Cahit Arf (1910, Thessaloniki – 26 December 1997, Istanbul) was a mathematician and humanist. He played a key role in the establishment of TÜBİTAK and served as its first Scientific Council chairman. He received the İnönü Prize in 1948 and the TÜBİTAK Science Award in 1974. In 1980 he received an honorary doctorate from Istanbul University and Karadeniz Technical University, and in 1981 from Middle East Technical University. In addition, in 1994 he was awarded the title of Commandeur des Palmes Académiques in France. Since 2009, Arf’s portrait has appeared on the reverse of the 10 Turkish lira banknotes.

Image of 10 Turkish Liras.

Early Life and Education

Cahit Arf was born in 1910 in Thessaloniki, then part of the Ottoman Empire. During the Balkan Wars, his family moved first to Istanbul and then to İzmir. He completed his primary education at the lowest level of secondary schools then known as “sultani.” A young teacher he encountered in fifth grade further strengthened his interest in mathematics. During his secondary education, his ability to solve problems that his peers could not easily solve drew the attention of both his teachers and family. As a result, he was sent to St. Louis High School in Istanbul to continue his education. He completed his three-year high school curriculum in just two years before returning to Turkey. In the same year, he passed a national examination organized by the Ministry of National Education to select students for overseas education and was accepted to École Normale Supérieure in France.

Career and Achievements

After completing his higher education, Cahit Arf returned to Turkey and taught mathematics at Galatasaray High School. In 1933 he began his academic career at the Mathematics Chair of Istanbul University and conducted research as an associate professor candidate. In 1937 he went to Germany to pursue his doctorate at the University of Göttingen, where he earned his doctorate for research on “non-commutative Class Field.” A significant portion of this work later became known in mathematical literature as the “Hasse-Arf Theorem.”

In 1938 Arf returned to Turkey and continued his academic rise at Istanbul University, becoming a professor in 1943 and a full professor in 1955. He taught at this university until 1962, after which he gave lectures at Robert College. Between 1964 and 1966 he held visiting scholar positions at the Institute for Advanced Study in Princeton and later at the University of California. In 1967 he returned to Turkey and joined the Department of Mathematics at Middle East Technical University, where he continued his research until his retirement in 1980.

Research and Contributions to Mathematical Science

Cahit Arf’s work focused primarily on algebra, number theory, differential equations, and mathematical analysis.

• Arf Theorem: Arf’s most famous contribution is the theorem bearing his name, the Arf theorem. This theorem is related to algebraic number theory and plays a significant role in understanding various mathematical structures. In particular, it has found broad application in the study of number theory, prime numbers, and their relationships. In this theorem, Arf demonstrated how a new structure can arise from the combination of two different types of algebraic structures.
• Arf Rings and Arf Algebras: Cahit Arf also made important contributions to the study of algebraic structures. Arf rings are a special structure widely used to solve numerous problems in linear algebra and algebraic geometry. These algebraic structures provide a framework for applying mathematical theory to more complex problems. In addition, Arf algebras have made significant contributions to research in abstract algebraic structures.
• Arf Numbers: Arf also contributed to number theory. Arf numbers are a concept defining a specific class of prime numbers. These numbers are widely recognized for helping solve many mathematical problems in number theory and modular arithmetic. Due to their theoretical and practical utility in addressing mathematical problems, Arf numbers hold an important place in mathematics.
• Work on Differential Equations: Arf conducted numerous studies on differential equations, which have extensive applications in physics, engineering, and other sciences. In this field, he developed new methods for solving both linear and nonlinear differential equations. His work enriched theoretical knowledge in differential equations and shed light on key problems in applied mathematics.
• Number Theory and Modular Arithmetic: Cahit Arf conducted in-depth research in more abstract areas of mathematics such as number theory and modular arithmetic. These theoretical advances include various mathematical techniques developed to understand the properties of large numbers and prime numbers. As a subfield of number theory, modular arithmetic forms the foundation of cryptography and encryption techniques. Arf’s work in this area enriched fundamental tools in mathematical analysis.
• Work on Mathematical Analysis and Functional Analysis: Arf also made significant contributions to functional analysis, a tool used to examine mathematical structures in greater depth, particularly focusing on concepts such as spaces and linear operators. In this field, Arf authored numerous important theoretical developments that made mathematical theories more comprehensible and applicable.
• Work on Topology and Geometry: He also conducted research in topology and geometry, mathematical disciplines that study the properties of shapes and spaces. Arf carried out in-depth studies in areas such as algebraic topology and differential geometry, developing new methods to understand the relationships between mathematical structures and spaces.
• Contributions to Mathematical Education and Teaching: In addition to his theoretical contributions, Cahit Arf made major contributions to the popularization of mathematical education. His method of teaching mathematics to students and researchers is considered one of his most important legacies to science. He developed various educational materials aimed at fostering mathematical thinking and enhancing students’ ability to think abstractly.

Personal Life and Death

In addition to mathematics, Cahit Arf was interested in music and had artistic inclinations such as playing the violin. In his free time, he enjoyed spending time in nature and reflecting on philosophical ideas. In his later years, Arf further solidified his position within Turkey’s mathematical community and continued his research. However, due to health problems, he was forced to retire from teaching. He passed away in Istanbul on 26 December 1997.

After Arf’s death, his mathematical contributions continued to be recognized internationally. His students and colleagues undertook various projects to honor and publish his work. Moreover, Arf’s contributions to mathematical thought have been incorporated into the curricula of many universities.