Constantin Carathéodory
Encyclopedia
Constantin Carathéodory (or Constantine Karatheodori) (Greek
: Κωνσταντίνος Καραθεοδωρή) (13 September 1873 – 2 February 1950) was a Greek
mathematician
. He made significant contributions to the theory of functions of a real variable
, the calculus of variations
, and measure theory. His work also 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.
to Greek
parents and grew up in Brussels
, where his father served as the Ottoman
ambassador to Belgium
. The Carathéodory family, originally from Bosnochori or Vyssa
, was well established and respected in Constantinople
, and its members held many important governmental positions.
The Carathéodory family spent 1874-75 in Constantinople, where Constantin's paternal grandfather lived, while 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 1895 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 École Militaire de Belgique 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 Turkey 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. 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.
at the Royal Military Academy
, where he was considered a charismatic and brilliant student.
University Career:
1900 Studies at University of Berlin.
1902 Completed graduation at University of Göttingen (1904 Ph.D, 1905 Habilitation)
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
(later, University of the Aegean
).
1922 Professor at University of Athens.
1922 Professor at Athens Polytechnic.
1924 Professor following Lindeman at University of Munich.
1938 Retirement from Professorship. Continued working from 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's contacts in Germany were many and included such famous names as: Minkowski
, Hilbert
, Klein
, Einstein, Schwarz, Fejér
. During the difficult period of World War II his close associates at the Bavarian Academy of Sciences were Perron and Tietze.
Academic contacts in Greece: 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 especially close friend and colleague in Athens was Nicolaous Kritikos who had attended his lectures at Gŏttingen, later going with him to Smyrna, then becoming professor at Athens Polytechnic. With Carathéodory he helped the famous topologist Christos Papakyriakopoulos
take a doctorate in topology at Athens University in 1943 under very 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.
. The method can also be extended to multiple integrals.
Real Analysis: He proved an existence theorem
for the solution to ordinary differential equations under mild regularity conditions.
Theory of measure: He is credited with the Carathéodory extension theorem which is fundamental to modern set theory. Later Carathéodory extended the theory from sets to Boolean algebras.
Theory of functions of a complex variable: He greatly extended the theory of conformal transformation 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 end
s.
Thermodynamics: In 1909, Carathéodory published a pioneering work "Investigations on the Foundations of Thermodynamics" in which he formulated the Laws of Thermodynamics axiomatically. It has been said that he was using only mechanical concepts and the theory of Pfaff's differential forms. But in reality he also relied heavily on the concept of an adiabatic process. The physical meaning of the term adiabatic rests on the concepts of heat and temperature. Thus, in Bailyn's survey of thermodynamics, Carathéodory's approach is called "mechanical", as distinct from "thermodynamic". Carathéodory's "first axiomatically rigid foundation of thermodynamics" was acclaimed by Max Planck
and Max Born
. In his theory he simplified the basic concepts, for instance heat is not an essential concept but a derived one. He formulated the axiomatic principle of irreversibility in thermodynamics stating that inaccessibility of states is related to the existence of entropy, where temperature is the integration function. The Second Law of Thermodynamics
was expressed via the following axiom: "In the neighbourhood of any initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state." In this connexion he coined the term adiabatic accessibility
.
Optics: Carathéodory's work in optics
is closely related to his method in the calculus of variations. In 1926 he gave a strict and general proof, that no system of lenses and mirrors can avoid aberration
, except for the trivial case of plane mirrors.
In his later work he gave the theory of the Schmidt telescope.
Historical: During the Second World War Carathéodory edited two volumes of Euler's Complete Works dealing with the Calculus of Variations which were submitted for publication in 1946.
A conjecture: He is credited with the authorship of the Carathéodory conjecture
claiming that a closed convex surface admits at least two umbilic points. As of 2007, this conjecture remained unproven despite having attracted a large amount of research.
See also
he submitted a plan on 20 October 1919 for the creation of a new University at Smyrna
in Asia Minor, to be named Ionian University. In 1920 Carathéodory was appointed Dean of the University and took a major part in establishing the institution, touring Europe to buy books and equipment. The university however never actually admitted students due to the War in Asia Minor
which ended in the Great Fire of Smyrna
. Carathéodory managed to save books from the library and was only rescued at the last moment by a journalist who took him by rowing boat to the battleship Naxos which was standing by. The present day University of the Aegean
claims to be a continuation of Carathéodory's original plan.
Carathéodory brought to Athens some of the university library and stayed in Athens, teaching at the university and technical school until 1924.
In 1924 Carathéodory was appointed professor of mathematics at the University of Munich, and held this position until retirement in 1938. He afterwards worked from the Bavarian Academy of Sciences until his death in 1950.
and French
were his first languages, and he mastered German
with such perfection, that his writings composed in the German language are stylistic masterworks. Carathéodory also spoke and wrote English
, Italian
, Turkish
, and the ancient languages without any effort. Such an impressive linguistic arsenal enabled him to communicate and exchange ideas directly with other mathematicians during his numerous travels, and greatly extend his fields of knowledge.
Much more than that, Carathéodory was a treasured conversation partner for his fellow professors in the Munich Department of Philosophy. The well-respected, German philologist
, professor of ancient languages Kurt von Fritz praised Carathéodory, saying that from him one could learn an endless amount about the old and new Greece, the old Greek language, and Hellenic mathematics. Fritz had an uncountable number of philosophical discussions with Carathéodory. Deep in his heart, Carathéodory felt himself Greek above all. The Greek language was spoken exclusively in Carathéodory's house – his son Stephanos and daughter Despina went to a German high school, but they obtained daily additional instruction in Greek language and culture from a Greek priest. At home, they were not allowed to speak any other language.
Known correspondence Carathéodory-Einstein can be seen as facsimile in Einstein Archives Online (11 items). Three letters concern mathematics and these are printed in vol.8 of Einstein's Collected Works (Princeton Univ. Press 1987)
The Greek authorities intended for a long time to create a museum honoring Karatheodoris in Komotini
, a major town of the northeastern Greek region which is close to where his family came from. On 21 March 2009 the museum "Karatheodoris"(Καραθεοδωρής) opened its gates to the public, in Komotini
.,
The coordinator of the Museum, Athanasios Lipordezis (Αθανάσιος Λιπορδέζης), noted that the museum gave home to original manuscripts of the mathematician of about 10,000 pages including correspondence of Carathéodory with the German mathematician Arthur Rozenthal for the algebraization of measure. Also visitors can view at the showcases the books " Gesammelte Mathematische Schriften Band 1,2,3,4 ", "Mass und Ihre Algebraiserung", " Reelle Functionen Band 1", " Zahlen/Punktionen Funktionen " and many more. Handwritten letters of C.Carathéodory to Albert Einstein
, Hellmuth Kneser
and photographs of the Carathéodory family are on display.
The effort to furnish the museum with more exhibits is continuous.
Conformal Representation, Cambridge 1932 (Cambridge Tracts in Mathematics and Physics)
Geometrische Optik, Berlin, 1937
Elementare Theorie des Spiegelteleskops von B. Schmidt (Elementary Theory of B. Schmidt's Reflecting Telescope), Leipzig Teubner, 1940 36 pp.; Ges. math. Schr. II 234-279
Functionentheorie I, II, Basel 1950, 1961 (Birkhäuser). English translation: Theory of Functions of a Complex Variable, 2 vols, New York, Chelsea Publishing Company, 1954
Mass und Integral und Ihre Algebraisierung, Basel 1956. English translation, Measure and Integral and their Algebraisation, New York, Chelsea Publishing Company, 1963
Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Leipzig, 1935. English translation, Calculus of Variations and Partial Differential Equations of the First Order, New York, Chelsea Publishing Company, 1965.
Gesammelte Mathematische Schriften München 1954-7 (Beck) I-V.
All of Carathéodory's books are written in a beautiful and lucid style; they have been studied by generations of mathematicians, and still being studied to great benefit. Carathéodory's books are unusual in the extent to which geometry is used in the exposition.
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...
: Κωνσταντίνος Καραθεοδωρή) (13 September 1873 – 2 February 1950) was a Greek
Greeks
The Greeks, also known as the Hellenes , are a nation and ethnic group native to Greece, Cyprus and neighboring regions. They also form a significant diaspora, with Greek communities established around the world....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He made significant contributions to the theory of functions of a real variable
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
, the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, and measure theory. His work also 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.
Origins
Constantin Carathéodory was born in BerlinBerlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
to Greek
Greeks
The Greeks, also known as the Hellenes , are a nation and ethnic group native to Greece, Cyprus and neighboring regions. They also form a significant diaspora, with Greek communities established around the world....
parents and grew up in Brussels
Brussels
Brussels , officially the Brussels Region or Brussels-Capital Region , is the capital of Belgium and the de facto capital of the European Union...
, where his father served as the Ottoman
Ottoman Empire
The Ottoman EmpireIt was usually referred to as the "Ottoman Empire", the "Turkish Empire", the "Ottoman Caliphate" or more commonly "Turkey" by its contemporaries...
ambassador to Belgium
Belgium
Belgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...
. The Carathéodory family, originally from Bosnochori or Vyssa
Vyssa
Vyssa is a former municipality in the Evros peripheral unit, East Macedonia and Thrace, Greece. Since the 2011 local government reform it is part of the municipality Orestiada, of which it is a municipal unit. Population 8,184 . The seat of the municipality was in Nea Vyssa. It is named after a...
, was well established and respected in Constantinople
Constantinople
Constantinople was the capital of the Roman, Eastern Roman, Byzantine, Latin, and Ottoman Empires. Throughout most of the Middle Ages, Constantinople was Europe's largest and wealthiest city.-Names:...
, and its members held many important governmental positions.
The Carathéodory family spent 1874-75 in Constantinople, where Constantin's paternal grandfather lived, while 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 1895 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
Pneumonia
Pneumonia is an inflammatory condition of the lung—especially affecting the microscopic air sacs —associated with fever, chest symptoms, and a lack of air space on a chest X-ray. Pneumonia is typically caused by an infection but there are a number of other causes...
in Cannes
Cannes
Cannes is one of the best-known cities of the French Riviera, a busy tourist destination and host of the annual Cannes Film Festival. It is a Commune of France in the Alpes-Maritimes department....
. 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
Italian Riviera
The Italian Riviera, or Ligurian Riviera is the narrow coastal strip which lies between the Ligurian Sea and the mountain chain formed by the Maritime Alps and the Apennines...
. 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 also studied at the École d'Application from 1893 to 1896. In 1897 a war broke out
Greco-Turkish War (1897)
The Greco-Turkish War of 1897, also called the Thirty Days' War and known as the Black '97 in Greece, was a war fought between the Kingdom of Greece and Ottoman Empire. Its immediate cause was the question over the status of the Ottoman province of Crete, whose Greek majority long desired union...
between Turkey 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
Camille Jordan
Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
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. 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.
Studies and University Career
Carathéodory studied engineering in BelgiumBelgium
Belgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...
at the Royal Military Academy
Royal Military Academy (Belgium)
The Royal Military Academy is the military university of Belgium. The school is responsible for the education of the officers of the four components of the Belgian defence . The school is located in Brussels in a building construct by the architects Henri Maquet and Henri Van Dievoet...
, where he was considered a charismatic and brilliant student.
University Career:
1900 Studies at University of Berlin.
1902 Completed graduation at University of Göttingen (1904 Ph.D, 1905 Habilitation)
1908 Dozent at Bonn
University of Bonn
The University of Bonn is a public research university located in Bonn, Germany. Founded in its present form in 1818, as the linear successor of earlier academic institutions, the University of Bonn is today one of the leading universities in Germany. The University of Bonn offers a large number...
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
Ionian University of Smyrna
The Ionian University of Smyrna was a university established by the local Greek authorities during the Greek Occupation of Smyrna , today Izmir, Turkey. The initiative for the organization of the institution was undertaken by the mathematician Constantin Carathéodory...
(later, University of the Aegean
University of the Aegean
The University of the Aegean is a state, multi-campus university located in Mytilene, Greece. The university was officially founded in 1984, although its historical roots date back to the early 1920s...
).
1922 Professor at University of Athens.
1922 Professor at Athens Polytechnic.
1924 Professor following Lindeman at University of Munich.
1938 Retirement from Professorship. Continued working from Bavarian Academy of Science
Doctoral students: Carathéodory had about 20 doctoral students among these being Hans Rademacher
Hans Rademacher
Hans Adolph Rademacher was a German mathematician, known for work in mathematical analysis and number theory.-Biography:...
, known for his work on analysis and number theory, and Paul Finsler
Paul Finsler
Paul Finsler was a German and Swiss mathematician.Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at the University of Göttingen, where he received his Ph.D. in 1919 under the supervision of Constantin Carathéodory...
known for his creation of Finsler space.
Academic contacts in Germany: Carathéodory's contacts in Germany were many and included such famous names as: Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
, Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
, Einstein, Schwarz, Fejér
Lipót Fejér
Lipót Fejér , was a Hungarian mathematician. Fejér was born Leopold Weiss, and changed to the Hungarian name Fejér around 1900....
. During the difficult period of World War II his close associates at the Bavarian Academy of Sciences were Perron and Tietze.
Academic contacts in Greece: 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 especially close friend and colleague in Athens was Nicolaous Kritikos who had attended his lectures at Gŏttingen, later going with him to Smyrna, then becoming professor at Athens Polytechnic. With Carathéodory he helped the famous topologist Christos Papakyriakopoulos
Christos Papakyriakopoulos
Christos Dimitriou Papakyriakopoulos, commonly known as "Papa" , was a Greek mathematician specializing in geometric topology. He worked in isolation at Athens University being awarded a Ph.D on the recommendation of Carathéodory...
take a doctorate in topology at Athens University in 1943 under very 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.
Works
Calculus of Variations: 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 closely related to light propagation in optics. The method became known as the royal road to the calculus of variations. More recently the same idea has been taken into the theory of optimal controlOptimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
. The method can also be extended to multiple integrals.
Real Analysis: He proved an existence theorem
Carathéodory's existence theorem
In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem...
for the solution to ordinary differential equations under mild regularity conditions.
Theory of measure: He is credited with the Carathéodory extension theorem which is fundamental to modern set theory. Later Carathéodory extended the theory from sets to Boolean algebras.
Theory of functions of a complex variable: He greatly extended the theory of conformal transformation proving his theorem
Carathéodory's theorem (conformal mapping)
In mathematical complex analysis, Carathéodory's theorem, proved by , states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann map...
about the extension of conformal mapping to the boundary of Jordan domains. In studying boundary correspondence he originated the theory of prime end
Prime end
In mathematics, the prime end compactification is a method to compactify a topological disc by adding a circle in an appropriate way....
s.
Thermodynamics: In 1909, Carathéodory published a pioneering work "Investigations on the Foundations of Thermodynamics" in which he formulated the Laws of Thermodynamics axiomatically. It has been said that he was using only mechanical concepts and the theory of Pfaff's differential forms. But in reality he also relied heavily on the concept of an adiabatic process. The physical meaning of the term adiabatic rests on the concepts of heat and temperature. Thus, in Bailyn's survey of thermodynamics, Carathéodory's approach is called "mechanical", as distinct from "thermodynamic". Carathéodory's "first axiomatically rigid foundation of thermodynamics" was acclaimed by Max Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...
and Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
. In his theory he simplified the basic concepts, for instance heat is not an essential concept but a derived one. He formulated the axiomatic principle of irreversibility in thermodynamics stating that inaccessibility of states is related to the existence of entropy, where temperature is the integration function. The Second Law of Thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
was expressed via the following axiom: "In the neighbourhood of any initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state." In this connexion he coined the term adiabatic accessibility
Adiabatic accessibility
Adiabatic accessibility denotes a certain relation between two equilibrium states of a thermodynamic system . The concept was coined by Constantin Carathéodory in 1909 and taken up 90 years later by Elliott Lieb and J. Yngvason in their axiomatic approach to the foundations of thermodynamics . It...
.
Optics: Carathéodory's work in optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
is closely related to his method in the calculus of variations. In 1926 he gave a strict and general proof, that no system of lenses and mirrors can avoid aberration
Aberration in optical systems
Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge...
, except for the trivial case of plane mirrors.
In his later work he gave the theory of the Schmidt telescope.
Historical: During the Second World War Carathéodory edited two volumes of Euler's Complete Works dealing with the Calculus of Variations which were submitted for publication in 1946.
A conjecture: He is credited with the authorship of the Carathéodory conjecture
Carathéodory conjecture
The Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924, [1]. Other early references are the Invited Lecture [3] of Stefan Cohn-Vossen at the International Congress of...
claiming that a closed convex surface admits at least two umbilic points. As of 2007, this conjecture remained unproven despite having attracted a large amount of research.
See also
- Carathéodory's theorem (disambiguation)
- Borel-Carathéodory theorem
- Carathéodory-Jacobi-Lie theoremCarathéodory-Jacobi-Lie theoremThe Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.-Statement:Let M be a 2n-dimensional symplectic manifold with symplectic form ω...
- Carathéodory metricCarathéodory metricIn mathematics, the Carathéodory metric is a metric defined on the open unit ball of a complex Banach space that has many similar properties to the Poincaré metric of hyperbolic geometry. It is named after the Greek mathematician Constantin Carathéodory....
- Carnot-Carathéodory metricSub-Riemannian manifoldIn mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces....
- Carathéodory's theorem (convex hull)Carathéodory's theorem (convex hull)In convex geometry Carathéodory's theorem states that if a point x of Rd lies in the convex hull of a set P, there is a subset P′ of P consisting of d+1 or fewer points such that x lies in the convex hull of P′. Equivalently, x lies in an r-simplex with vertices in P, where r \leq d...
The Smyrna years
At the invitation of the Greek Prime Minister Eleftherios VenizelosEleftherios Venizelos
Eleftherios Venizelos was an eminent Greek revolutionary, a prominent and illustrious statesman as well as a charismatic leader in the early 20th century. Elected several times as Prime Minister of Greece and served from 1910 to 1920 and from 1928 to 1932...
he submitted a plan on 20 October 1919 for the creation of a new University at Smyrna
Smyrna
Smyrna was an ancient city located at a central and strategic point on the Aegean coast of Anatolia. Thanks to its advantageous port conditions, its ease of defence and its good inland connections, Smyrna rose to prominence. The ancient city is located at two sites within modern İzmir, Turkey...
in Asia Minor, to be named Ionian University. In 1920 Carathéodory was appointed Dean of the University and took a major part in establishing the institution, touring Europe to buy books and equipment. The university however never actually admitted students due to the War in Asia Minor
Greco-Turkish War (1919-1922)
The Greco–Turkish War of 1919–1922, known as the Western Front of the Turkish War of Independence in Turkey and the Asia Minor Campaign or the Asia Minor Catastrophe in Greece, was a series of military events occurring during the partitioning of the Ottoman Empire after World War I between May...
which ended in the Great Fire of Smyrna
Great Fire of Smyrna
The Great Fire of Smyrna or the Catastrophe of Smyrna was a fire that destroyed much of the port city of Izmir in September 1922. Eye-witness reports state that the fire began on 13 September 1922 and lasted until it was largely extinguished on September 22...
. Carathéodory managed to save books from the library and was only rescued at the last moment by a journalist who took him by rowing boat to the battleship Naxos which was standing by. The present day University of the Aegean
University of the Aegean
The University of the Aegean is a state, multi-campus university located in Mytilene, Greece. The university was officially founded in 1984, although its historical roots date back to the early 1920s...
claims to be a continuation of Carathéodory's original plan.
Carathéodory brought to Athens some of the university library and stayed in Athens, teaching at the university and technical school until 1924.
In 1924 Carathéodory was appointed professor of mathematics at the University of Munich, and held this position until retirement in 1938. He afterwards worked from the Bavarian Academy of Sciences until his death in 1950.
Linguistic talent
Carathéodory excelled at languages, much like many members of his family did. GreekGreek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
and French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
were his first languages, and he mastered German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
with such perfection, that his writings composed in the German language are stylistic masterworks. Carathéodory also spoke and wrote English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
, Italian
Italian language
Italian is a Romance language spoken mainly in Europe: Italy, Switzerland, San Marino, Vatican City, by minorities in Malta, Monaco, Croatia, Slovenia, France, Libya, Eritrea, and Somalia, and by immigrant communities in the Americas and Australia...
, Turkish
Turkish language
Turkish is a language spoken as a native language by over 83 million people worldwide, making it the most commonly spoken of the Turkic languages. Its speakers are located predominantly in Turkey and Northern Cyprus with smaller groups in Iraq, Greece, Bulgaria, the Republic of Macedonia, Kosovo,...
, and the ancient languages without any effort. Such an impressive linguistic arsenal enabled him to communicate and exchange ideas directly with other mathematicians during his numerous travels, and greatly extend his fields of knowledge.
Much more than that, Carathéodory was a treasured conversation partner for his fellow professors in the Munich Department of Philosophy. The well-respected, German philologist
Philology
Philology is the study of language in written historical sources; it is a combination of literary studies, history and linguistics.Classical philology is the philology of Greek and Classical Latin...
, professor of ancient languages Kurt von Fritz praised Carathéodory, saying that from him one could learn an endless amount about the old and new Greece, the old Greek language, and Hellenic mathematics. Fritz had an uncountable number of philosophical discussions with Carathéodory. Deep in his heart, Carathéodory felt himself Greek above all. The Greek language was spoken exclusively in Carathéodory's house – his son Stephanos and daughter Despina went to a German high school, but they obtained daily additional instruction in Greek language and culture from a Greek priest. At home, they were not allowed to speak any other language.
Legacy
The Greek authorities intended for a long time to create a museum honoring Karatheodoris in Komotini
Komotini
Komotini is a city in Thrace, northeastern Greece. It is the capital of the region of East Macedonia and Thrace and of the Rhodope regional unit. It is also the administrative center of the Rhodope-Evros super-prefecture. The city is home to the Democritus University of Thrace, founded in 1973...
, a major town of the northeastern Greek region which is close to where his family came from. On 21 March 2009 the museum "Karatheodoris"(Καραθεοδωρής) opened its gates to the public, in Komotini
Komotini
Komotini is a city in Thrace, northeastern Greece. It is the capital of the region of East Macedonia and Thrace and of the Rhodope regional unit. It is also the administrative center of the Rhodope-Evros super-prefecture. The city is home to the Democritus University of Thrace, founded in 1973...
.,
The coordinator of the Museum, Athanasios Lipordezis (Αθανάσιος Λιπορδέζης), noted that the museum gave home to original manuscripts of the mathematician of about 10,000 pages including correspondence of Carathéodory with the German mathematician Arthur Rozenthal for the algebraization of measure. Also visitors can view at the showcases the books " Gesammelte Mathematische Schriften Band 1,2,3,4 ", "Mass und Ihre Algebraiserung", " Reelle Functionen Band 1", " Zahlen/Punktionen Funktionen " and many more. Handwritten letters of C.Carathéodory to Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
, Hellmuth Kneser
Hellmuth Kneser
Hellmuth Kneser was a German mathematician, who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifolds...
and photographs of the Carathéodory family are on display.
The effort to furnish the museum with more exhibits is continuous.
Publications of Carathéodory
A complete list of Carathéodory's publications can be found in his Collected Works (Ges. Math. Schr.). Notable publications are:- Über die diskontinuierlichen Lösungen in der Variationsrechung. Diss. Göttingen Univ. 1904; Ges. Math. Schr. I 3-79.
- Über die starken Maxima und Minima bei einfachen Integralen. Habilitationschrift Göttingen 1905; Math. Annalen 62 1906 449-503; Ges. Math. Schr. I 80-142.
- Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann. 67 (1909) pp. 355–386; Ges. Math. Schr. II 131-166.
- Über das lineare Mass von Punktmengen - eine Verallgemeinerung des Längenbegriffs., Gött. Nachr. (1914) 404-406; Ges. Math. Schr. IV 249-275.
- Elementarer Beweis für den Fundamentalsatz der konformen Abbildungen. Schwarzsche Festschrift, Berlin 1914; Ges. Math. Schr.IV 249-275.
- Zur Axiomatic der speziellen Relativitätstheorie. Sitzb. Preuss. Akad. Wiss. (1923) 12-27; Ges. Math. Schr. II 353-373.
- Variationsrechnung in Frank P. & von Mises (eds): Die Differential= und Integralgleichungen der Mechanik und Physik, Braunschweig 1930 (Vieweg); New York 1961 (Dover) 227-279; Ges. Math. Schr. I 312-370.
- Entwurf für eine Algebraisierung des Integralbegriffs, Sitzber. Bayer. Akad. Wiss. (1938) 27-69; Ges. Math. Schr. IV 302-342.
Books by Carathéodory
Vorlesungen über reelle Funktionen. (Lectures on Real Functions) Leipzig-Berlin 1918, 1927,1939 (Teubner); rpr. New York 1948; 3rd corrected ed. 1968 (Chelsea)Conformal Representation, Cambridge 1932 (Cambridge Tracts in Mathematics and Physics)
Geometrische Optik, Berlin, 1937
Elementare Theorie des Spiegelteleskops von B. Schmidt (Elementary Theory of B. Schmidt's Reflecting Telescope), Leipzig Teubner, 1940 36 pp.; Ges. math. Schr. II 234-279
Functionentheorie I, II, Basel 1950, 1961 (Birkhäuser). English translation: Theory of Functions of a Complex Variable, 2 vols, New York, Chelsea Publishing Company, 1954
Mass und Integral und Ihre Algebraisierung, Basel 1956. English translation, Measure and Integral and their Algebraisation, New York, Chelsea Publishing Company, 1963
Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Leipzig, 1935. English translation, Calculus of Variations and Partial Differential Equations of the First Order, New York, Chelsea Publishing Company, 1965.
Gesammelte Mathematische Schriften München 1954-7 (Beck) I-V.
All of Carathéodory's books are written in a beautiful and lucid style; they have been studied by generations of mathematicians, and still being studied to great benefit. Carathéodory's books are unusual in the extent to which geometry is used in the exposition.
Books
- Maria Georgiadou, Constantin Carathéodory: Mathematics and Politics in Turbulent Times, Berlin-Heidelberg:Springer Verlag, 2004. ISBN 3-540-44258-8 MAA Book review
- Themistocles M. RassiasThemistocles M. RassiasThemistocles M. Rassias is a Greek mathematician, and a professor at the National Technical University of Athens , Greece. He has published more than 220 papers, 6 research books and 30 edited volumes in research Mathematics as well as 4 textbooks in Mathematics for university students...
(editor) (1991) Constantin Caratheodory: An International Tribute, Teaneck, NJ: World Scientific Publishing Co., ISBN 981-02-0544-9 (set) - Nicolaos K. Artemiadis; translated by Nikolaos E. Sofronidis [2000](2004), History of Mathematics: From a Mathematician's Vantage Point, Rhode Island, USA: American Mathematical Society, pp. 270–4, 281, ISBN 0-8218-3403-7
- Constantin Carathéodory in his...origins. International Congress at Vissa-Orestiada, Greece, 1-4 September 2000. Proceedings: T Vougiouklis (ed.), Hadronic Press, Palm Harbor FL 2001.
Biographical Articles
- C. Carathéodory, Autobiographische Notizen, (In German) Wiener Akad. Wiss. 1954-57, vol.V, pp. 389–408. Reprinted in Carathéodory's Collected Writings vol.V. English translation in A. Shields, Carathéodory and conformal mapping, The Mathematical Intelligencer 10 (1) (1988), 18-22.
- O. PerronOskar PerronOskar Perron was a German mathematician.He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951...
, Obituary: Constantin Carathéodory, Jahresberichte der Deutschen Mathematiker Vereinigung 55 (1952), 39-51. - N. Sakellariou, Obituary: Constantin Carathéodory (Greek), Bull. Soc. Math. Grèce 26 (1952), 1-13.
- H TietzeHeinrich Franz Friedrich TietzeHeinrich Franz Friedrich Tietze was an Austrian mathematician, famous for the Tietze extension theorem. He also developed the Tietze transformations for group presentations, and was the first to pose the group isomorphism problem.He was born in Schleinz, Austria and died in Munich,...
, Obituary: Constantin Carathéodory, Arch. Math. 2 (1950), 241-245. - H. Behnke, Carathéodorys Leben und Wirken, in A. Panayotopolos (ed.), Proceedings of C .Carathéodory International Symposium, September 1973, Athens (Athens, 1974), 17-33.
- Bulirsch R., Hardt M., (2000): Constantin Carathéodory: Life and Work, International Congress: "Constantin Carathéodory", 1–4 September 2000, Vissa, Orestiada, Greece
Encyclopaedias — reference
- Chambers Biographical Dictionary (1997), Constantine Carathéodory, 6th ed., Edinburgh: Chambers Harrap Publishers Ltd, pp 270–1, ISBN 0-550-10051-2, * Also available online.
- The New Encyclopædia Britannica (1992), Constantine Carathéodory, 15th ed., vol. 2, USA: The University of Chicago, Encyclopædia Britannica, Inc., pp 842, ISBN 0-85229-553-7 * New edition Online entry
- H Boerner, Biography of Carathéodory in Dictionary of Scientific Biography (New York 1970-1990).
Conferences
- International Conference: C. Carathéodory Symposium, Athens, Greece September 1973. Proceedings edited by A. Panayiotopoulos (Greek Mathematical Society) 1975.
- Conference on Advances in Convex Analysis and Global Optimization (Honoring the memory of C. Carathéodory) June 5–9, 2000, Pythagorion, Samos, Greece.
- International Congress: Carathéodory in his ... origins, September 1–4, 2000, Vissa Orestiada, Greece. Proceedings edited by Thomas Vougiouklis (Democritus University of Thrace), Hadronic Press FL USA, 2001. ISBN 1-57485-053-9.