Jean-Pierre Serre

For other uses, see Serre (disambiguation).

Jean-Pierre Serre
Born	(1926-09-15) 15 September 1926 Bages, Pyrénées-Orientales, France
Nationality	French
Alma mater	École Normale Supérieure University of Paris
Awards	Abel Prize (2003) Fields Medal (1954) Wolf Prize in Mathematics (2000) Balzan Prize (1985)
Scientific career
Fields	Mathematics
Institutions	Centre National de la Recherche Scientifique Collège de France
Doctoral advisor	Henri Cartan
Doctoral students	Michel Broué John Labute

Jean-Pierre Serre (born 15 September 1926) is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.

Biography

Early years

Born in Bages, Pyrénées-Orientales, France, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994.

Career

From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, in the context of sheaf theory and homological algebra techniques. Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was considered as the major problem in topology.

In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl praised Serre in seemingly extravagant terms, and also made the point that the award was for the first time awarded to an algebraist. Serre subsequently changed his research focus. However, Weyl's perception that the central place of classical analysis had been challenged by abstract algebra has subsequently been justified, as has his assessment of Serre's place in this change.

Algebraic geometry

In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents (FAC), on coherent cohomology, and Géometrie Algébrique et Géométrie Analytique (GAGA).

Even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients. Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients.

Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties — those that become trivial after pullback by a finite étale map — are important. This acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique (SGA) 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures.

Other work

From 1959 onward Serre's interests turned towards group theory, number theory, in particular Galois representations and modular forms.

Amongst his most original contributions were: his "Conjecture II" (still open) on Galois cohomology; his use of group actions on Trees (with H. Bass); the Borel-Serre compactification; results on the number of points of curves over finite fields; Galois representations in ℓ-adic cohomology and the proof that these representations have often a "large" image; the concept of p-adic modular form; and the Serre conjecture (now a theorem) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free. This question led to a great deal of activity in commutative algebra, and was finally answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976. This result is now known as the Quillen-Suslin theorem.

Honors and awards

Serre, at twenty-seven in 1954, is the youngest ever to be awarded the Fields Medal. In 1985, he went on to win the Balzan Prize, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000, and was the first recipient of the Abel Prize in 2003.

He is a foreign member of several scientific Academies (France, US, Norway, Sweden, Russia, ...) and has received about a dozen honorary degrees (Cambridge, Oxford, Harvard, ...).

See also

• Bass–Serre theory
• Serre duality
• Serre's multiplicity conjectures
• Serre's property FA
• Serre fibration
• Serre twist sheaf
• Thin set in the sense of Serre
• Quillen–Suslin theorem (sometimes known as "Serre's Conjecture")
• Nicolas Bourbaki
• Serre's Conjecture concerning Galois representations
• Serre's "Conjecture II" concerning linear algebraic groups

Bibliography

• Groupes Algébriques et Corps de Classes (1959), translated in English as Algebraic Groups and Class Fields (1988)
• Corps Locaux (1962), as Local Fields (1980)
• Cohomologie Galoisienne (1964) Collège de France course 1962–63, as Galois Cohomology (1997)
• Algèbre Locale, Multiplicités (1965) Collège de France course 1957–58, as Local Algebra (2000)
• Serre, Jean-Pierre (2006) [1965], Lie algebras and Lie groups, Lecture Notes in Mathematics, vol. 1500, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-70634-2, ISBN 978-3-540-55008-2, MR 2179691
• Algèbres de Lie Semi-simples Complexes (1966), as Complex Semisimple Lie Algebras (1987)
• Abelian ℓ-Adic Representations and Elliptic Curves (1968)
• Cours d'arithmétique (1970), as A Course in Arithmetic (1973)
• Représentations linéaires des groupes finis (1971), as Linear Representations of Finite Groups (1977)
• Arbres, amalgames, SL₂(1977) as Trees (1980)
• Oeuvres/Collected Papers in four volumes (1986) Vol. IV in 2000
• Lectures on the Mordell-Weil Theorem (1990)
• Topics in Galois Theory (1992)
• Motives (1994) two volumes, editor with Uwe Jannsen and Steven Kleiman
• "Cohomological Invariants in Galois Cohomology (2003) with Skip Garibaldi and Alexander Merkurjev
• "Exposés de séminaires 1950–1999" (2001), SMF
• Grothendieck–Serre Correspondence (2003), bilingual edition, edited with Pierre Colmez

External links

• O'Connor, John J.; Robertson, Edmund F., "Jean-Pierre Serre", MacTutor History of Mathematics Archive, University of St Andrews
• Jean-Pierre Serre, Collège de France, biography and publications.
• Jean-Pierre Serre at the Mathematics Genealogy Project
• Jean-Pierre Serre at the French Academy of Sciences, in French.
• Interview with Jean-Pierre Serre in Notices of the American Mathematical Society.
• An Interview with Jean-Pierre Serre by C.T. Chong and Y.K. Leong, National University of Singapore.
• How to write mathematics badly a public lecture by Jean-Pierre Serre on writing mathematics.
• Biographical page (in French)