Jean-Pierre Serre (French: [sɛʁ]; born 15
September 1926) is a French mathematician
who has made contributions to algebraic topology,
algebraic geometry, and algebraic number theory.
He was awarded the Fields Medal in 1954 and
the inaugural Abel Prize in 2003.
== Biography ==
=== 
Personal life ===
Born in Bages, Pyrénées-Orientales, France,
to pharmacist parents, 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. His wife, Professor Josiane Heulot-Serre,
was a chemist; she also was the director of
the Ecole Normale Supérieure de Jeunes Filles.
Their daughter is the former French diplomat,
historian and writer Claudine Monteil. The
French mathematician Denis Serre is his nephew.
He practices skiing, table tennis, and rock
climbing (in Fontainebleau).
=== 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, where he introduced 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 one of the major problems in topology.
In his speech at the Fields Medal award ceremony
in 1954, Hermann Weyl gave high praise to
Serre, and also made the point that the award
was for the first time awarded to a non-analyst.
Serre subsequently changed his research focus.
==== 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 by Pierre Deligne.
==== 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 Hyman 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. He went
on to win the Balzan Prize in 1985, the Steele
Prize in 1995, the Wolf Prize in Mathematics
in 2000, and was the first recipient of the
Abel Prize in 2003. He has been awarded other
prizes, such as the Gold Medal of the French
National Scientific Research Centre (Centre
National de la Recherche Scientifique, CNRS).
He is a foreign member of several scientific
Academies (France, US, Norway, Sweden, Russia,
the Royal Society, Royal Netherlands Academy
of Arts and Sciences (1978)) and has received
many honorary degrees (from Cambridge, Oxford,
Harvard, and others). In 2012 he became a
fellow of the American Mathematical Society.Serre
has been awarded the highest honors in France
as Grand Cross of the Legion of Honour (Grand
Croix de la Légion d'Honneur) and Grand Cross
of the Legion of Merit (Grand Croix de l'Ordre
National du Mérite).
== See also ==
List of things named after Jean-Pierre Serre
Nicolas Bourbaki
== Bibliography ==
Groupes Algébriques et Corps de Classes (1959),
Hermann, translated into English as Algebraic
Groups and Class Fields (1988), Springer-Verlag
Corps Locaux (1962), Hermann, as Local Fields
(1980), Springer-Verlag
Cohomologie Galoisienne (1964) Collège de
France course 1962–63, as Galois Cohomology
(1997), Springer-Verlag
Algèbre Locale, Multiplicités (1965) Collège
de France course 1957–58, as Local Algebra
(2000), Springer-Verlag
"Lie algebras and Lie groups" (1965) Harvard
Lectures, Springer-Verlag.
Algèbres de Lie Semi-simples Complexes (1966),
as Complex Semisimple Lie Algebras (1987),
Springer-Verlag
Abelian ℓ-Adic Representations and Elliptic
Curves (1968), CRC Press, reissue. Addison-Wesley.
1989.
Cours d'arithmétique (1970), PUF, as A Course
in Arithmetic (1973), Springer-Verlag
Représentations linéaires des groupes finis
(1971), Hermann, as Linear Representations
of Finite Groups (1977), Springer-Verlag
Arbres, amalgames, SL2 (1977), SMF, as Trees
(1980), Springer-Verlag
Oeuvres/Collected Papers in four volumes (1986)
Vol. IV in 2000, Springer-Verlag
Lectures on the Mordell-Weil Theorem (1990),
Vieweg
Topics in Galois Theory (1992), CRC Press
"Cohomological Invariants in Galois Cohomology
(2003) with Skip Garibaldi and Alexander Merkurjev,
AMS
"Exposés de séminaires 1950–1999" (2001),
SMF
Grothendieck–Serre Correspondence (2003),
bilingual edition, edited with Pierre Colmez,
SMF-AMS
"Lectures on N_X(p)" (2012), AK Peters, CRC
Press
Correspondance Serre-Tate (2015), edited with
Pierre Colmez, SMF
"Finite Groups: an Introduction" (2016), Higher
Education Press & International PressA list
of corrections, and updating, of these books
can be found on his home page at College de
France.
== Notes ==
== External links ==
O'Connor, John J.; Robertson, Edmund F., "Jean-Pierre
Serre", MacTutor History of Mathematics archive,
University of St Andrews.
Jean-Pierre Serre at the Mathematics Genealogy
Project
Jean-Pierre Serre, Collège de France, biography
and publications.
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)
