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Caracciodori is one of the most influential mathematicians in the 20th century, a Greek mathematician respected by Einstein as a teacher. His education and research experience combined with two European cultures, experienced two Greek-Turkish wars and two world wars, and was sincere for the whole life of Greece. In the academic aspect, Karaciodori has made important contributions to the mathematical theories such as variational method, complex analysis, real variable function theory and measurement theory, and has done outstanding work on the axiomatization of physics. With great talent, he is proficient in many foreign languages, has a wide range of research, and is an excellent orator. What is more unknown is that he has two Chinese students. There are few Chinese materials about this master, and this article is carefully combed by the author to provide dinner for the readers.

Einstein's general theory of relativity completely changed man's perception of space-time in the universe, but many people did not know that he had the help of a Greek-German mathematician. In his last public speech before his death in 1955, Ai Weng said: "you ask me to answer all kinds of questions, but no one wants to know who is my teacher and who shows me the way to a higher level of mathematical science, thought and research." I just want to say that my great teacher is the unparalleled Greek Constantine Caracciodori, and I owe it all to him. " Constantine Caracciodory (English: Constantin Carath é odory, Greek: "ω" σ tau α "tau ν" α ρ α ε "δ ω ρ", 1873-1950) was born in Greece, was born in Berlin, grew up in Brussels and spent most of his career in Germany. He spans two cultures and is one of the most influential mathematicians of the 20th century.

The dual cultural background of Constantine Karaciodori Greece is the birthplace of western philosophy, literature, drama, sculpture, architecture, historiography, politics, mathematics and science, and is known as "the cradle of European civilization". The ancient Greeks made great achievements in mathematics and science, starting from the mythological historical view of creating fairy tales to explain the world. Until Pythagoras, one of the three Greek philosophers, Euclid, the scientific giant Archimedes, and so on, they were excellent. According to Einstein, one of the two major contributions of western civilization to human society is the deductive system invented by ancient Greek philosophers, and the other is the empirical tradition developed during the Renaissance. Greek was the first language in the world with vowels, which eventually promoted the democracy, science, drama and philosophy of ancient Greece to the whole world. For mathematicians all over the world, the special meanings represented by Greek letters always make people laugh, such as π for pi, φ for golden ratio, ε for infinitesimal, and so on.

In the 2nd century BC, ancient Greece became a part of the Roman Empire after its demise. The Eastern Roman Empire took Byzantium as its center and gave birth to the oriental Greek culture. After the Ottoman Empire captured Constantinople (now Istanbul) in 1453, Greece entered the Ottoman Turkish Empire and did not officially declare its independence until 1832. In 1897, the first Greek-Turkish War broke out between the Kingdom of Greece and the Ottoman Empire because of the ownership of Crete. From 1920 to 1922, the second Greek-Turkish War broke out because of the Asia minor incident. The Fanals are Orthodox Greek residents who traditionally lived in the Farnard district of Constantinople. They originated from some wealthy Greek businessmen (mostly Roman remnants) in the late 16th century and received a good western education. They have a sense of identity with their Greek identity. Since the end of the 17th century, the Fanaer people gradually gained the importance of the Ottoman central government, and many of them held important positions and became translators and middlemen in the communication between the Empire and European countries. The Fanals also played an important role in the Greek independence movement in the 19th century and the complex political environment in Europe.

Caracciodori comes from a learned Greek aristocratic family in the Farnard district of Constantinople. Many members of the family have held important imperial government positions and diplomatic envoys abroad. Caracciodori's father was a lawyer and a long-time Ottoman diplomat in Belgium, Russia and Germany. His mother's ancestor is a wealthy merchant family on Greece's fifth largest island, Sios, which is said to be the hometown of Homer, the ancient Greek bard, and Hippocrates, the "father of medicine". In 1822, during the Greek War of Independence, the Ottoman army carried out a massacre on the island of Sios, and Karacciodori's matrilineal family moved to Italy and Marseilles, France, so he had a dual cultural background of the East and the West. Karaciodori dabbled in the fields of physics, linguistics, philosophy, politics, engineering and archaeology. He experienced two Greek-Turkish wars and two world wars in his life, which was closely related to the changes of the world in the first half of the 20th century.

Caracciodori's mother died of illness at the age of six, and he and his sister were raised by his grandmother. Caracciodori grew up with outstanding figures in the fields of diplomacy, science, law and music from different countries of the East and the West. when he was in high school, he showed an interest in geometry and twice won the first prize in the all-German high school math competition. After graduating from high school, following the family tradition since the first half of the 19th century, he trained as a civil engineer and educated in mathematics at the Royal Belgian military Academy. In 1895, Caracciodori met Venizelos, one of the most famous politicians in modern Greece, on the island of Crete, and the two became lifelong friends. When the first Greek-Turkish War broke out in 1897, Karashiodori made clear his political stand. He later went to Egypt to participate in dam construction and surveying pyramids, taught himself mathematics in his spare time, and wrote a book on the historical geography of Egypt as a farewell gift to this charming country.

From left to right: Caracciodori, his father, brother-in-law and sister (pictured in 1898) embark on a professional path. 1900 is the end of the old century and the beginning of the new century. David Hilbert made a famous speech entitled "Mathematical problems" at the second International Congress of mathematicians in Paris, and put forward 23 important mathematical problems, which became the highest program for the development of mathematics in the 20th century. It was in this year that 27-year-old Kara Theodore entered the University of Berlin to study mathematics, and two years later he went to the University of Gottingen, then the center of world mathematics, for a PhD. The Gottingen School led by Hilbert created the mathematical glory of the first 30 years of the 20th century, when nearly half of the world's famous mathematicians were from the Gottingen School. The loudest slogan among mathematics students in the world is: "pack your backpack and go to Gottingen."

When Karaciodori came to Gottingen, he was taught by great mathematicians such as Hilbert and Felix Klein, and Hans Hahn's lectures on variational methods broadened his horizons. Under the guidance of another math guru, Hermann Minkowski, a good friend of Hilbert, Karaciodori received his doctorate in 1904 for his thesis on discontinuous solutions in variational methods, and in the same year he attended the third International Congress of mathematicians in Heidelberg. At the suggestion of Hilbert, Karaciodori obtained the Habilitation Chartered Teaching qualification in 1905-the highest academic title in Germany, and served as a professor at colleges or universities in Bonn, Hanover, Breslau, Gottingen, Berlin and other places.

The picture on the left shows Karaciodori (left) and the Hungarian mathematician Lip ó t Fej é r, who worked together to prove the Carath é odory-Fejer theorem on harmonic functions in 1911; the picture on the right shows Karaciodori (right) and Hilbert. In the autumn of 1919, Caracciodori's life changed dramatically. He accepted the invitation of Venizelos, who has become prime minister of Greece, resigned his teaching post at the University of Berlin and went to Smyrna in Asia minor to build a university to support his old friends' dream of rebuilding the Ionian coast that was the symbol of Hellenism. Caracciodori tried to take the Anglo-Saxon university as a model to create the most perfect university in the East and a unique collection-rich library in the whole East. So he traveled all over Europe to buy books and equipment for new universities. His vision did not become a reality when Smyrna was burned after being occupied by the Turks before the end of the second Greek-Turkish War in August 1922. Karaciodori was one of the last Greeks in Smyrna. Under his arrangement, all the books and instruments of the university were moved to Athens, and the Greek civilization left the land with him. In 1922 and 1923, Karaciodori was appointed Professor of Mathematics at the University of Athens and Professor of Mechanics at the National University of Technology of Athens, respectively.

In 1924, Caracciodori returned to Germany and applied for the position of professor and dean of the University of Munich, where he resumed his brilliant career and stayed there for the rest of his life. Caracciodori's interest in mathematics is deeply rooted in geometry and mechanics, and he is a talented analyst with a high taste for the simplicity, nature and elegance of mathematics. With his flying spirit and unremitting pursuit, Caracciodori has made outstanding contributions in many fields of mathematics. He has published more than 60 academic papers and written seven books in his life, solving the mathematical problems that have plagued mathematical masters such as Carl Friedrich Gauss and Leonhard Euler for hundreds of years. At least 17 mathematical definitions or theorems are named after Caracciodori, and there are more than 700 articles with his name in the American Mathematical Society's Mathematical Review magazine. He is so far the only Greek to win the title of academician of the German Academy of Sciences.

Caracciodori's main research field of fruitful achievements in mathematical physics is the variational method, that is, the field of mathematics dealing with extreme functional. The variational method originates from some specific physical problems, which can be traced back to Newton's minimum resistance problem in 1687 and John Bernoulli's fastest path problem in 1696. In the 18th century, Euler began to study this kind of problem systematically, and determined the name of the corresponding discipline in his work variational principles. In the 1750s, Euler and Joseph Lagrange put forward the Euler-Lagrange equation, which was later named by Euler and Lagrange, which integrated Euler's geometric method and Lagrange's analytical method, and became the key theorem for solving functional critical value functions, but the disadvantage is that they can not distinguish the maximum value, the minimum value or both. In the late 19th century, Karl Weierstrass successively obtained the sufficient conditions for the minimum of weak variation and the maximum of strong variation, which was considered to be of epoch-making significance.

In his doctoral thesis and follow-up work, Karaciodori extended the theory of smooth curve in variational method to angular curve. He constructed extreme value field based on Hamilton-Yakubi equation, proved the sufficient condition of extreme functional, and derived Euler-Lagrange equation and Weierstrass condition. Karaciodori restudied the relationship between variational methods and partial differential equations. In recent years, his methods have been applied to optimal control and dynamic programming theory. During World War II, Caracciodori edited Euler's two volumes on variational methods, which were published in 1946. Karaciodori's work on geometrical optics is closely related to his variational method. He proved that no lens except a planar lens can avoid aberration, and he also proposed the theory of the Schmidt telescope. In his 1937 work geometrical Optics, Caracciodori demonstrated the equivalence of Huygens principle and Fermat principle based on Cauchy's characteristic theory.

Complex analysis is a mathematical theory for the study of complex functions, especially meromorphic functions and complex analytic functions. Bernhard Riemann's doctoral thesis "General theoretical basis of simple complex functions" in 1851 contains the sprout of the main part of modern complex function theory, and paves the way for his later study of Riemannian geometry. In this paper, Riemann stated one of the most profound results of complex analysis-Riemann mapping theorem, but its proof is not complete. William Fogg Osgood gave the first strict proof of this theorem in 1900. in 1913, Caracciodori gave another strict and complete proof and generalized the theorem, which was also the first proof that depended solely on function theory rather than potential theory, so it became the standard proof method in textbooks.

On the cover of two books of Caracciodori, the English version of Conformal representation on the left was published in 1932, and the German version of variational method and first order partial differential equation on the right was published in 1935. The two-volume English version was published in 1965 and 1967 respectively. Since the 19th century, mathematical analysis has entered a rigorous stage and began to construct an axiom system. With Weierstrass's basic research on real analysis and complex analysis and the publication of Georg Cantor's set theory, the classical Riemannian integral theory has exposed great limitations. Henri Lebesgue established the measure and integral theory later named after him in his doctoral thesis in 1902, which made a great contribution to the field of mathematics in the 20th century. Karaciodori studied the axiom of measure theory, put forward the criterion of Lebesgue measurable set and the extension theorem of measure, and extended it to Boolean algebra, which became a powerful tool of abstract measure theory and the basis of modern measure theory. Karaciodori also studied ordinary differential equations, functions of multiple complex variables, convex geometry, symplectic geometry and so on.

The sixth of Hilbert's 23 mathematical problems is axiomatic Physics. In his textbook basis of Geometry published in 1899, he established a strict axiomatic basis of geometry to avoid some of the weaknesses of the traditional Euclidean axiom. In 1909, Karaciodori published the groundbreaking paper "basic Research of Thermodynamics" in the Mathematical Yearbook. He gave the mathematical definitions of the basic concepts of thermodynamics, such as equilibrium, state and thermodynamic coordinates, put forward the axiom principle of irreversibility in thermodynamics, and pointed out that the non-integrability of states is related to the existence of entropy. He expressed the second law of thermodynamics through the following axiom: "in the adjacent state of any equilibrium state of a system, there is a state that cannot be achieved by a reversible adiabatic process." Caracciodori's work on measure theory provided the basis for the improvement of continuum physics and thermodynamic formulas, determined ergodicity as metric transitivity by George Birkhoff, and paved the way for Bernard Koopman and von Neumann to introduce spectral analysis of dynamical systems in Hilbert space.

Karaciodori's doctoral mentor, Minkowski, was also a teacher of Einstein when he was studying at the Federal Institute of Technology in Zurich, Switzerland, and Minkowski's four-dimensional space-time provided a mathematical structure for Einstein's special theory of relativity. During his tenure as an academician of the Prussian Academy of Sciences in 1916, Einstein expected to have a deeper understanding of the mathematical basis of his own general theory of relativity, especially about the geodesic corresponding to the closed trajectories of light and free particles in the static universe. So he got in touch with Caracciodori to consult "classical mathematical problems" such as Hamilton-Jacobi equation and canonical transformation, and soon got a reply and detailed explanation from Kalaciodori. All of their communications from 1916 to 1930 were collected at the Einstein Museum in Bern, Switzerland, and at the Caracciodori Museum in Comotini, Greece. In 1928, as the first visiting scholar of the American Mathematical Society, Caracciodori became a visiting professor at Harvard University. He visited the United States again in 1936-37 and was invited to deliver a speech at the annual meeting of the American Mathematical Society.

A mural on the street of Athens shows Caracciodori on the left and Einstein on the right. Karaciodori loved ancient Greek cultural traditions and heritage so much that he proposed that Smyrna establish a new Greek university in the 1920s. This idea was realized with the establishment of the University of Aristotle in Thessaloniki, Greece's second largest city in 1925. In 1930, Caracciodori returned to Greece again at the invitation of Venizelos and took two years to reorganize the University of Athens and the University of Aristotle. Although usually introduced as a German professor during his visit to the United States, Caracciodori often mentioned his Greek ancestry and was proud of his contribution to improving Greece's reputation in the United States. Karaciodori is an outstanding language master. In addition to his native language, Greek and French, he is also proficient in German, English, Italian, Turkish and ancient languages. Karaciodori is also a talented orator. In 1936, he was appointed as one of the five members of the Fields Prize jury at the International Congress of mathematicians in Oslo, Norway. He delivered a speech at the conference and awarded the first Fields Medal.

An interesting study by Caracciodori, who participated in the centenary of the Greek Archaeological Society and published his paper in 1937, is about the curve and spacing of the pillars on the north side of the Parthenon. Built in the fifth century BC, the Parthenon is a shrine for Athena, the patron saint of the Athenian city-state. Named after Athena's nickname Parthenon, it stands at the top of the Acropolis. The Parthenon is a symbol of democracy in ancient Greece and Athens, representing the highest level of ancient Greek architecture and sculpture. Since the 19th century, a number of British architects have measured and studied the pillar curve of the Parthenon, most of which are considered to be a parabola. Caracciodori used the latest measurements at the time to study it mathematically. He concluded that the pedestal curve of the Parthenon is an arc with a radius of 5560 meters, because the conic curve was discovered by the ancient Greek mathematician Menahimus a hundred years after the temple was built.

When Karaciodori came to Munich in 1924, it had provided fertile ground for nationalism and anti-Semitism, but no one could have foreseen the catastrophe of Nazi control of Germany nearly a decade later. Karaciodori was discriminated against to a certain extent because of his Greek origin during World War II and, like the German middle class, remained silent about the Nazi Holocaust in Greece and the expulsion of Jews from scientific institutions. However, not everyone can be a "hero". As a person standing between two peoples, Caracciodori always lend a helping hand when others need it. He used his connections and reputation in the international academic community to help his "non-Aryan" counterparts find teaching posts or escape from Germany, including some famous mathematicians and scientists. Caracciodori's children are only allowed to speak Greek at home, and his daughter Despina clearly remembers that she received the only slap in the face of her father because she had made positive comments about the Nazis at home.

Caracciodori has maintained a close relationship with Greek academia all his life. He was an undergraduate student Evangelos Stamatis when he taught at the University of Athens and later made great achievements in classical Greek mathematics. During the most difficult times of World War II, Karachiodori helped Greek topologist Christos Papakyriakopoulos earn his doctorate from the University of Athens. After the war, Karashiodori did his best to contribute to Germany's re-integration into the civilized world. Caracciodori was a well-bred and respected man who completely got rid of the vanity and jealousy that were common in academia and felt pure joy in the achievements of others. Until a few days before his death at the age of 77, Caracciodori was still sitting at his desk studying mathematics. Caracciodori died in Munich in 1950.

The picture on the left shows two Greek stamps issued in 1994, the top shows Karashiodori and the lower Thales, and on the right is the graveyard of him, his wife Euprhrosyne and son Stephanos in the forest cemetery south of Munich. The top of the tombstone is designed with a Greek Ionic stigma. Two Chinese students, Karaci Odori, mentored more than 20 doctoral students all his life, and many of them became accomplished mathematicians. The earliest and most famous two were Hans Rademacher and Paul Finsler, who graduated from the University of Bonn. The former is famous for its research in analytic number theory, mathematical genetics, real analysis and quantum theory, while the latter defines Finsler manifolds in differential geometry and proves Hadwiger-Finsler inequality. Karaciodori formed a harmonious Golden Triangle with Heinrich Tietze and Oskar Perron at the University of Munich, and the largest lecture hall of the University of Munich's School of Mathematics was named after him. In Munich, Caracciodori directed ten doctoral dissertations as the first tutor and seven as the second mentor. his last student was Karl Weigand, who graduated in 1947. There are also two Chinese on Caracciodori's student list, Li Da (Ta Li,1905-1997) who graduated in 1933 and Xu Ruiyun (S ü e-yung Kiang,1915-1969) who graduated in 1940. Xu Ruiyun is married and her husband's surname is "Jiang".

Li Da, Zhong Heng, a native of Pingjiang, Hunan, was admitted to Southeast University in 1924. In 1933, under the guidance of Peron and Caracciodori, he received his doctorate from the University of Munich for his thesis on the Stability of differential equations. From September 4 to 12, 1932, Li Da, who was studying for his doctorate, together with Xiong Qinglai and Xu Guobao, attended the 9th ICM International Congress of mathematicians in Zurich, Switzerland, and became one of the first Chinese mathematicians to participate in ICM. Li Da wrote the 5,000-word minutes of the World Conference of mathematicians on the third day after the meeting, which was published in Nanjing's Science World and Changsha's World Journal on November 1. This article is the earliest and most detailed report of Chinese mathematicians on international mathematical academic activities in the first half of the 20th century, especially recording the origin of the Fields Prize: on the 12th, "announced that the Association would receive the Fields Medal Fund and set up a committee with Caratheodory,Cartan Severi,Takagi (Takagi Sadaku) as a member to discuss who the medal should be given and how it should be given."

After returning to China in 1935, Li Da taught at Tsinghua, Shandong, Tongji, Lantian normal College and Chongqing University. In 1942, he became dean of the School of Science and Dean of the Department of Mathematics and Science of Fudan University in Beibei, Chongqing, and became the highest-level professor. According to the first issue of Fudan Mathematical and physical Communication in 1948, the department was founded in the autumn of 32. Mr. Li as dean of science and head of the department …... it was extremely difficult, but through its painstaking planning, the foundation of the department was laid. " The Victory School of the Anti-Japanese War moved back to Shanghai. "all of you have an indefatigable spirit of teaching, so there is a harmonious relationship between teachers and students and a strong research atmosphere." Li Da signed "Li Zhongheng" in 1948 and published "arithmetic in China in the past 30 years" in Science magazine, which is the first comprehensive review of the research achievements of various branches of mathematics in China since 1918. In 1947, Li Da went to the United States for visiting research and settled in the United States. He was good at mathematical physics. He independently presided over an aerospace research department and a number of important projects, and was elected as a communications academician of the American Academy of Aeronautics and Astronautics.

Li Da publishes the Proceedings of the World Conference of mathematicians in the World Journal (left). In 1955, the higher Education Press published two volumes of the Soviet mathematician Na Tangsong's Theory of Real variable functions (right), translated by Xu Ruiyun. Xu Ruiyun, a native of Cixi, Zhejiang Province, was born in Shanghai. She entered Wuben Girls' Middle School, the first girls' school run by Chinese people in Shanghai in 1927, and was admitted to the Mathematics Department of Zhejiang University in 1932. She studied under famous mathematicians Chen Jiangong and Su Buqing and stayed in school after graduation. In 1937, Xu Ruiyun and her new husband Jiang Ximing won a scholarship and came to the University of Munich to study for a doctorate. Under the guidance of Caracciodori, she completed her doctoral thesis "Fourier expansion of singular functions in Lebesgue decomposition" in 1940 and became the second female PhD in mathematics in China after Liu Shuting. Editor's note: for a long time, Xu Ruiyun is considered to be the first female PhD in mathematics in China, but the latest research result in recent years is Shu Ting Hsia Liu. After graduating and returning to China, Xu Ruiyun taught at Zhejiang University and later founded the Mathematics Department of Hangzhou University, making great efforts to train professionals in function theory. In 1955, the higher Education Press published two volumes of the real variable function Theory written by the Soviet mathematician Na Tangsong, translated by Xu Ruiyun and Chen Jiangong School. Xu Ruiyun also instructed Zhao Yanda to translate Karassiodori's last painstaking book, the Theory of complex functions (Volume 1). Unfortunately, Xu Ruiyun had been dead for 16 years when the book was published in 1985.

In pursuit of the beauty of mathematics and poetry, I first heard the name of Carath é odory when I was an undergraduate at Fudan University more than 40 years ago. In the class of real variable function theory, I was impressed when I learned the Caracciodori criterion for distinguishing Lebesgue measurable sets. In this theorem, the outer measure is like an ingenious scalpel, which cuts the point set E arbitrarily, and the necessary and sufficient condition for E to be measurable is that the sum of the outer measures of the two parts after cutting is equal to the outer measure of E itself. On the one hand, this rule and its proof are really beautiful, and it is one of the author's favorite theorems in college; on the other hand, Caracciodori's name is so musical that it is easy to read, and the textbook at that time was translated as "Karatai House alone." However, the author knows nothing about Caracciodori's life, let alone heard of his Chinese students Li Da and Xu Ruiyun, two previous mathematicians who are not far away from us. Although the author has almost finished all the exercises in the book "Real variable function Theory" written by Tang Song, I have never paid attention to the name of the translator.

Another 20th century Greek, on a par with Caracciodori, was George Sephelis, winner of the 1963 Nobel Prize for Literature (English: Giorgos Seferis, Greek: epi, ρ η, 1900-1971). He became the first Greek to win the Nobel Prize for his "excellent lyric poems full of deep feelings for the cultural heritage of ancient Greece". Born in the city of Smyrna in Asia minor, Sephelis was shocked when the poet's hometown and Smyrna were merged into Turkey during the second Greek-Turkish War. Mathematical research and poetry creation are similar, both need rich intuition and imagination, and the common pursuit is simplicity and harmony. I don't know if Kara Theodore has ever been in contact with Sephelis, but their love for their motherland is the same, and their hometown Greece is the source and destination of their spiritual torrent. They look for their hometown in mathematics and poetry, which is "all kinds of feelings and thoughts raised under the exotic sky".

Athens and Berlin are two cities inextricably bound to the life of Caracciodori. This article ends with the Parthenon in Athens and the Brandenburg Gate in Berlin, which is the first neo-Greek building in Germany. The central sculpture on the top of the door is Nike, the goddess of victory in Greek mythology. The two buildings are built in a thick and majestic Doric style, and the Temple of Tenon enjoys the reputation of "the most perfect Doric temple in history".

The Parthenon in Athens (left) and the Brandenburg Gate in Berlin (right) Professor Ding Jiu of the University of Southern Mississippi in the United States read the first draft of this article and corrected it. The rest of the pictures in the article are from the Internet.

reference

1. M. Georgiadou,Constantin card é odory: Mathematics and Politics in Turbulent Times, Springer-Verlag, 2004.

2. H. J. Pesch, Constantin Carath é odory, a Greek-German Professor of Mathematics at the University of Munich in the Years 1924-1945-- A Personal Statement, 2017.

3. G. P. Stevens, A Curve of the North Stylobate of the Parthnon, American School of Classical Studies at Athen, 1937.

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