Constantin Carathéodory (; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He is the author of significant contributions to real and complex analysis, the calculus of variations, and measure theory and of an axiomatic formulation of thermodynamics. Carathéodory is considered one of the greatest mathematicians of his era and the most renowned Greek mathematician since antiquity.
Origins
Constantin Carathéodory was born in 1873 in Berlin to Greek parents and grew up in Brussels. His father , a lawyer, served as the Ottoman ambassador to Belgium, St. Petersburg and Berlin. His mother, Despina, , was from the island of Chios. The Carathéodory family, originally from Bosna, was well established and respected in Constantinople, and its members held many important governmental positions. His grandfather, the Ottoman Greek physician Constantinos Caratheodory, was the personal doctor to Sultan Abdülmecit I.
The Carathéodory family spent 1874–75 in Constantinople, where Constantin's paternal grandfather lived, while his father Stephanos was on leave. Then 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 who taught the children to speak German. Constantin was already 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 then spent time with his father on a visit to Berlin, and also 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 Belgian Royal Military Academy from October 1891 to May 1895 and he also studied at the École d'Application from 1893 to 1896. In 1897 a war broke out between the Ottoman Empire and 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 also visited the Cheops pyramid and made measurements which he wrote up and published in 1901.Brussells 1901 (Hayez);Ges. math. Schr. V. 273-281 He also published a book on Egypt in the same year which contained a wealth of information on the history and geography of the country.H Aigyptos, Syllogos Ophelimon Biblion, no 14, 118 pp Athens 1901, 1928, New York 1920
Studies and university career
Carathéodory studied engineering in Belgium at the Royal Military Academy, where he was considered a charismatic and brilliant student.
University career
1900 Studies at Humboldt University of Berlin. 1902 Completed graduation at University of Göttingen (1904 Ph.D., 1905 Habilitation) 1908 Dozent at University of 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 Humboldt University of Berlin 1919 Elected to Prussian Academy of Science. 1920 University Dean at Ionian University of Smyrna (later, University of the Aegean). 1922 Professor at University of Athens. 1922 Professor at Athens Polytechnic. 1924 Professor following Lindemann at Ludwig-Maximilians-Universität München (LMU). 1938 Retirement from professorship. Continued working at Bavarian Academy of Science
Doctoral students Carathéodory had about 20 doctoral students among these being Hans Rademacher, known for his work on analysis and number theory, and Paul Finsler known for his creation of Finsler space.
Academic contacts in Germany Carathéodory had numerous contacts in Germany. They included such famous names as Hermann Minkowski, David Hilbert, Felix Klein, Albert Einstein, Edmund Landau, Hermann Amandus Schwarz, and Lipót Fejér. During the difficult period of World War II, his close associates at the Bavarian Academy of Sciences were Perron and Tietze.
Einstein, then a member of the Prussian Academy of Sciences in Berlin, was working on his general theory of relativity when he contacted Carathéodory for clarifications on the Hamilton-Jacobi equation and canonical transformations. He wanted to see a satisfactory derivation of the former and the origins of the latter. Einstein told Carathéodory his derivation was "beautiful" and recommended its publication in the Annalen der Physik. Einstein employed the former in a 1917 paper titled Zum Quantensatz von Sommerfeld und Epstein (On the Quantum Theorem of Sommerfeld and Epstein). Carathéodory explained some fundamental details of the canonical transformations and referred Einstein to E.T. Whittaker's Analytical Dynamics. Einstein was trying to solve the problem of "closed time-lines" or the geodesics corresponding to the closed trajectory of light and free particles in a static universe, which he introduced in 1917.
Landau and Schwarz stimulated his interest in the study of complex analysis.
Academic contacts in Greece While in Germany, Carathéodory retained numerous links with the Greek academic world, details of which can be found in Georgiadou's book. He was directly involved with the reorganization of Greek universities. An especially close friend and colleague in Athens was Nicolaos Kritikos who had attended his lectures at Göttingen, later going with him to Smyrna, then 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 very difficult circumstances. While teaching at Athens University, Carathéodory had Evangelos Stamatis as an undergraduate student, who subsequently achieved considerable distinction as a scholar of ancient Greek mathematical classics.J P Christianidis & N Kastanis: In memoriam Evangelos S Stamatis (1898–1990) Historia Mathematica 19 (1992) 99-105
Works Calculus of variations In his doctoral dissertation, Carathéodory showed how to extend solutions to discontinuous cases and studied isoperimetric problems.
Previously, between the mid-1700s to the mid-1800s, Leonhard Euler, Adrien-Marie Legendre, and Carl Gustav Jacob Jacobi were able to establish necessary but insufficient conditions for the existence of a strong relative minimum. In 1879, Karl Weierstrass added a fourth that does indeed guarantee such a quantity exists. Carathéodory constructed his method for deriving sufficient conditions based on the use of the Hamilton–Jacobi equation to construct a field of extremals. The ideas are closely related to light propagation in optics. The method became known as Carathéodory's method of equivalent variational problems or the royal road to the calculus of variations.H. Boerner, Carathéodory und die Variationsrechnung, in A Panayotopolos (ed.), Proceedings of C. Carathéodory International Symposium, September 1973, Athens (Athens, 1974), 80–90. A key advantage of Carathéodory's work on this topic is that it illuminates the relation between the calculus of variations and partial differential equations. It allows for quick and elegant derivations of conditions of sufficiency in the calculus of variations and leads directly to the Euler-Lagrange equation and the Weierstrass condition. He published his Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung (Calculus of Variations and First-order Partial Differential Equations) in 1935.
More recently, Carathéodory's work on the calculus of variations and the Hamilton-Jacobi equation has been taken into the theory of optimal control and dynamic programming.Bellman for his Dynamic programming in its continuous-time form used Carathéodory's work in the form of the Hamilton–Jacobi–Bellman equation. Kálmán also explicitly used Carathéodory's formulation in his initial papers on optimal control. See e.g. R. E. Kalman: Contributions to the theory of optimal control. Boletin de la Sociedad Matematica Mexicana 1960
Convex geometry Carathéodory's theorem in convex geometry states that if a point of lies in the convex hull of a set , then can be written as the convex combination of at most points in . Namely, there is a subset of consisting of or fewer points such that lies in the convex hull of . Equivalently, lies in an -simplex with vertices in , where . The smallest that makes the last statement valid for each in the convex hull of P is defined as the Carathéodory's number of . Depending on the properties of , upper bounds lower than the one provided by Carathéodory's theorem can be obtained.
He is credited with the authorship of the Carathéodory conjecture claiming that a closed convex surface admits at least two umbilic points. The conjecture was proven in 2024 by Brendan Guilfoyle and Wilhelm Klingenberg.
Real analysis He proved an existence theorem for the solution to ordinary differential equations under mild regularity conditions.
Another theorem of his on the derivative of a function at a point could be used to prove the Chain Rule and the formula for the derivative of inverse functions.
Complex analysis He greatly extended the theory of conformal transformationA. Shields: Carathéodory and Conformal Mapping Math. Intelligencer vol.10(1), 1988 proving his theorem about the extension of conformal mapping to the boundary of Jordan domains. In studying boundary correspondence he originated the theory of prime ends. He exhibited an elementary proof of the Schwarz lemma.
Carathéodory was also interested in the theory of functions of multiple complex variables. In his investigations on this subject he sought analogs of classical results from the single-variable case. He proved that a ball in is not holomorphically equivalent to the bidisc.
Theory of measure He is credited with the Carathéodory extension theorem which is fundamental to modern measure theory. Later Carathéodory extended the theory from sets to Boolean algebras.
Thermodynamics