Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. He has received numerous awards and honors for his mathematical research and exposition, including the Fields Medal in 1954 and the Abel Prize in 2003.

Biography

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. He is currently a professor at the Collège de France.

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; he apparently thought that homotopy theory, where he had started, was already overly technical. 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.

The Weil conjectures

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.

In later years Serre was sometimes a source of counterexamples to over-optimistic extrapolations. He also had a close working relationship with Pierre Deligne, who eventually finished the proof of the Weil conjectures.

Other work

From 1959 onward Serre's interests turned towards number theory, in particular class field theory and the theory of complex multiplication.

Amongst his most original contributions were: the concept of algebraic K-theory; the Galois representation theory for ?-adic cohomology and the conceptions that these representations were "large"; and the Serre conjecture on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

Honours 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 is the first recipient of the Abel Prize in 2003. Serre and John Thompson are the only laureates of all three of the Fields Medal, the Wolf Prize, and the Abel Prize.

See also

• Serre duality
• Serre's multiplicity conjectures
• Serre's property FA
• Serre conjecture (number theory)
• Serre spectral sequence
• Serre fibration
• Serre twist sheaf
• Thin set in the sense of Serre
• Quillen-Suslin theorem
• Nicolas Bourbaki

Bibliography

• Groupes Algébriques et Corps de Classes (1959) as Algebraic Groups and Class Fields (1988)
• Corps Locaux (1962) as Local Fields (1980)
• Cohomologie Galoisienne (1964) Collège de France course 1962?3, as Galois Cohomology (1997)
• Algèbre Locale, Multiplicités (1965) Collège de France course 1957?8, as Local Algebra (2000)
• Lie Algebras and Lie Groups (1965) 1964 Harvard lectures
• Algèbres de Lie Semi-simples Complexes (1966) as Complex Semisimple Lie Algebras (1987)
• Abelian ?-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 L. Kleiman
• Cohomological Invariants in Galois Cohomology (2003) with Skip Garibaldi and Alexander Merkurjev
• Grothendieck?Serre Correspondence (2003) edited with Pierre Colmez

External links

• Jean-Pierre Serre at the French Academy of Sciences, in French.
• Jean-Pierre Serre at the Collège de France, in French.

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