Constantin Carathéodory was a Greek mathematician who spent most of his professional career in Germany. He made significant contributions to the theory of functions of a real variable, the calculus of variations, measure theory, his work includes important results in conformal representations and in the theory of boundary correspondence. In 1909, Carathéodory pioneered the Axiomatic Formulation of Thermodynamics along a purely geometrical approach. Constantin Carathéodory grew up in Brussels, his father Stephanos, a lawyer, served as the Ottoman ambassador to Belgium, St. Petersburg and Berlin, his mother, Despina, née Petrokokkinos, was from the island of Chios. The Carathéodory family from Bosnochori or Vyssa, was well established and respected in Constantinople, its members held many important governmental positions; the Carathéodory family spent 1874–75 in Constantinople, where Constantin's paternal grandfather lived, while his father Stephanos was on leave. In 1875 they went to Brussels when Stephanos was appointed there as Ottoman Ambassador.
In Brussels, Constantin's younger sister Julia was born. The year 1879 was a tragic one for the family since Constantin's paternal grandfather died in that year, but much more tragically, Constantin's mother Despina died of pneumonia in Cannes. Constantin's maternal grandmother took on the task of bringing up Constantin and Julia in his father's home in Belgium, they employed a German maid. Constantin was bilingual in French and Greek by this time. Constantin began his formal schooling at a private school in Vanderstock in 1881, he left after two years and spent time with his father on a visit to Berlin, spent the winters of 1883–84 and 1884–85 on the Italian Riviera. Back in Brussels in 1885 he attended a grammar school for a year where he first began to become interested in mathematics. In 1886, he entered the high school Athénée Royal d'Ixelles and studied there until his graduation in 1891. Twice during his time at this school Constantin won a prize as the best mathematics student in Belgium.
At this stage Carathéodory began training as a military engineer. He attended the École Militaire de Belgique from October 1891 to May 1895 and he studied at the École d'Application from 1893 to 1896. In 1897 a war broke out between Ottoman Greece; this put Carathéodory in a difficult position since he sided with the Greeks, yet his father served the government of the Ottoman Empire. Since he was a trained engineer he was offered a job in the British colonial service; this job took him to Egypt where he worked on the construction of the Assiut dam until April 1900. During periods when construction work had to stop due to floods, he studied mathematics from some textbooks he had with him, such as Jordan's Cours d'Analyse and Salmon's text on the analytic geometry of conic sections, he visited the Cheops pyramid and made measurements which he wrote up and published in 1901. He published a book on Egypt in the same year which contained a wealth of information on the history and geography of the country.
Carathéodory studied engineering in Belgium at the Royal Military Academy, where he was considered a charismatic and brilliant student. 1900 Studies at University of Berlin. 1902 Completed graduation at University of Göttingen 1908 Dozent at Bonn 1909 Ordinary Professor at Hannover Technical High School. 1910 Ordinary Professor at Breslau Technical High School. 1913 Professor following Klein at University of Göttingen. 1919 Professor at University of Berlin 1919 Elected to Prussian Academy of Science. 1920 University Dean at Ionian University of Smyrna. 1922 Professor at University of Athens. 1922 Professor at Athens Polytechnic. 1924 Professor following Lindemann at University of Munich. 1938 Retirement from Professorship. Continued working from Bavarian Academy of Science Carathéodory had about 20 doctoral students among these being Hans Rademacher, known for his work on analysis and number theory, Paul Finsler known for his creation of Finsler space. Carathéodory's contacts in Germany were many and included such famous names as: Minkowski, Klein, Schwarz, Fejér.
During the difficult period of World War II his close associates at the Bavarian Academy of Sciences were Perron and Tietze. While in Germany Carathéodory retained numerous links with the Greek academic world about which detailed information may be found in Georgiadou's book, he was directly involved with the reorganization of Greek universities. An close friend and colleague in Athens was Nicolaos Kritikos who had attended his lectures at Gŏttingen going with him to Smyrna becoming professor at Athens Polytechnic. Kritikos and Carathéodory helped the Greek topologist Christos Papakyriakopoulos take a doctorate in topology at Athens University in 1943 under difficult circumstances. While teaching in Athens University Carathéodory had as undergraduate student Evangelos Stamatis who subsequently achieved considerable distinction as a scholar of ancient Greek mathematical classics. In his doctoral dissertation Carathéodory originated his method based on the use of the Hamilton–Jacobi equation to construct a field of extremals.
The ideas are related to light propagation in optics. The method became known as the royal road to the calculus of variations. More the same idea has been taken into the theory of optimal control; the method can be extended to multiple integrals. He proved an existence theorem for the solution to ordinary differential equations under mild regularity conditio