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 a professor at the Collège de France.
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 refers to his dissertation on 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.
While Serre subsequently moved field — at this point he apparently thought that homotopy theory where he had started was already over-technical — 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.
Serre had early on perceived a need to construct more general and refined cohomology theories to tackle these conjectures. In simple terms, 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.
Grothendieck in SGA4 eventually delivered a full technical development. Around 1958 Serre had suggested that isotrivial covers of algebraic varieties should be important — those that become trivial after pullback by a finite covering map. This was one important step towards the eventual étale covering theory.
In the later developments Serre was sometimes a source instead 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.
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.
- O'Connor, John J., and Edmund F. Robertson. "JeanPierre Serre". MacTutor History of Mathematics archive.