# Arf, Cahit

# ARF, CAHIT

(*b*. Selanik, Ottoman Empire [later Thessaloníki, Greece], 11 October 1910; *d*. Istanbul, Turkey, 26 December 1997)

*mathematics*,*algebra, algebraic number theory*.

Arf was the leading Turkish mathematician of the twentieth century. His research was mainly in algebraic number theory and related fields, although he also contributed to elasticity theory and analysis. An invariant for quadratic forms over a field of characteristic 2 introduced by Arf in his early works has, as it turned out, important applications in algebraic and differential topology. This invariant appears in mathematics literature as the Arf invariant. Concepts such as Arf rings, Arf closure, and Arf characters also carry his name. Arf was one of the founding members of the Turkish Mathematical Society in 1948 and played a major role in the establishment of the Scientific and Technical Research Council of Turkey (TÜBITAK) in 1963.

The story of Arf’s childhood runs parallel to the history of the turbulent final years of the Ottoman Empire. With the outbreak of the Balkan War in 1912, his middle-class family had to migrate from Selanik to Istanbul. The Ottoman Empire entered World War I as one of the Central Powers, which lost the war. Istanbul was occupied by an Entente force, and a Greek army landed in Izmir in 1919. The Ottoman government was compelled to sign the Treaty of Sèvres in 1920. Immediately afterwards a national assembly convened in Ankara, led by Mustafa Kemal, refused to accept the terms of the Sèvres treaty. Arf’s family went from Istanbul to Ankara, via Kastamonu. The Ankara government assigned his father to reorganize the postal services in Adana after this southern city was retaken from the French. From Adana, Arf’s family came back to Ankara and returned to Istanbul after the war between Turkey and Greece (May 1919–August 1922) ended. Finally, the family settled in Izmir. The Ottoman Empire was dissolved and Turkey was declared a republic in 1923.

Arf’s extraordinary gift for mathematics was discovered by his schoolteacher in Izmir. The teacher encouraged Arf by regularly asking him to produce his own proofs of classical theorems in geometry without consulting books. His father bought French francs when that currency was devalued in order to send his son to France. Arf went to Paris and graduated from the Lycée St.-Louis. With a scholarship he continued his education in that city at the École Normale Supérieure. After his graduation, he refused offers to continue towards a doctorate, because he wanted above all to return to Turkey and be a schoolteacher.

Although Arf asked to be assigned as a mathematics teacher to a school in the provincial city of Kastamonu, the Ministry of Education instead appointed him to a prominent school, Galatasaray, in Istanbul. In 1933 he entered Istanbul University and decided to pursue an academic career in mathematics as an assistant professor. In 1937 he went to Göttingen University in Germany for his doctorate. His supervisor was Helmut Hasse. Because Arf was already mathematically mature, he completed his doctoral studies in 1938. The main result of his thesis was later known as the Hasse-Arf theorem. Upon advice of Hasse, he stayed in Göttingen during the difficult period leading to World War II, studying quadratic forms over a field of characteristic 2. He introduced a complete invariant for such forms, which is known as the Arf invariant in the literature. This invariant turned out to be very important in algebraic and differential topology.

From Göttingen, Arf returned to Istanbul University in 1939, where he worked until 1962. He continued his research on invariants of certain algebraic structures over fields of characteristic 2, worked on multiplicity sequences of algebraic branches, and published a series of papers on elasticity theory. Arf was promoted to professor in 1943 and to ordinarius professor in 1955. He spent a year (1951) at the University of Maryland and was elected a corresponding member of the Mainz Academy in Germany.

After leaving Istanbul University in 1962, he taught at Robert College in Istanbul, spent two years (1964– 1966) at the Institute for Advanced Study in Princeton, New Jersey, and then one year at the University of California at Berkeley. Upon his final return to Turkey in 1967, he entered the recently established Middle East Technical University in Ankara, where he worked until retiring in 1980.

Arf generally avoided administrative duties in his academic career, although he served as the president of TÜBITAK from 1967 until 1971 and was the president of the Turkish Mathematical Society from 1985 until 1989. When the government tried to impose stricter control over Middle East Technical University in 1977, Arf— who believed in autonomy for Turkish universities—led a group of professors in opposition. His influence on Turkish mathematics was profound. He was a constant source of inspiration and encouragement, especially for younger mathematicians.

Robert Langlands was a young mathematician when he was visiting Middle East Technical University from 1967–1968. Langlands’ year at Ankara was quite decisive for his own research, in particular through his contact with Cahit Arf. Arf had pointed out to him a paper by Helmut Hasse who had proved the first results in the direction that Langlands was working at that time.

Arf received numerous awards for his contributions to mathematics and for his stance in support of scientific excellence and academic freedom. Among them are the Inönü Award (1948), the TÜBITAK Science Award (1974), and the Commandeur des Palmes Académiques (1994). He was an honorary member of the Turkish Academy of Sciences and received honorary doctorates from Black Sea Technical University, Middle East Technical University, and Istanbul Technical University. To commemorate his legacy, Middle East Technical University instituted the Cahit Arf Lectures in 2001.

## BIBLIOGRAPHY

*A complete bibliography of Arf’s works is included in* The Collected Papers of Cahit Arf, *cited below*.

### WORKS BY ARF

“Untersuchungen über quadratische Formen in Körpern der Charakteristik 2” (Research on quadratic forms over fields of characteristic 2). *Journal für die reine und angewante Mathematik* 183 (1941): 148–167.

“Une interprétation algébrique de la suite des ordres de multiplicité d’une branche algébrique” (An algebraic interpretation of the multiplicity orders of an algebraic branch).*Proceedings of the London Mathematical Society* (2) 50 (1948): 256–287.

“On the Determination of Multiply Connected Domains of an Elastic Plane Body, Bounded by Free Boundaries with Constant Tangential Stresses.” *American Journal of Mathematics* 74 (1952): 797–820.

*The Collected Papers of Cahit Arf*. Edited by Tosun Terzioğlu. Ankara: Turkish Mathematical Society, 1990.

### OTHER SOURCES

Ikeda, Masatoshi G. “Cahit Arf’s Contribution to Algebraic Number Theory and Related Fields.” *Turkish Journal of Mathematics* 22 (1998): 1–14.

Langlands, Robert. “Benim tanidigim Cahit Arf” (Recollections of a year in Turkey with Cahit Arf). 2004. Available from http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/intro.html.

Middle East Technical University. Department of Mathematics. “Cahit Arf Lectures. Available from http://www.math.metu.edu.tr/arflectures

O’Connor, John J., and E. F. Robertson. “Cahit Arf.” *MacTutor History of Mathematics*. September 1998. Available from http://www-history.mcs.st-andrews.ac.uk/biographies/arf.html

Roquette, Peter J. “Introduction of Langlands at the Arf Lecture.” 2004. Available from http://www.rzuser.uniheidelberg.de/ci3.

Terzioglu, Tosun. *Cahit Arf*. Ankara: Middle East Technical University, 1981.

———, and Akin Yilmaz, eds. *“Anlamak” Tutkunu Bir Matemakçi Cahit Arf*(Life story of Cahit Arf). Ankara: Türkiye Bilimler Akademisi, 2005.

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