# Cahit Arf

### Quick Info

Born
11 October 1910
Salonika, Ottoman Empire (now Thessaloniki, Greece)
Died
26 December 1997
Istanbul, Turkey

Summary
Cahit Arf was a Turkish mathematician. He is known for the Arf invariant in topology.

### Biography

Cahit Arf was born in Thessaloniki which, in 1910, was a part of the Ottoman Empire. However, the Balkan War was fought by Serbia, Bulgaria, Greece, and Montenegro against the Ottoman Empire when Arf was two years old. Montenegro declared war on Turkey on 8 October 1912, and the other members of the Balkan league declared war on Turkey 10 days later. The Balkan allies were soon victorious. The Bulgarians defeated the main Ottoman forces, advancing towards Istanbul (then called Constantinople), and the Greeks occupied Thessaloniki. With the outbreak of fighting, Arf's family escaped to Istanbul.

Arf writes in [2] that he went to school in Istanbul at the age of four:-
I never played with other pupils in school. I was shy. Then I continued my education in the Besiktas Sultanisi. After a fire, we left Besiktas and we began to move from one place to another. At last we rented a house in Suleymaniye. Then I transferred to Istanbul Sultanisi. The same thing continued there too. My parents did not let me go out but the school was going well.
In 1919 Arf's family moved again, this time to Ankara, but they returned to Istanbul for a short period before they finally settled in Izmir. Cahit Arf's interest in mathematics was stimulated during his school years in Izmir by a teacher who encouraged him to solve problems in euclidean geometry. In 1926 Arf's father bought French francs when the currency was devalued and it became a cheaper option for the family to send Arf to school in France. His parents [2]:-
... sent me to France to live with my uncle's friends. I registered at the St Louis Lycée there. I did not know much French. All I knew was the French that was taught in school. ... I got the best score in the mathematics exam and that's why I finished that three year lycée in two years but then my father did not have any more francs. I came back to Turkey.
Arf won a scholarship to continued his education in Paris and he returned to France, graduating from the École Normale Supérieure after spending two years there.

Returning to Istanbul to be a school teacher rather than to complete his doctorate, Arf taught at the Galatasaray High School during 1932 [2]:-
I was going to take the place of a teacher from France ... and do the work that he did. My salary was 60 Turkish Liras but the person whom I took the place of was getting paid 600 Turkish Liras.... I worked as an idealist for a year. Some teachers felt sorry for me. They were saying "poor him. ... He is working for 60 Liras". Probably because of that, I lost my idealism. ... At that time reforms were being made in universities so they offered me a position as an assistant professor. I accepted it and then I thought myself that I was successful.
Arf joined the Mathematics Department of Istanbul University. However he decided to continue his study of mathematics [2]:-
In the Lycée, I would ask myself which [geometric problems] could be solved with a ruler and which ones could not. Later, I learned the Galois theorem and then I understood. ... At that time, I was thinking about making a list of the algebraic equations or Galois algebraic equations that could be solved. That was my problem. ... [Jordan] found all the groups that could be solved. He wrote a thick book about that. I tried to read that book ... I could not read books. ... Anyway, I considered this problem as a project. It was only a project. I had done nothing about it yet. While I was busy with all these ideas, time passed. ... I thought that I could not deal with this project in Istanbul so I obtained permission from the university and went to Göttingen.
In 1937 he went to the University of Göttingen to study for his doctorate under the supervision of Helmut Hasse. He completed his doctoral studies in 1938 obtaining, among other results, the theorem now known as the Hasse-Arf theorem. He had studied at Göttingen through the very difficult period leading up to World War II but Hasse asked him to remain there for another year to continue his work and during this period Arf's work produced what are today called the Arf invariants.

Arf returned from Germany to Istanbul University where he worked until 1962. He was promoted to professor in 1943 and to ordinarius professor in 1955. During this period he spent one year as a visiting professor at the University of Maryland and was honoured by being elected a corresponding member of the Mainz Academy.

In 1963 Arf taught at Robert College in Istanbul. Then, between 1964 and 1966, Arf worked at the Institute for Advanced Study at Princeton in the United States. After remaining in the United States for a further year spent visiting the University of California at Berkeley, he returned to Turkey in 1967 and joined the Middle East Technical University in Ankara. Arf retired in 1980 and lived in Istanbul.

Arf played a prominent role in establishing TUBITAK in 1971, the Scientific and Technical Research Council of Turkey. He served as its president for many years from the time that it was established. From 1985 until 1989 he was the president of the Turkish Mathematical Society.

Arf received many awards and prizes for his outstanding contributions to mathematics and for his most distinguished career including the Inonu award. Among the honours he received were honorary doctorates from the Black Sea Technical University, the Middle East Technical University and Istanbul Technical University.

Arf contributed to the education of many of the present day mathematicians in Turkey, not only by his lectures but also through illuminating discussions in conferences and seminars. Those who had the opportunity to come into close contact with Arf, were deeply influenced by his sincere devotion to mathematics and to science in general. Especially keen to help young mathematicians, he gave them very sound advice and generous encouragement. Arf's approach to mathematics was described by M G Ikeda [1]:-
To every problem, he has his own idea of approach. The characteristic of his approach is thoroughness; he always seeks invariants, and prefers explicit constructions rather than combination of existing theories. Once he determines his approach, he energetically tackles the problem and never gives up until he achieves his aim. If one studies Cahit Arf's works, which are full of originality and painstaking computations, one will surely wonder where Professor Arf gets his inspirations, and how he gets insight into most complicated computations.
Much of Arf's most important work was in algebraic number theory and he invented Arf invariants which have many applications in topology. His early work was on quadratic forms in fields, particularly fields of characteristic 2. His name is not only attached to Arf invariants but he is also remembered for the Hasse-Arf Theorem which plays an important role in class field theory and in Artin's theory of $L$-functions. In ring theory, Arf rings are named after him.

In addition Arf worked in applied mathematics writing several papers on elastic plane bodies bounded by free boundaries and a paper on the algebraic structure of the cluster expansion in statistical mechanics. Arf described this digression into applied mathematics had been done for the wrong reasons [2]:-
I looked for applause. That's why I talked with engineers and tried to understand their work. ... Mustafa Inan was given a problem while he was making his doctorate. A bridge had collapsed in Belgium. The reason was not known. ... [Mustafa] built a model of that bridge from some material, he put loads on it and found the spots where it began to crack. It was possible to see the places where the tensions increased on that material. ...I took that problem ... I gave the formulas that would build that kind of profiles. ... I wrote five or six essays, which completed each other, about that problem. I was applauded ... but to do things for applause is not nice.
Arf presented a paper On a generalization of Green's formula and its application to the Cauchy problem for a hyperbolic equation to the volume Studies in mathematics and mechanics presented to Richard von Mises in 1954. Arf had met von Mises in 1933 in Istanbul.

An International Symposium on Algebra and Number Theory was held in Arf's honour in Silivri from 3 to 7 September 1990. At an earlier conference on Rings and Geometry held in Istanbul in 1984, Arf had presented a paper The advantage of geometric concepts in mathematics.

Arf died after a heart attack and was buried in Istanbul, following a ceremony at Istanbul University.

### References (show)

1. C Arf, The collected papers of Cahit Arf : published on the occasion of his 80th birthday (Turkish Mathematical Society, Ankara, 1991).
2. C Arf, Speech given on 13 September 1980 (unpublished).
3. A Dvnmez, Matematik tarihi (Turkish) (Ankara, 1986).