André Lichnerowicz Explained

André Lichnerowicz
Birth Date:January 21, 1915
Birth Place:Bourbon l'Archambault, France
Death Place:Paris
Nationality:French
Fields:Mathematics
Workplaces:University of Paris
College de France
Alma Mater:École Normale Supérieure
Thesis Title:Problemes Globaux en Mécanique Relativiste
Thesis Year:1939
Doctoral Advisor:Georges Darmois
Doctoral Students:Thierry Aubin
Claude Berge
Edmond Bonan
Marcel Berger
Yvonne Choquet-Bruhat

Paul Gauduchon
Richard Kerner
Yvette Kosmann

Charles-Michel Marle
Jean-Marie Souriau
Marie-Hélène Schwartz
Known For:Lichnerowicz conjecture
Lichnerowicz Laplacian
Lichnerowicz formula
Poisson manifold
Awards:Peccot Lectures, 1944
Prix de la langue française, 1988

André Lichnerowicz (January 21, 1915, Bourbon-l'Archambault – December 11, 1998, Paris) was a French differential geometer and mathematical physicist. He is considered the founder of modern Poisson geometry.[1] [2] [3]

Biography

His grandfather Jan fought in the Polish resistance against the Prussians. Forced to flee Poland in 1860, he finally settled in France, where he married a woman from Auvergne, Justine Faure. Lichnerowicz's father, Jean, held agrégation in classics and was secretary of the Alliance française, while his mother, a descendant of paper makers, was one of the first women to earn the agrégation in mathematics. Lichnerowicz's paternal aunt, Jeanne, was a novelist and translator known under the pseudonym .[4]

André attended the Lycée Louis-le-Grand and then the École Normale Supérieure in Paris, gaining agrégation in 1936. After two years, he entered the Centre national de la recherche scientifique (CNRS) as one of the first researchers recruited by this institution.

Lichnerowicz studied differential geometry under Élie Cartan. His doctoral dissertation, completed in 1939 under the supervision of Georges Darmois, was entitled "Problemes Globaux en Mécanique Relativiste" (Global problems in relativistic mechanics).[5]

His academic career began under the cloud of Nazi occupation, during World War II. In 1941 he started teaching at the University of Strasbourg, which was moved to Clermont Ferrand and only returned to Strasbourg in 1945, after the end of the war. In November 1943 he was arrested during a raid but managed to escape. During 1944 he was invited to give a Cours Peccot at the Collège de France.

From 1949 to 1952 he held a position at the University of Paris, and in 1952 he was appointed professor at the Collège de France, where he worked until his retirement in 1986.[6]

Lichnerowicz served as president of the Société mathématique de France during 1959.[7] He was elected member of several national and international academies: the Accademia dei Lincei in 1962, the Académie des Sciences in 1963, the Real Academia de Ciencias in 1968,[8] the Académie Royale de Belgique in 1975, the Pontifical Academy of Sciences in 1981,[9] and the in 1984.[10]

In 1988 he was awarded the Prix de la langue française for having illustrated the quality and the beauty of French language in his works.[11] In 2001 he received posthumous (together with his co-authors Alain Connes and Marco Schutzenberger) the Peano Prize for his work Triangle of Thoughts.[12] [13]

In 2008 the André Lichnerowicz Prize was created to reward progresses in Poisson geometry, a research field where Lichnerowicz made pioneering contributions.

Lichnerowicz was a believing Catholic[14] who served as vice-president of the Centre Catholique des Intellectuels Français.[15]

Research

In an interview in his last years, Lichnerowicz self-described his research interests as "Differential geometry and global analysis on manifolds", "the relations between mathematics and physics" and "the mathematical treatment of Einstein’s theory of gravitation".[16] Indeed, his works contributed, among others, to many areas of Riemannian geometry, symplectic geometry and general relativity.

His research in general relativity began with his PhD thesis, where he described necessary and sufficient conditions for a metric of hyperbolic signature to be a global solution of the Einstein field equations. In a series of papers in 1940 with Raymond Marrot, he provided a mathematical formulation of the relativistic kinetic theory.[17] [18] [19] He later worked on gravitational radiation,[20] spinor fields,[21] and propagators[22] on curved space-time, obtaining results which preluded his later works on quantisation and deformation.

Among his contributions to Riemannian geometry, in 1944 he formulated a conjecture about locally harmonic 4-manifolds,[23] which has been later generalised and is now known as Lichnerowicz conjecture. In 1952 he showed, together with Armand Borel, that the restricted holonomy group of a Riemannian manifold is compact.[24] [25] He proved the now standard equivalence of the various definitions of Kähler manifold and he worked on the classification of compact homogeneous Kähler spaces.[26] [27] In 1958 he was one of the first to introduce a relation between the spectrum of the Laplacian and the curvature of the metric.[28] After formalising Cartan’s and Weyl’s theory of spinors in a rigorous framework, he proved in 1963 the Lichnerowicz formula relating the Dirac operator and the Laplace–Beltrami operator acting on spinors.[29]

In the 1970s his interests turned to symplectic geometry and dynamical systems, with many pioneering papers which, in the next decades, would give rise to the modern field of Poisson geometry. Indeed, starting in 1974, together with Moshé Flato and Daniel Sternheimer, Lichnerowicz formulated the first definitions of a Poisson manifold in terms of a bivector, the counterpart of a (symplectic) differential 2-form.[30] [31] [32] He showed later that the same philosophy can be used to generalise contact structures to Jacobi manifolds.[33] In a 1976 paper one can already find the classical formula

[df,dg]=d\{f,g\}

for the Lie algebroid bracket of

T*M

on exact 1-forms via the Poisson bracket of functions. In 1977 Lichnerowicz introduced the operator defining what is now called Poisson cohomology.[34] His 1978 papers on the deformation of the algebra of smooth functions on a Poisson manifold established the new research area of deformation quantisation.[35] [36]

Lichnerowicz published more than 350 papers and supervised 24 Ph.D. students. A collection of scientific contributions from several of his collaborators was published in his honour in occasion of his 60th birthday.[37] In 1982 a personal selection of his own works was published by Hermann.[38]

Pedagogy of mathematics

While pursuing an active research career, Lichnerowicz had a deep interest in mathematics education and pedagogy. From 1963 to 1966 he was President of the International Commission on Mathematical Instruction of the International Mathematical Union.[39] [40] In 1967 the French government created the Lichnerowicz Commission made up of 18 teachers of mathematics. The commission recommended a curriculum based on set theory and logic with an early introduction to mathematical structures. It recommended introduction to complex numbers for seniors in high school, less computation-based instruction, and more development from premises (the axiomatic approach). These reforms have been called New Math and have been repeated internationally.[41] However, the reforms faced stern backlash from parents, who had trouble helping their children with homework,[42] teachers, who found themselves ill-prepared and ill-equipped,[43] and scholars from various disciplines, who deemed the New Math to be simply unsuitable and impractical.[44] [45] [46] Lichnerowicz resigned and the commission was disbanded in 1973. Nevertheless, the influence of the proposed reforms in mathematics education had endured, as the Soviet mathematician Vladimir Arnold recalled in a 1995 interview.[47]

Works in French

Works in English translation

See also

Notes and References

  1. Revuz. André. Berger. Marcel. Choquet-Bruhat. Yvonne. Marle. Charles-Michel. 1999. Andre Lichnerowicz (1915-1998). Gazette des Mathématiciens. French. 82. 90–108.
  2. Revuz. André. Berger. Marcel. Choquet-Bruhat. Yvonne. Marle. Charles-Michel. Bourguignon. Jean-Pierre. 1999. Andre Lichnerowicz (1915-1998). Notices Amer. Math. Soc.. 46. 11. 1387–1396.
  3. Kosmann-Schwarzbach. Yvette. 2009. Tribute to André Lichnerowicz (1915–1998). Notices of the AMS. 56. 2. 244–246.
  4. Web site: André Lichnerowicz - Biography. 2021-07-23. Maths History. en.
  5. Web site: André Lichnérowicz - The Mathematics Genealogy Project. 2021-07-23. www.genealogy.math.ndsu.nodak.edu.
  6. Web site: Biographie. 2021-07-23. www.college-de-france.fr. fr.
  7. Web site: 2016-10-24. Anciens Présidents Société Mathématique de France. 2021-07-23. https://web.archive.org/web/20161024023304/http://smf.emath.fr/content/anciens-presidents. 2016-10-24.
  8. Web site: Académicos Históricos - Real Academía de Ciencias Exactas, Físicas y Naturales. 2021-07-23. rac.es.
  9. Germain. Paul. 2000. Commemoration of academicians - André Lichnerowicz. Science and the Future of Mankind. Pontificiae Academiae Scientiarum Scripta Varia. Proceeding of the Jubilee Plenary Session 10–13 November 2000. 40–43. 88-7761-075-1. 23 July 2021. 16 November 2019. https://web.archive.org/web/20191116170648/http://www.pas.va/content/dam/accademia/pdf/sv99/sv99.pdf. dead.
  10. Web site: André LICHNEROWICZ. 2021-07-23. www.accademiadellescienze.it.
  11. Web site: André Lichnerowicz. 2021-07-23. www.prix-litteraires.net. 2021-07-23. https://web.archive.org/web/20210723214448/https://www.prix-litteraires.net/detail_prix_auteur.php?auteur=1207_Andre_Lichnerowicz. dead.
  12. Web site: Premio Peano Associazione Subalpina Mathesis. 3 November 2015 . 2021-07-23. it-IT.
  13. [Alain Connes]
  14. Web site: Andre Lichnerowicz.
  15. Book: Les intellectuels catholiques dans la société française : Le Centre catholique des intellectuels français (1941-1976). 9782753524286. Chapitre 9. Des années riches de promesses. Histoire. 24 February 2015. 181–199. Presses universitaires de Rennes.
  16. Book: Schmitt, Marian. Hommes de science : 28 portraits. 1990. Hermann éditeurs des sciences et des arts. 978-2-7056-6124-3. Paris. 804198208.
  17. Lichnerowicz. André. Marrot. Raymond. 1940. Remarques sur l'équation intégrodifferentielle de Boltzmann. C. R. Acad. Sci. Paris. French. 210. 391–393. 0023.32302.
  18. Lichnerowicz. André. Marrot. Raymond. 1940. Propriétés statistiques des ensembles de particules en relativite restreinte. C. R. Acad. Sci. Paris. French. 210. 759–761. 0024.37704.
  19. Lichnerowicz. André. Marrot. Raymond. 1940. Sur l'équation intégrodifferentielle de Boltzmann. C. R. Acad. Sci. Paris. French. 211. 531–533. 0024.31804.
  20. Lichnerowicz. André. 1960-12-01. Ondes et radiations électromagnétiques et gravitationelles en relativité générale. Annali di Matematica Pura ed Applicata. fr. 50. 1. 1–95. 10.1007/BF02414504. 183813776. 1618-1891. free.
  21. Lichnerowicz. André. 1964. Champs spinoriels et propagateurs en relativité générale. Bulletin de la Société Mathématique de France. fr. 92. 11–100. 10.24033/bsmf.1604. free.
  22. Lichnerowicz. André. 2018-10-16. Republication of: Propagators, commutators and anti-commutators in general relativity. General Relativity and Gravitation. en. 50. 11. 145. 10.1007/s10714-018-2433-x. 2018GReGr..50..145L. 125130979. 1572-9532.
  23. Lichnerowicz. André. 1944. Sur les espaces riemanniens complètement harmoniques. Bulletin de la Société Mathématique de France. fr. 72. 146–168. 10.24033/bsmf.1359. free.
  24. Borel. Armand. Lichnerowicz. André. 1952. Espaces riemanniens et hermitiens symétriques. C. R. Acad. Sci. Paris. French. 234. 2332–2334. 0046.39803.
  25. Borel. Armand. Lichnerowicz. André. 1952. Groupes d'holonomie des variétés riemanniennes. C. R. Acad. Sci. Paris. French. 234. 1835–1837. 0046.39801.
  26. Lichnerowicz. André. 1953. Espaces homogènes kählériens. Colloques Internat. Centre Nat. Rech. Sci.. French. 52. 171–184. 0053.11603.
  27. Lichnerowicz. André. 1990. Groupes kähleriens. Comptes Rendus de l'Académie des Sciences, Série I. French. 310. 9. 671–676. 0706.22006.
  28. Lichnerowicz . André . Géométrie des groupes de transformations . Travaux et Recherches Mathématiques . 1958 . 3 . Dunod, Paris . fr . 911753544. 124009. 0096.16001.
  29. Lichnerowicz. André. 1963. Spineurs harmoniques. C. R. Acad. Sci. Paris. French. 257. 7–9. 0136.18401.
  30. Flato. Moshé. Lichnerowicz. André. Sternheimer. Daniel. 1974. Déformations 1-différentiables d'algèbres de Lie attachees à une variété symplectique ou de contact. Comptes Rendus de l'Académie des Sciences, Série A. French. 279. 877–881. 0289.53031.
  31. Flato. Moshé. Lichnerowicz. André. Sternheimer. Daniel. 1975. Déformations 1-différentiables des algèbres de Lie attachees à une variété symplectique ou de contact. Compos. Math.. French. 31. 47–82. 0317.53039.
  32. Flato. Moshé. Lichnerowicz. André. Sternheimer. Daniel. 1976. Algèbres de Lie attachees à une variété canonique. J. Math. Pures Appl. . 9e Série. French. 54. 445–480. 0318.53041.
  33. Lichnerowicz. André. Les variétés de Jacobi et leurs algèbres de Lie associees.. J. Math. Pures Appl. . 9e Série. French. 57. 453–488. 0407.53025.
  34. Lichnerowicz. André. 1977-01-01. Les variétés de Poisson et leurs algèbres de Lie associées. Journal of Differential Geometry. 12. 2. 10.4310/jdg/1214433987. 0022-040X. free.
  35. 1978-03-01. Deformation theory and quantization. I. Deformations of symplectic structures. Annals of Physics. en. 111. 1. 61–110. 10.1016/0003-4916(78)90224-5. 0003-4916. Bayen. F.. Flato. M.. Fronsdal. C.. Lichnerowicz. A.. Sternheimer. D.. 1978AnPhy.111...61B.
  36. 1978-03-01. Deformation theory and quantization. II. Physical applications. Annals of Physics. en. 111. 1. 111–151. 10.1016/0003-4916(78)90225-7. 0003-4916. Bayen. F.. Flato. M.. Fronsdal. C.. Lichnerowicz. A.. Sternheimer. D.. 1978AnPhy.111..111B.
  37. Book: Differential Geometry and Relativity: A Volume in Honour of André Lichnerowicz on His 60th Birthday. 1976. Springer Netherlands. 978-94-010-1510-3. Cahen. M.. Dordrecht. en. 10.1007/978-94-010-1508-0. Flato. M..
  38. Book: Lichnerowicz, André. Choix d'œuvres mathématiques. 1982. Hermann. 2-7056-5946-3. Paris. 9555359.
  39. Book: Búrigo . E. Z. . Researching the History of Mathematics Education: An International Overview . 2018 . Springer . Real Numbers in School: 1960s Experiments in France and Brazil . 9783319682945 . 30 . Fulvia . Furinghetti . Alexander . Karp . https://books.google.com/books?id=GZxBDwAAQBAJ&pg=PA30.
  40. Web site: Historical Sketch of ICMI International Mathematical Union (IMU). 2021-07-24. www.mathunion.org.
  41. Book: Mashaal, Maurice. Bourbaki : a secret society of mathematicians. 2006. American Mathematical Society. 0-8218-3967-5. Providence, RI. 63297898.
  42. News: Knudson . Kevin . 2015 . The Common Core is today's New Math – which is actually a good thing . The Conversation . September 9, 2015.
  43. Web site: Gispert . Hélène . L'enseignement des mathématiques au XXe siècle dans le contexte français . live . https://web.archive.org/web/20170715164210/http://culturemath.ens.fr/histoire%20des%20maths/htm/Gispert08-reformes/Gispert08.htm . July 15, 2017 . November 4, 2020 . CultureMATH . FR.
  44. Feynman . Richard P. . Richard Feynman . 1965 . New Textbooks for the 'New' Mathematics . Engineering and Science . XXVIII . 6 . 9–15 . 0013-7812.
  45. Book: Kline, Morris . Why Johnny Can't Add: The Failure of the New Math . Why Johnny Can't Add . . 1973 . 0-394-71981-6 . New York . 17, 98 . Morris Kline.
  46. Book: Simmons, George F. . Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry: Geometry, Algebra, Trigonometry . . 2003 . 9781592441303 . 33 . Algebra  - Introduction . George F. Simmons . https://books.google.com/books?id=dN1KAwAAQBAJ&pg=PA33.
  47. Lui . S.H. . 1995 . An Interview with Vladimir Arnold . Notices of the American Mathematical Society . 44 . 4 . 432–8.
  48. 10.1063/1.3061795. Review of Théories Relativistes de la Gravitation et de l'Électromagnétisme by A. Lichnerowicz. 1955. Teichmann, T.. Physics Today. 8. 10. 24–26. 1955PhT.....8R..24L.
  49. Le Corbeiller, P.. Philippe Le Corbeiller. Review of Theories Relativistes de la Gravitation et de l'electromagnétisme by A. Lichnerowicz. Science. 122. 3166. 1955. 424. 10.1126/science.122.3166.424.b. 239870403 .
  50. Web site: Stenger, Allen. December 1, 2016. Review of Dover reprint of Elements of Tensor Calculus by A. Lichnerowicz. MAA Reviews, Mathematical Association of America.
  51. Bergmann, Peter G.. Peter Bergmann. Review of Relativistic Hydrodynamics and Magnetohydrodynamics: Lectures on the Existence of Solutions. Physics Today. 21. 4. 1968. 103–105. 0031-9228. 10.1063/1.3034878.
  52. Mackey, George W.. George Mackey. Review: Algèbre et analyse linéaires, by A. Lichnerowicz. Bull. Amer. Math. Soc.. 1948. 54. 11, Part 1. 1094–1095. 10.1090/s0002-9904-1948-09110-8. free.
  53. Chern, S. S.. Shiing-Shen Chern. Review: Théorie globale des connexions et des groupes d'holonomie, by A. Lichnerowicz. Bull. Amer. Math. Soc.. 1957. 63. 1. 57–59. 10.1090/s0002-9904-1957-10076-7. free.
  54. Manin, Yuri I.. Yuri Manin. Book Review: Triangle of Thoughts. Notices of the American Mathematical Society. March 2002. 49. 3. 325–327.

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