Luc Illusie

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Luc Illusie
University of Paris-Sud
Doctoral advisorAlexander Grothendieck[2]
Doctoral studentsGérard Laumon

Luc Illusie (French: [ilyzi]; born 1940)[2] is a French mathematician, specializing in algebraic geometry. His most important work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry.[2] In 2012, he was awarded the Émile Picard Medal of the French Academy of Sciences.

Biography

Luc Illusie entered the

Alexandre Grothendieck, collaborating with him on two volumes of the latter's Séminaire de Géométrie Algébrique du Bois Marie. In 1970, Illusie introduced the concept of the cotangent complex
.

A researcher in the

University of Paris-Sud, retiring as emeritus professor in 2005.[3] Between 1984 and 1995, he was the director of the arithmetic and algebraic geometry group in the department of mathematics of that university. Torsten Ekedahl [sv] and Gérard Laumon
are among his students.

Thesis

In May 1971, Illusie defended a

University of Paris-Sud
, in front of a jury composed of
Henri Cartan.[4]

The thesis was published in French by Springer-Verlag as a two-volume book (in 1971[5] & 1972[6]). The main results of the thesis are summarized in a paper in English (entitled "Cotangent complex and Deformations of torsors and group schemes") presented in Halifax, at Dalhousie University, in January 1971 as part of a colloquium on algebraic geometry.[4] This paper, originally published by Springer-Verlag in 1972,[7] also exists in a slightly extended version.[4]

Illusie's construction of the cotangent complex generalizes that of Michel André[8] and Daniel Quillen[9] to morphisms of ringed topoi. The generality of the framework makes it possible to apply the formalism to various first-order

deformation problems: schemes, morphisms of schemes
, group schemes and torsors under group schemes. Results concerning commutative group schemes in particular were the key tool in Grothendieck's proof of his existence and structure theorem for infinitesimal deformations of
Mordell conjecture
. In Chapter VIII of the second volume of the thesis, Illusie introduces and studies derived de Rham complexes.

Awards

Illusie has received the Langevin Prize of the French Academy of Sciences in 1977 and, in 2012, the Émile Picard Medal of the French Academy of Sciences for "his fundamental work on the cotangent complex, the Picard–Lefschetz formula, Hodge theory and logarithmic geometry".[1]

Selected works

  • Complexe cotangent et déformations, Lecture Notes in Mathematics 239 et 283, Berlin and New York,
    Springer
    , 1971–1972.
  • (ed.) Cohomologie ℓ-adique et fonctions L, Séminaire de Géométrie Algébrique du Bois-Marie 1965–66, SGA 5, dir. A. Grothendieck, Lecture Notes in Mathematics 589, Berlin and New York,
    Springer
    , 1977.
  • (with
    Springer
    , 1971.
  • "Complexe de de Rham–Witt et cohomologie cristalline", Annales Scientifiques de l'École Normale Supérieure, 1979, ser. 4, vol. 12, 4, pp. 501–661, url=http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1979_4_12_4/ASENS_1979_4_12_4_501_0/ASENS_1979_4_12_4_501_0.pdf.
  • (coed. with
    Springer
    , 1981.
  • (with Michel Raynaud), "Les suites spectrales ssociées au complexe de De Rham–Witt", Publ. Math. IHÉS, vol. 57, 1983, pp. 73–212.
  • (with Pierre Deligne),"Relèvements modulo p2 et décomposition du complexe de de Rham", Inv. math. (1987), vol. 89, pp. 247–270.
  • "Sur la formule de Picard–Lefschetz", in Algebraic Geometry 2000, ed. Azumino (Hotaka), Advanced Studies in Pure Mathematics 36, 2002, pp. 249–268, Mathematical Society of Japan, Tokyo.

References

  1. ^ a b "Médaille Émile Picard (Mathématique): lauréats – Prix de l'Académie des sciences" (PDF). French Academy of Sciences. 3 October 2012. Retrieved 27 July 2016.
  2. ^ a b c d "Luc Illusie. Mathématicien". CNRS Le journal. Retrieved 27 July 2016.
  3. ^ "Luc Illusie". Mathematics Department, Université Paris-Sud. Retrieved 27 July 2016.
  4. ^ a b c Illusie, Luc (1971). "Complexe cotangent; application à la théorie des déformations, Thèses présentées au Centre d'Orsay de l'Université Paris-Sud pour obtenir le grade de docteur es-sciences [Orsay – Série A, n° 749], Publications mathématiques d'Orsay 23, Bibliothèque de la Faculté des sciences Mathématique, 20415" (PDF).
  5. ISSN 0075-8434
    .
  6. ISSN 0075-8434
    .
  7. ISBN 978-3-540-37609-5
    .
  8. ^ André, Michel (1974). Homologie des algèbres commutatives. Springer-Verlag. p. 287.
  9. ISBN 9780821814178
    .
  10. ^ Illusie, Luc (1985). "Déformations de groupes de Barsotti–Tate (d'après A. Grothendieck)". Seminar on Arithmetic Bundles: The Mordell Conjecture (Paris, 1983/84). Astérisque. 127: 151–198.

External links

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