Jean-Pierre Serre

from Wikipedia, the free encyclopedia
Jean-Pierre Serre at the summer school on the Serre conjecture at the CIRM in Luminy, July 19, 2007

Jean-Pierre Serre (born September 15, 1926 in Bages in the French Pyrénées-Orientales department ) is one of the leading mathematicians of the 20th century. He is considered a pioneer of modern algebraic geometry , number theory and topology . Serre is the recipient of the Fields Medal and the Abel Prize . He was awarded the Fields Medal at the age of 27 and is thus the youngest recipient of this award so far (2018).

Life

Serre's parents were pharmacists. He went to the Nîmes high school (Lycée de Nîmes), won the national Concours général in mathematics in 1944 and studied from 1945 to 1948 at the École normal supérieure in Paris . He received his doctorate from the Sorbonne in 1951. During this time he became a member of the mathematicians' circle Nicolas Bourbaki . From 1948 to 1954 he worked at the Center National de la Recherche Scientifique (CNRS) in Paris, first as Attaché de Recherches and later as Maître de Recherches . 1954-1956 he was Maître de conférences at the University of Nancy and was then since 1956 professor at the Collège de France in Paris (Chair of Algebra and Geometry). He has held an honorary professorship there since 1994.

He was visiting professor at Harvard and often at the Institute for Advanced Study (first 1955 to 1957).

His hobbies include skiing, rock climbing and table tennis.

From 1983 to 1986 he was with Ludwig Faddejew Vice President of the International Mathematical Union . In 1970 he was President of the Société Mathématique de France .

Jean-Pierre Serre (3rd from left) with Josiane Serre (behind him), René Thom (left) and others in Oberwolfach in 1949

He was married to the chemist Josiane Heulot-Serre (1922-2004), the former director of the École normal supérieure de jeunes filles in Sèvres . In 1949 their daughter Claudine Monteil was born. As a feminist author and writer, she wrote the biographies of Simone de Beauvoir and Charles Chaplin and his wife Oona .

The mathematician Denis Serre is his nephew.

Works

From a very young age, Serre was one of Henri Cartan's most outstanding students . Around 1950 he dealt with algebraic topology and applied Jean Leray's spectral sequences to the fiber bundle spaces from a topological space X as the basis and the space of the paths in X as the fiber (loop space method). In this way he was able to find relationships between the homology groups in fiber bundle spaces and between homology and homotopy groups. The application in the determination of the homotopy groups of spheres, a notoriously difficult area, caused a sensation at the time (dissertation 1951). He proved that the m-th homotopy group of the n-dimensional sphere is finite for m> n, except for the case n is even and m = 2n – 1.

Serre 2009

After a stay in Princeton in 1952, where he a. a. attended the Artin - Tate seminar on class field theory, after returning to Paris in the Cartan seminar he turned to the current topics of functional theory of several variable and algebraic geometry , which he developed with the help of Jean Leray's sheaf theory and the methods of algebraic topology ( cohomology theory ) placed on a new foundation. At first this happened for the results just obtained by Cartan and Oka in the function theory of several variables. Work on generalizations of the Riemann-Roch theorem (which simultaneously pushed Hirzebruch and Kodaira) in 1953 finally led him to algebraic geometry from 1954 . From the discussions in the Cartan seminar in the mid-1950s, the foundation stone for Alexander Grothendieck's theory of schemes emerged , on which Grothendieck and his school rebuilt algebraic geometry. Two of Serre's best-known articles from this period are FAC ( Faisceaux Algébriques Cohérents , on the coherence of coherent module sheaves ) from 1955 and GAGA ( Géometrie Algébrique et Géométrie Analytique ) from 1956. With "analytical geometry" is meant the functional theory of several complex variables. Serre's principle of duality is well known . Grothendieck and Serre worked closely together from the 1950s to the late 1960s.

From 1959 Serre dealt mainly with number theory , especially with the expansion of Galois cohomology for the class field theory and the Galois representations in the theory of elliptic curves over the rational numbers. Here he formulated the Serre conjecture in the theory of "two-dimensional" representations of the " absolute Galois group ". The aim of his work was the formal representation of an absolute Galois group of any number field , that is, the group of its automorphisms. Therefore, special representations (places of action) of this group are examined, e.g. B. in the "n- torsion points" (rational points of the curve, which are "added" n times using the secant / tangent method of Poincaré zero) elliptical curves. Since these are geometrically (since twice periodic) in the shape of a torus , one speaks of "two-dimensional representation". In 1972 Serre proved his Open image theorem for elliptic curves (without complex multiplication ) over algebraic number fields . It says that the representations of the Galois group of body extensions of , which were formed by adding the torsion points, are "as large as possible" in the group .

He also initiated the theory of p-adic modular forms together with Nicholas Katz around 1972 .

His book A course in Arithmetic brings in a tight space both a discussion of square forms and the theory of modular forms (with application to lattices ). He received the Leroy P. Steele Prize for this .

Serre also made an important contribution to the chain of evidence that led from Gerhard Frey to Ken Ribet to Andrew Wiles to prove the Fermat conjecture .

His friendship with Armand Borel also resulted in his interest in Lie groups and their algebras , discrete groups and their geometry and representation theory of groups. It was then only natural that he should also publish the collected works of Ferdinand Georg Frobenius .

Serre is also known for various guesswork. In addition to the above-mentioned conjecture about Galois representations, for example for a conjecture in commutative algebra , which was independently proven by Andrei Alexandrowitsch Suslin and Daniel Quillen (that projective modules over polynomial rings are free).

Awards and honors

He has multiple honorary doctorates : Cambridge (1978), Stockholm (1980), Glasgow (1983), Athens (1996), Harvard (1998), Durham (2000), London (2001), Oslo (2002), Oxford (2003), Bucharest (2004), Barcelona (2004), Madrid (2006), Mc-Gill University (2008).

Quotes

  • Precision combined with informal brevity - that is the ideal in books as well as in lectures (interview with Leong, Chong 1985)
  • Some mathematicians have clear and far-reaching "programs" ... I never had such a program, not even a small one
  • Regarding the question of how one could motivate students for mathematics: I have the theory that one should discourage young people from wanting to become mathematicians. There is no need for too many mathematicians. But if they still want to study math after that, they should indeed be encouraged and helped. In high school, the most important point to show students that math exists and that it is not a dead science (there is a tendency for many to assume that the only open problems are in physics and biology). The main flaw in standard math classes is that teachers never mention these open questions, which is a shame because there are many of them, especially in number theory, that students could understand very well. In an interview at the ICM in 2006, when asked what advice he would give a good math student, the answer was: A good student doesn't need advice.

Fonts

Books:

  • Algèbre locale, multiplicités. Cours professé au Collège de France, 1957–1958. Rédigé by Pierre Gabriel . sn, sl 1958, (English: Local algebra. Springer, Berlin et al. 2000, ISBN 3-540-66641-9 ).
  • Groupes algébriques and corps de classes. Cours au Collège de France (= Publications de l'Institut de Mathématique de l'Université de Nancago. 7 = Actualités Scientifiques et Industrielles. 1264). Herrmann, Paris 1959, (English: Algebraic groups and class fields (= Graduate Texts in Mathematics . 117). Springer, New York NY et al. 1988, ISBN 0-387-96648-X ).
  • Corps locaux (= Publications de l'Institut de Mathématique de l'Université de Nancago. 8 = Actualités Scientifiques et Industrielles. 1296). Hermann, Paris 1962, (English: Local fields (= Graduate Texts in Mathematics. 67). Springer, New York NY 1979, ISBN 0-387-90424-7 ).
  • Cohomology Galoisienne. Cours au Collège de France, 1962–1963. Collège de France, Paris 1963, (English: Galois cohomology. Springer, Berlin et al. 1997, ISBN 3-540-61990-9 ).
  • Lie algebras and Lie groups. 1965, lectures given at Harvard University. Benjamin, New York NY et al. 1965.
  • Algèbres de Lie semi-simple complexes. Benjamin, New York NY et al. 1966, (English: Complex semisimple Lie algebras. Springer, New York NY et al. 1987, ISBN 3-540-96569-6 ).
  • Représentations linéaires des groupes finis. II, ENS, Cours aux Carrés, April – May 1966. Rédigé by Yves Balasko. Ecole normal supérieure, Paris 1966, (English: Linear representations of finite groups (= Graduate Texts in Mathematics. 42). Springer, New York NY et al. 1977, ISBN 0-387-90190-6 ).
  • Abelian- adic representations and elliptic curves. McGill University lecture notes. Benjamin, New York NY et al. 1968.
  • Cours d'arithmétique (= Collection SUP. Le Mathématicien. 2, ZDB -ID 185116-0 ). Presses Universitaires de France, Paris 1970, (English: A course in arithmetic (= Graduate Texts in Mathematics. 7). Springer, New York NY et al. 1973, ISBN 0-387-90041-1 ).
  • Arbres, an amalgamated, . Cours au Collège de France (= Astérisque. 46, ISSN  0303-1179 ). Société Mathématique de France, Paris 1977, (English: Trees. Springer, Berlin et al. 1980, ISBN 0-387-10103-9 ).
  • Autour du théorème de Mordell-Weil. Cours au Collège de France, January – mars 1980. and: Cours au Collège de France, October 1980 – January 1981 (= Publications Mathematiques de l'Universite Pierre et Marie Curie. 62 and 65, ISSN  1151-1745 ). Notes rédigées by Michel Waldschmidt . 2 volumes. Université Pierre et Marie Curie, Paris 1980–1981, (English: Lectures on the Mordell-Weil theorem (= Aspects of Mathematics. E, 15). Vieweg, Braunschweig et al. 1989, ISBN 3-528-08968-7 ).
  • Oeuvres. = Collected Works. 4 volumes. Springer, Berlin et al. 1986-2000, ISBN 3-540-15621-6 .
  • Topics in Galois theory (= Research Notes in Mathematics. 1). Notes written by Henri Damon. 1992, Jones and Bartlett, Boston MA et al. 1992, ISBN 0-86720-210-6 .
  • Grothendieck-Serre correspondence. Editors Pierre Colmez , Jean-Pierre Serre. Bilingual edition. American Mathematical Society, Providence RI 2004, ISBN 0-8218-3424-X (the numerous telephone conversations between the two, especially during their simultaneous presence in Paris, are not recorded).

Some essays and interviews:

  • Chi Tat Chong, Yap Kok Leong: An interview with Jean-Pierre Serre. In The Mathematical Intelligencer . Volume 8, No. 4, 1986, pp. 8-13, (online here ( Memento from July 19, 2011 in the Internet Archive )).
  • Martin Raussen, Christian Skau: Interview with Jean-Pierre Serre. In: Notices of the American Mathematical Society. Volume 51, No. 2, 2004, pp. 210-214 .

literature

  • Jean-Pierre Serre. In: Shiing S. Chern , Friedrich Hirzebruch (Ed.) Wolf Prize in Mathematics. Volume 2. World Scientific, Singapore et al. 2001, ISBN 981-02-3946-7 , pp. 523-551.
  • Pilar Bayer : Jean-Pierre Serre, Medalla Fields. In: La Gaceta de la Real Sociedad Matemática Española. Volume 4, No. 1, 2001, pp. 211-247 .
  • Pilar Bayer: Jean-Pierre Serre. An overview of his work. In: Helge Holden , Ragni Piene (Eds.): The Abel Prize. 2003-2007. The First Five Years. Springer, Berlin et al. 2010, ISBN 978-3-642-01372-0 , pp. 27-84, (with interview with Marc Kirsch (Serre: My first fifty years at the College de France) and list of publications).

Web links

Remarks

  1. ↑ For sufficiently large there is a surjective mapping.

Individual evidence

  1. ↑ The last letter from the Grothendieck-Serre correspondence, which was published in 2001 by the Societe Mathematique de France, dates from January 1969, only to be continued with a few letters in the mid-1980s
  2. ↑ In: Inventiones Mathematicae. Volume 15, No. 4, 1972, pp. 259-331.
  3. ^ Serre: Formes modulaires et fonctions zêta p-adiques. In: Willem Kuijk, Jean-Pierre Serre (Ed.): Modular functions of one variable III. Proceedings International Summer School, University of Antwerp, RUCA, July 17 - August 3, 1972 (= Lecture Notes in Mathematics. 350). Springer, Berlin et al. 1973, ISBN 3-540-06483-4 , pp. 191-268.
  4. ↑ Manfred Lindinger: Master of all numbers. The mathematician Jean-Pierre Serre turns ninety. In: Frankfurter Allgemeine Zeitung , September 15, 2016, p. 14.
  5. ↑ Interview with Leong, Chong, Singapore 1985, some mathematicians have clear and far-ranging “programs” ... [he mentions Grothendieck and Langlands as examples] ... I never had such a program, not even a small size one. I just work on things which happen to interest me at the moment.
  6. ↑ Interview with Leong, Chong in: The Mathematical Intelligencer. 1986, I have a theory on this, which is that one should first discourage young people from doing mathematics. There is no need for too many mathematicians. But if after that they still insist on studying mathematics, then one should indeed encourage and help them. As for high school students, the main point is to make them understand that mathematics exists, that it isn't dead (they have a tendency to think that the only open questions remaining are in physics and biology). The defect in the traditional way of teaching mathematics is that the teacher never mentions these questions. That's a pity. There are many such, for instance in number theory, that teenagers could very well understand.
  7. ↑ Interview with Jean Pierre Serre, Fields Medal and Abel Prize Winner. In: ICM2006. Bulletin. 18, ( online ).
personal data
SURNAME Serre, Jean-Pierre
BRIEF DESCRIPTION French mathematician
DATE OF BIRTH September 15, 1926
PLACE OF BIRTH Bages