Charles Ehresmann Explained

Charles Ehresmann
Birth Date:19 April 1905
Birth Place:Straßburg, Alsace-Lorraine, German Empire (today Strasbourg, Alsace, France)
Death Place:Amiens, Picardy, France
Nationality:French
Fields:Mathematics
Workplaces:University of Strasbourg
Paris Diderot University
Alma Mater:École Normale Supérieure
Thesis Title:Sur la topologie de certains espaces homogènes
Thesis Url:http://www.numdam.org/article/THESE_1934__162__391_0.pdf
Thesis Year:1934
Doctoral Advisor:Élie Cartan
Known For:Almost complex manifold
Ehresmann's fibration theorem
Ehresmann connection
Jet bundle
Lie groupoid
Sketch
Strict 2-category
Awards:Prix Francoeur
Spouse:Andrée Ehresmann

Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory.

He was an early member of the Bourbaki group, and is known for his work on the differential geometry of smooth fiber bundles, notably the introduction of the concepts of Ehresmann connection and of jet bundles, and for his seminar on category theory.

Life

Ehresmann was born in Strasbourg (at the time part of the German Empire) to an Alsatian-speaking family; his father was a gardener. After World War I, Alsace returned part of France and Ehresmann was taught in French at Lycée Kléber.[1]

Between 1924 and 1927 he studied at the École Normale Supérieure (ENS) in Paris and obtained agrégation in mathematics. After one year of military service, in 1928-29 he taught at a French school in Rabat, Morocco. He studied further at the University of Göttingen during the years 1930–31, and at Princeton University in 1932–34.

He completed his PhD thesis entitled Sur la topologie de certains espaces homogènes (On the topology of certain homogeneous spaces) at ENS in 1934 under the supervision of Élie Cartan.

From 1935 to 1939 he was a researcher with the Centre national de la recherche scientifique and he contributed to the seminar of Gaston Julia, which was a forerunner of the Bourbaki seminar. In 1939 Ehresmann became a lecturer at the University of Strasbourg, but one year later the whole faculty was evacuated to Clermont-Ferrand due to the German occupation of France. When Germany withdrew in 1945, he returned to Strasbourg.

From 1955 he was Professor of Topology at Sorbonne, and after the reorganization of Parisian universities in 1969 he moved to Paris Diderot University (Paris 7).

Ehresmann was President of the Société Mathématique de France in 1965.[2] He was awarded in 1940 the Prix Francoeur for young researchers in mathematics[3] and in 1967 an honorary doctorate by the University of Bologna.[4] He also held visiting chairs at Yale University, Princeton University, in Brazil (São Paulo, Rio de Janeiro), Buenos Aires, Mexico City, Montreal, and the Tata Institute of Fundamental Research in Bombay.[5]

After his retirement in 1975 and until 1978 he gave lectures at the University of Picardy at Amiens, where he moved because his second wife, Andrée Charles-Ehresmann, was a professor of mathematics there. He died at Amiens in 1979.

Mathematical work

In the first part of his career Ehresmann introduced many new mathematical objects in differential geometry and topology, which gave rise to entire new fields, often developed later by his students.[6] [7]

In his first works he investigated the topology and homology of manifolds associated with classical Lie groups, such as Grassmann manifolds and other homogeneous spaces.[8] [9]

He developed the concept of fiber bundle, and the related notions of Ehresmann connection and solder form,[10] [11] [12] [13] building on the works by Herbert Seifert[14] and Hassler Whitney[15] in the 1930s. Norman Steenrod was working in the same direction from a topological point of view,[16] but Ehresmann, influenced by Cartan's ideas, was particularly interested in differentiable (smooth) fiber bundles, and in the differential-geometric aspects of these. This approach led him also to the notion of almost complex structure,[17] which was introduced independently also by Heinz Hopf.[18]

In order to obtain a more conceptual understanding of completely integrable systems of partial differential equations, in 1944 Ehresmann inaugurated the theory of foliations,[19] [20] which will be later developed by his student Georges Reeb.[21] With the same perspective, he pioneered the notions of jet and of Lie groupoid.[22] [23] [24] [25]

Since the 1960s, Ehresmann's research interests moved to category theory, where he introduced the concepts of sketch and of strict 2-category.[26] [27]

His collected works, edited by his wife, appeared in seven volumes in 1980–1983 (four volumes published by Imprimerie Evrard, Amiens, and the rest in the journal Cahiers de Topologie et Géométrie Différentielle Catégoriques, which he had founded in 1957).[28] His publications include also the books Catégories et structures (Dunod, Paris, 1965) and Algèbre (1969).

Jean Dieudonné described Ehresmann's personality as "... distinguished by forthrightness, simplicity, and total absence of conceit or careerism. As a teacher he was outstanding, not so much for the brilliance of his lectures as for the inspiration and tireless guidance he generously gave to his research students ... "

He had 76 PhD students, including Georges Reeb, Wu Wenjun (吴文俊), André Haefliger, Valentin Poénaru, and Daniel Tanré. His first student was Jacques Feldbau.[29] [30]

References

  1. Web site: Dieudonné . Jean . Jean Dieudonné . 2007 . Ehresmann, Charles . 2022-02-18 . . Charles Scribner's Sons.
  2. Web site: Liste anciens présidents Société Mathématique de France . 2022-02-18 . smf.emath.fr.
  3. Web site: Charles Ehresmann . 2022-02-18 . serge.mehl.free.fr . fr.
  4. Web site: Lauree Honoris Causa – Archivio Storico . 2022-02-26 . archiviostorico.unibo.it . it.
  5. Web site: CTHS – EHRESMANN Charles . 2022-02-18 . cths.fr.
  6. Libermann . Paulette . Paulette Libermann . 2007 . Charles Ehresmann's concepts in differential geometry . Geometry and Topology of Manifolds . Banach Center Publications . 76 . en . Będlewo, Poland . Institute of Mathematics Polish Academy of Sciences . 35–50 . 10.4064/bc76-0-2. free .
  7. Ehresmann . Andrée C. . Andrée Ehresmann . 2007 . How Charles Ehresmann's vision of geometry developed with time . Geometry and Topology of Manifolds . Banach Center Publications . 76 . en . Będlewo, Poland . Institute of Mathematics Polish Academy of Sciences . 29–34 . 10.4064/bc76-0-1. free .
  8. Ehresmann . Charles . 1934 . Sur la Topologie de Certains Espaces Homogenes . Annals of Mathematics . 35 . 2 . 396–443 . 10.2307/1968440 . 1968440 . 0003-486X.
  9. Ehresmann . Charles . 1936 . Sur les espaces localement homogènes . On locally homogeneous spaces . Enseign. Math. . fr . 35 . 317–333 . 62.1473.03.
  10. Marle . Charles-Michel . Charles-Michel Marle . 2007 . The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles . Geometry and Topology of Manifolds . Banach Center Publications . 76 . en . Będlewo, Poland . Institute of Mathematics Polish Academy of Sciences . 65–86 . 10.4064/bc76-0-4. 43455645 . free . 1401.8272 .
  11. Ehresmann . Charles . 1947 . Sur les espaces fibrés différentiables . On differentiable fibred bundles . . fr . 224 . 1611–1612 . 0029.42001.
  12. Ehresmann . Charles . 1949 . Sur la théorie des espaces fibres . On the theory of fibred bundles . Colloques Internat. Centre Nat. Rech. Sci. (Topologie Algebrique) . fr . 12 . 3–15 . 0039.39703.
  13. Ehresmann . Charles . 1955 . Les prolongements d'un espace fibré différentiable . The prolongations of a differentiable fibred bundle . . fr . 240 . 1755–1757.
  14. Seifert . Herbert . 1933 . Topologie Dreidimensionaler Gefaserter Räume . Topology of tridimensional fibred space . Acta Mathematica . de . 60 . 147–238 . 10.1007/BF02398271 . 122571694 . 0001-5962 . Project Euclid. free .
  15. Whitney . Hassler . 1935-07-01 . Sphere-Spaces . Proceedings of the National Academy of Sciences of the United States of America . en . 21 . 7 . 464–468 . 10.1073/pnas.21.7.464 . 0027-8424 . 1076627 . 16588001. 1935PNAS...21..464W . free .
  16. Book: Steenrod, Norman . The Topology of Fibre Bundles. (PMS-14), Volume 14 . . 1951 . 978-1-4008-8387-5 . en . 10.1515/9781400883875 . Norman Steenrod.
  17. Ehresmann . Charles . 1952 . Sur les variétés presques complexes . On almost complex manifolds . Proc. Internat. Congr. Math. . fr . 2 . 412–419 . 0049.12904.
  18. Hopf . Heinz . Heinz Hopf . 1948 . Zur Topologie der komplexen Mannigfaltigkeiten . On the topology of complex manifolds . Studies and Essays Presented to R. Courant on His 60th Birthday . de . New York . 167–185.
  19. Haefliger . André . André Haefliger . 2005 . Naissance des feuilletages, d'Ehresmann-Reeb à Novikov . Birth of foliations, from Ehresmann-Reeb to Novikov . Géométrie au XXe Siècle, 1930-2000: Histoire et Horizons . fr . Hermann . 257–272. 10.1.1.694.1226 .
  20. Ehresmann . Charles . Charles Ehresmann . 1951 . Sur la théorie des variétés feuilletées . On the theory of foliated manifolds . . fr . . Rome . 10 . 5 . 64–82.
  21. Reeb . Georges . Georges Reeb . 1952 . Sur certaines propriétés topologiques des variétés feuilletées . On certain topological properties of foliated manifolds . Actualités Scientifiques et Industrielles . fr . Hermann . Paris . 1183 . 5–89, 155–156 . 0055692.
  22. Pradines . Jean . 2007 . In Ehresmann's footsteps: from group geometries to groupoid geometries . Geometry and Topology of Manifolds . Banach Center Publications . 76 . en . Będlewo, Poland . Institute of Mathematics Polish Academy of Sciences . 87–157 . 10.4064/bc76-0-5. 0711.1608 . 17743849 .
  23. Tulczyjew . Włodzimierz M. . 2007 . Evolution of Eresmann's jet theory . Geometry and Topology of Manifolds . Banach Center Publications . 76 . en . Będlewo, Poland . Institute of Mathematics Polish Academy of Sciences . 159–176 . 10.4064/bc76-0-6. 17456748 . free .
  24. Ehresmann . Charles . Charles Ehresmann . 1959 . Catégories topologiques et categories différentiables . Topological categories and differentiable categories . Colloque de Géométrie différentielle globale . French . CBRM, Bruxelles . 137–150.
  25. Ehresmann . Charles . Charles Ehresmann . 1963 . Catégories structurées . Structured categories . Annales scientifiques de l'École Normale Supérieure . fr . 80 . 4 . 349–426 . 10.24033/asens.1125 . free.
  26. Ehresmann . Charles . 1968 . Esquisses et types des structures algébriques . Sketches and types of algebraic structures . Bul. Inst. Politeh. Iaşi . New Series . fr . 14 . 18 . 1–14 . 0238918 . 0196.03102.
  27. Ehresmann . Charles . Bastiani . Andrée . Andrée Ehresmann . 1972 . Categories of sketched structures . . en . 13 . 2 . 105–214 . 0263.18009.
  28. Web site: Oeuvres de Charles Ehresmann. 2022-02-18. ehres.pagesperso-orange.fr.
  29. Audin. Michèle. Michèle Audin. 2009. Publier sous l'Occupation. Autour du cas de Jacques Feldbau et de l'Académie des Sciences. Publish under the Occupation. On the case of Jacques Feldbau and the Academy of Sciences. Revue d'histoire des mathématiques. fr. 15. 1. 7–57.
  30. Audin. Michèle. Michèle Audin. March 2008. Differential Geometry, Strasbourg, 1953. Notices of the AMS. 55. 366–370.

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