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

• The story of two Greek mathematicians of “modern times”

• Maurolico & Carathéodory

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

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.

• 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

• 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).

• 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.

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 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

• 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).

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.