The story of two Greek mathematicians of

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) F?a???s???
?a????????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.

4
Maurolicos scientific work

• Public lectures at the Univ. of Messina (mainly
  Elements of Euclid).
• Appointed professor in 1569.
• Published Cosmographia, Aristotles 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 Polyas Mathematical discovery
  and in Bourbakis Set Theory.
• Arithmeticorum Libri Duo (1575)
• The sum of the first n odd integers equals
• the square of n

6
Constantin Carathéodory (1873-1950)???sta?t????
?a?a?e?d????
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 odes.
• 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 Einsteins 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
Einsteins letter (on display in Einsteins
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
Carathéodorys 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!

13
Theorem

• Let S be any set of points and directions in Rn,
  and
• let Cconv S. Then x belongs to C if and only if
  x can be
• expressed as a convex combination of n1 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.a
  c.uk/history).
• Britannica.com.
• Galileo project (_at_rice.edu).
• The Mathematics Genealogy Project.
• Mathematical Reviews (several articles w/
  Carathéodory in title).
• Google and other search engines.