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

1. Since Ancient Greece The story of two Greek mathematicians of “modern times” Maurolico & Carathéodory

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

3. Francesko Maurolico (1494-1575)ΦραγκίσκοςΜαυρόλυκοςClarissimum Siciliae lumen • 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. • Means of support: personal, church, academia, government. • Scientific interests: Math, astronomy, optics.

4. Maurolico’s scientific work • Public lectures at the Univ. of Messina (mainly Elements of Euclid). • Appointed professor in 1569. • 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.

5. First complete inductive proofcredited 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

6. Constantin Carathéodory (1873-1950)ΚωνσταντίνοςΚαραθεοδωρής

7. Constantin Carathéodory - Chronology • 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. • 1904-1909: Univ. Of Hanover (Full Professor). 1910-1913: Univ. of Breslau. 1913-1918: Univ. of Göttingen. 1918-1920: Univ. of Berlin.

8. Chronology continued… • 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. • 1922-1924: Taught at the National Technical Univ. of Athens. • 1924-1950: Invited and returned to Germany: Univ. of Munich.

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

10. 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 the theory of canonical substitutions can be most easily derived in my opinion in the following way." Mathematical expressions from Hamilton-Jacobi Theory follow.

11. 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 the 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.

12. Carathéodory’s legacy • 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

13. 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).

14. Facts and anecdotes • The “birth, rise, development & fortunes of the 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).

15. Some more facts… • Married with two children (Despina and Stephanos) . • Influenced the “Harvard school” (Birkhoffs, Marshal Stone, Ahlfors). • Was on the Fields committee that awarded a medal to Garrett Birkhoff. • “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.

16. …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.

17. References & sources • Greek Scientists 1453-1821 (in Greek), Spandagos and Travlou. • Convex Analysis, Rockafellar. • McTutor 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.