The story of two Greek mathematicians
of “modern times”
Maurolico & Carathéodory
Greece through the ages
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3000 to 1400BC
1600 to 1100BC
1100 to 800BC
800 to 500BC
1100BC to 700AD
284AD to 1453AD
1453 to 1821
1821 to 1945
1920 to 1922
1922 to 1945
1945 to 1950
1967 to 1974
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
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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.
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Means of support: personal, church, academia, government.
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Scientific interests: Math, astronomy, optics.
Maurolico’s scientific work
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Public lectures at the Univ. of Messina (mainly Elements of Euclid).
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Appointed professor in 1569.
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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
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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…
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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.
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1922-1924: Taught at the National Technical Univ. of Athens.
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1924-1950: Invited and returned to Germany: Univ. of Munich.
Mathematical achievements
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Calculus of variations/theory of discontinuous solutions of ode’s.
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Point set measure theory & probability theory.
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Function theory: conformal representation of simply connected
regions on the unit circle; theory of boundary correspondence.
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Thermodynamics.
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Geometrical optics.
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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
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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
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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…
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Married with two children (Despina and Stephanos) .
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Influenced the “Harvard school” (Birkhoffs, Marshal Stone, Ahlfors).
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Was on the Fields committee that awarded a medal to Garrett Birkhoff.
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“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
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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.