The story of two Greek mathematicians of modern

The story of two Greek mathematicians of “modern times” Maurolico & Carathéodory

Greece through the ages n n n n 3000 to 1400 BC 1600 to 1100 BC 1100 to 800 BC 800 to 500 BC 1100 BC to 700 AD 284 AD to 1453 AD 1453 to 1821 to 1945 1920 to 1922 to 1945 to 1950 1967 to 1974 to present Minoan Crete Mycenean Greeks; Bronze Age Pre-classic period; Iron Age Classical period Hellenic Civilization Byzantine Civilization Ottoman Rule Building of Greek nation Greek-Turkish War Absorption of Asia Minor Refugees Depression & the German occupation Greek Civil War Coup of Colonels; Military Junta Republic of Greece

Francesko Maurolico (1494 -1575) Φραγκίσκος Μαυρόλυκος Clarissimum Siciliae lumen n Born in Messina, Sicily. Father: Antonios Maroulis - Greek physician who fled Constantinople; affluent, aristocrat. Learned Greek, Math & Astronomy from his father and from Constantinos Laskaris. n Means of support: personal, church, academia, government. n Scientific interests: Math, astronomy, optics.

Maurolico’s scientific work n Public lectures at the Univ. of Messina (mainly Elements of Euclid). n Appointed professor in 1569. n n Published: Cosmographia, Aristotle’s Mechanical Problems, Classical Greek Geometry. Published works on music, the islands of the world, discovered a star in 1572, involved in military engineering.

First complete inductive proof credited to Maurolico Supported by writings of Pascal (letter to Carcavi): “Çela est aise par Maurolic” Also claimed in Polya’s Mathematical discovery and in Bourbaki’s Set Theory. Arithmeticorum Libri Duo (1575): The sum of the first n odd integers equals the square of n

Constantin Carathéodory (1873 -1950) Κωνσταντίνος Καραθεοδωρής

Constantin Carathéodory - Chronology q q q q Born in Berlin (to Greek parents: his father was a Turkish diplomat at the time Greeks could attain high office). Raised by his Grandmother in Brussels. Educated in Brussels (civil engineer-Belgian officer). Worked in a British dam project in Egypt, road planning in Greece. 1900: Enters Univ. of Berlin to study mathematics. 1902: Starts Ph. D. at Univ. of Göttingen (under Hermann Minkowski). Receives degree in 1904 -1909: Univ. Of Hanover (Full Professor). 1910 -1913: Univ. of Breslau. 1913 -1918: Univ. of Göttingen. 1918 -1920: Univ. of Berlin.

Chronology continued… q q q 1919: Admitted to Prussian Academy of Sciences (dedication by Max Plank). 1920: Accepts post at the Univ. of Smyrna which the Greeks under Eleftherios Venizelos were setting up in Anatolia (now Izmir in Turkey). When the Turks razed Smyrna in 1922, Carathéodory saved the university library and moved it to Athens. q 1922 -1924: Taught at the National Technical Univ. of Athens. q 1924 -1950: Invited and returned to Germany: Univ. of Munich.

Mathematical achievements q Calculus of variations/theory of discontinuous solutions of ode’s. q Point set measure theory & probability theory. q Function theory: conformal representation of simply connected regions on the unit circle; theory of boundary correspondence. q Thermodynamics. q Geometrical optics. q Helped develop Einstein’s theory of special relativity.

Correspondence with Einstein September 1916 "Would you think a little bit about the problem of closed time trajectories? Here lies the essence of this still unsolved part of the space-time problem. I wish you all the best from yours truly, A. Einstein. “ December 1916 "Dear colleague, the main points in theory of canonical substitutions can be most easily derived in my opinion in the following way. " Mathematical expressions from Hamilton-Jacobi Theory follow.

Einstein’s letter (on display in Einstein’s museum in Jerusalem) Dear colleague! I find your derivation wonderful, now I understand everything. At first, the small writing mistakes on the second page had caused me some difficulties. Now, however, I understand everything. You should publish theory in this new form in the Annals of Physics since the physicists do not normally know anything about this subject as was also the case with me. With my letter I must have come across to you like a Berliner who had just discovered Grunewald and wondered whether people were already living there. If you wouldn't mind also making the effort to present to me the canonical transformations, you'll find in me a grateful and attentive audience. If you, however, answer the question about the closed time trajectories, I will appear before you with my hands folded. The underlying truth, though, is well worth some perspiration. Best regards, yours Albert Einstein.

Carathéodory’s legacy n n n n n Carathéodory-Finsler manifold Carnot-Carathéodory metric/problem Carathéodory-Fejer method Carathéodory-Toeplitz theorem/method Carathéodory criterion Integer Carathéodory property Carathéodory-Pesin structure Carathéodory-von Neumann algebraic probability Carathéodory topology Carathéodory superposition of multivalued maps Carathéodory matrix coefficient problem Carathéodory-Schur interpolation problem Osgood-Taylor-Carathéodory theorem Carathéodory extension theorem Julia-Carathéodory theorem Carathéodory-Rieffen distance Borel-Carathéodory inequality 700 items in Math Reviews with Carathéodory in title! 4090 items with Carathéodory

Theorem Let S be any set of points and directions in R^n, and let C=conv S. Then x belongs to C if and only if x can be expressed as a convex combination of n+1 of the points and directions in S (not necessarily distinct).

Facts and anecdotes n n The “birth, rise, development & fortunes of theory & axiomatization of thermodynamics” is generally attributed to him. Command of French, Greek, German, English, Turkish, Italian. Math Genealogy Project: 6 students/286 descendants. Retired from Chair of the dept in Munich (1938). Long quarrel arose as to who would replace him. He proposed Herglotz, Van der Waerden or Siegel (opposing certain Nazi sympathizers).

Some more facts… n Married with two children (Despina and Stephanos). n Influenced the “Harvard school” (Birkhoffs, Marshal Stone, Ahlfors). n Was on the Fields committee that awarded a medal to Garrett Birkhoff. n n “Carathéodory was completely free of the widespread faults of vanity and jealousy found frequently in the academic world. He felt pure joy for others who made great accomplishments. “ (Erhard Schmidt). He was able to give several of his "non-Arian" colleagues a chance for a future by arranging for them an opportunity to emigrate.

…February 2, 1950 Nobody could have said it as well as another famous member of the Bavarian Academy of Sciences , the Geheimrat Oskar Perron: Carathéodory, one of the most magnificent mathematicians, substantially enriched and vitally influenced the sciences. . . a man of unusually extensive education. As a member of the Greek nation, with his soaring spirit and restless pursuit, he continued the recognition of the tradition and legacy of classical Greek culture.

References & sources n n n n Greek Scientists 1453 -1821 (in Greek), Spandagos and Travlou. Convex Analysis, Rockafellar. Mc. Tutor history site (www-history. mcs. st-andrews. ac. uk/history). Britannica. com. Galileo project (@rice. edu). The Mathematics Genealogy Project. Mathematical Reviews (several articles w/ Carathéodory in title). Google and other search engines.