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. With the Fields Medal in 1954 and the Abel Prize in 2003, he has received two of the highest honors in mathematics.
Life and career
Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the Ecole 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 now a professor at the Collège de France.
Early work
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.
In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl praised Serre in apparently 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.
Foundational work in algebraic geometry and 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 FAC (Faisceaux Algébriques Cohérents, on coherent cohomology) and 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. In simple terms, 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 (1954/55) was one based on Witt vector coefficients.
Around 1958 Serre suggested that isotrivial covers of algebraic varieties — those that become trivial after pullback by a finite covering map — should be important. This was one significant step towards the eventual étale covering theory. Grothendieck in SGA4 eventually delivered a full technical development.
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 l-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.
Awards
Serre was awarded the Fields Medal in 1954, and was the first recipient of the Abel Prize in 2003. He also received the Balzan Prize (1985), the Steele Prize (1995), and the Wolf Prize (2000).
