met A Speiser in Zürich when he visited there in 1928 and became interested in classification problems of Riemann surfaces. Two years later Nevanlinna was offered Weyl's chair after he had left Zürich to take up the chair at Göttingen, but Nevanlinna refused the chair. It was through the second half of the 1920s that Lars Ahlfors studied with Nevanlinna, obtaining his doctorate in 1930, and he had accompanied his thesis advisor on the trip to Zürich. Ahlfors' proof of the Denjoy Conjecture, which led to him being awarded a Fields Medal in 1936, followed suggestions made by Nevanlinna while in Zürich. The geometric ideas marked a new element in Nevanlinna's research and is evident in papers he published in the early 1930s. They are set out in detail in his the book *Eindeutige analytische Funktionen* (1936) which discusses ideas related to his invention of harmonic measure. Hayman writes that this book [12]:-

... has inspired function theorists ever since it was written and which, like most of the later books, has been translated into Russian and English as well as having several German editions.

He continued to work at Helsinki, but also continued his love for research visits, spending 1936-37 as a visiting professor at Göttingen. Returning to Finland, Nevanlinna and his wife Mary bought a summer home in Lohja, about 45 km west of Helsinki, in 1937 and this became a place where many visiting mathematicians spent time during the summers. He became rector of the University of Helsinki in 1941, retaining this position for the duration of World War II. During the war he contributed to the war effort by improving ballistic tables and was awarded the Cross of Liberty.

From his student days when he had wanted German help to free Finland of Russian domination, Nevanlinna was known to have German links. After the defeat of Germany in 1945, the authorities at Helsinki University felt that it would be inappropriate to have a rector who was known to have German sympathies so he was asked to resign from that role. He retained his chair at Helsinki, however, but in October 1946 he again visited Zürich. This marks the beginning of a fifteen year period during which he was a guest lecturer in Zürich, a period which is described in detail in [7]. Lecture courses which Nevanlinna gave in Zürich and in Helsinki formed the basis for his book

*Uniformisierung* (1953). M Heins begins a detailed review of the book with the following paragraph:-

Since the appearance of Weyl's celebrated 'Idee der Riemannschen Fläche'(1913)the theory of abstract Riemann surfaces has experienced a considerable development, especially in the last quarter of a century. In spite of this fact, apart from the monograph of Stoilow, very little has appeared on this subject in book form up to the present. The treatise of Nevanlinna, which is based upon his lectures at Zürich and Helsinki, is the first of a new generation of books to appear on the subject. The objective of the author is to provide an introduction to the theory of abstract Riemann surfaces with uniformization theory being the focal point of interest.

After World War II, during his association with Zürich, his mathematical interests moved to the calculus of variations and applications to physics. For example he wrote the paper *Calculus of variation and partial differential equations* (1967).

Throughout his life Nevanlinna took an active part role in musical events. It was while he was setting up a chamber music society in 1945 that he met the author and translator Sinikka Kallio-Visapaa. After his first marriage had been dissolved, he married Sinikka in 1958. They had a daughter Kristiina.

We have already mentioned the mathematical collaboration between Rolf Nevanlinna and his brother Frithiof. Walter Hayman writes of their collaboration in the 1920s [12]:-

Evenings and Sundays were excellent times for research, and Nevanlinna theory was greatly influenced by Rolf's discussions with Frithiof, which continued all their lives. They would walk up and down each side of a big square, talking mathematics.

They wrote papers together, for example: *Über die Eigenschaften analytischer Funktionen in der Umgebung einer singulären Stelle oder Linie* (1922), *Über die Nullstellen der Riemannschen Zetafunktion* (1924), and *Über die Nullstellen der Riemannschen Zetafunktion* (1925). Years later, they also collaborated in writing the book *Absolute analysis* (1959). An English translation was published in 1973. Nevanlinna coauthored the book *Einführung in die Funktionentheorie* (1965) with V Paatero. A J Lohwater begins a review:-

This book is a masterful account of the traditional first course in complex variables given at Helsinki University, and reflects in its first seven chapters the influence of E Lindelöf's introductory text.

As well as his textbooks and monographs, Nevanlinna published two books and over fifty articles which were intended to introduce mathematical ideas to non-mathematicians. One such example is his book *Raum, Zeit und Relativität* (Space, time and relativity) (1964). An example of his ideas on teaching mathematics, which became of considerable interest to him, is given in his paper *Reform in teaching mathematics* (1966).

Nevanlinna received many honours for his contributions. He was awarded the International Wihuri Prize for Scientists and Artists (1958) and the Henrik Steffens Prize for Nordic culture (1967). He was made an honorary member of the Deutsche Akademie (1938); the Finnish Mathematical Society (1955); the London Mathematical Society (1959); the Swiss Mathematical Society (1962); the Society of Actuaries of Finland (1965); the Teachers of Mathematics and Physics (1965); the Göttingen Academy of Sciences (1967); the Royal Swedish Academy of Sciences (1967); the Danish Academy of Sciences (1967); the German Academy of Scientists Leopoldina (1967); the Hungarian Academy of Sciences (1970); the Institut de France (1967); and the Finnish Academy of Science and Letters (1975). He was awarded honorary degrees by the universities of Heidelberg (1936), Bucharest (1942), Giessen (1952), Berlin (1955), Jyväskylä (1969), Glasgow (1969), Uppsala (1974) and Istanbul (1976). He was president of the International Mathematical Union from 1959 to 1962.

Nevanlinna continued to publish papers right up to the time of his death in 1980 from cancer. Since 1982, an award, the Rolf Nevanlinna Prize, is presented at the International Congress of Mathematicians. The one prize per congress is for young mathematicians dealing with the mathematical aspects of information science.

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