• Mathematician Jean - Pierre Serre, 1926
  

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 - Mac Lane 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.

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 could not capture as much topology as singular cohomology with integer coefficients. Among 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.

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

Among 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.

Serre, at twenty seven in 1954, was 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 was a foreign member of several scientific Academies and received about a dozen honorary degrees.

Many theories and result bear his name. E.g., Bass - Serre theory, Serre duality, Serre's multiplicity conjectures, Serre's property FA, Serre fibration, Serre - Tate theorem, Serre twist sheaf, Thin set in the sense of Serre, Quillen - Suslin theorem (sometimes known as "Serre's Conjecture"), Nicolas Bourbaki (member), Serre's Conjecture concerning Galois representations, Serre's "Conjecture II" concerning linear algebraic groups.