## The story of two Greek mathematicians of “modern times” ## Maurolico & Carathéodory
## 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 ## 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
## Francesko Maurolico (1494-1575) * Φραγκίσκος Μαυρόλυκος Clarissimum 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.
__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. ## 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__ ## 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. -
__Chronology continued…__ ## 1919: Admitted to Prussian Academy of Sciences ## (dedication by Max Plank). ## 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.
__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.
__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.
__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.
__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 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__ ## 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 ## arose as to who would replace him. He proposed Herglotz, ## Van der Waerden or Siegel ## (opposing certain Nazi sympathizers).
__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.
__…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 ## 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.
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