Jean-Pierre Serre
Encyclopedia
Jean-Pierre Serre is a French
mathematician
. He has made contributions in the fields of algebraic geometry
, number theory
, and topology
.
, Pyrénées-Orientales
, France
, to pharmicist 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.
, 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. 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.
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.
, 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 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.
. 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 is a foreign member of several scientific Academies (France, US, Norway, Sweden, Russia, ...) and has received about a dozen honorary degrees (Cambridge, Oxford, Harvard, ...).
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He has made contributions in the fields of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
.
Early years
Born in BagesBages, Pyrénées-Orientales
Bages is a commune in the Pyrénées-Orientales department in southern France.-References:*...
, Pyrénées-Orientales
Pyrénées-Orientales
Pyrénées-Orientales is a department of southern France adjacent to the northern Spanish frontier and the Mediterranean Sea. It also surrounds the tiny Spanish enclave of Llívia, and thus has two distinct borders with Spain.- History :...
, France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
, to pharmicist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure
École Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
. He was awarded his doctorate from the Sorbonne
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique
Centre national de la recherche scientifique
The National Center of Scientific Research is the largest governmental research organization in France and the largest fundamental science agency in Europe....
in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
. In 1956 he was elected professor at the Collège de France
Collège de France
The Collège de France is a higher education and research establishment located in Paris, France, in the 5th arrondissement, or Latin Quarter, across the street from the historical campus of La Sorbonne at the intersection of Rue Saint-Jacques and Rue des Écoles...
, a position he held until his retirement in 1994.
Career
From a very young age he was an outstanding figure in the school of Henri CartanHenri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
, working on algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
and then commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, in the context of sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
theory and homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
techniques. Serre's thesis concerned the Leray–Serre spectral sequence
Serre spectral sequence
In mathematics, the Serre spectral sequence is an important tool in algebraic topology...
associated to a fibration
Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...
, 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
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
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
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
has subsequently been justified, as has his assessment of Serre's place in this change.
Algebraic geometry
In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander GrothendieckAlexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
led to important foundational work, much of it motivated by the Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents (FAC), on coherent cohomology, and Géometrie Algébrique et Géométrie Analytique (GAGA
Gaga
- Entertainment :* Lady Gaga , American recording artist* Gaga , a dancing technique* Rara, a type of Caribbean music called Gaga in the Dominican Republic* GNZ-004 Gaga, a mobile weapon in Mobile Suit Gundam 00- Sports :...
).
Even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
over a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
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
Witt vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the ring of p-adic integers.-Motivation:Any p-adic...
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
Étale topology
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic...
and the corresponding theory of étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
. 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.
Other work
From 1959 onward Serre's interests turned towards group theoryGroup theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, in particular Galois representations and modular forms.
Amongst his most original contributions were: his "Conjecture II
Serre's conjecture II (algebra)
In mathematics, Jean-Pierre Serre conjectured the following result regarding the Galois cohomology of a simply connected semisimple algebraic group...
" (still open) on Galois cohomology; his use of group actions on Trees
Bass–Serre theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees...
(with H. Bass
Hyman Bass
Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...
); 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
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
; and the Serre conjecture
Serre conjecture (number theory)
In mathematics, Jean-Pierre Serre conjectured the following result regarding two-dimensional Galois representations. This was a significant step in number theory, though this was not realised for at least a decade.-Formulation:...
(now a theorem) on mod-p representations that made Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
a connected part of mainstream arithmetic geometry.
In his paper FAC, Serre asked whether a finitely generated projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
over a polynomial ring is free. This question led to a great deal of activity in commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, and was finally answered in the affirmative by Daniel Quillen and Andrei Suslin
Andrei Suslin
Andrei Suslin is a Russian mathematician who has made major contributions to the field of algebra, especially algebraic K-theory and its connections with algebraic geometry. He is currently a Trustee Chair and Professor of mathematics at Northwestern University.He was born on December 27, 1950,...
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 MedalFields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
. In 1985, he went on to win the Balzan Prize
Balzan Prize
The International Balzan Prize Foundation awards four annual monetary prizes to people or organisations who have made outstanding achievements in the fields of humanities, natural sciences, culture, as well as for endeavours for peace and the brotherhood of man.-Rewards and assets:Each year the...
, the Steele Prize in 1995, the Wolf Prize in Mathematics
Wolf Prize in Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts...
in 2000, and was the first recipient of the Abel Prize
Abel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
in 2003.
He is a foreign member of several scientific Academies (France, US, Norway, Sweden, Russia, ...) and has received about a dozen honorary degrees (Cambridge, Oxford, Harvard, ...).
See also
- Bass–Serre theoryBass–Serre theoryBass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees...
- Serre dualitySerre dualityIn algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
- Serre's multiplicity conjecturesSerre's multiplicity conjecturesIn mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry...
- Serre's property FASerre's property FAProperty FA is a property of mathematical groups. Jean-Pierre Serre defined property FA in his book Arbres, amalgames, SL_2 ....
- Serre fibration
- Serre twist sheaf
- Thin set in the sense of SerreThin set (Serre)In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or...
- Quillen–Suslin theoremQuillen–Suslin theoremThe Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra about the relationship between free modules and projective modules over polynomial rings...
(sometimes known as "Serre's Conjecture") - Nicolas BourbakiNicolas BourbakiNicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
- Serre's ConjectureSerre conjecture (number theory)In mathematics, Jean-Pierre Serre conjectured the following result regarding two-dimensional Galois representations. This was a significant step in number theory, though this was not realised for at least a decade.-Formulation:...
concerning Galois representations - Serre's "Conjecture II"Serre's conjecture II (algebra)In mathematics, Jean-Pierre Serre conjectured the following result regarding the Galois cohomology of a simply connected semisimple algebraic group...
concerning linear algebraic groupLinear algebraic groupIn mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
s
External links
- Jean-Pierre Serre, Collège de France, biography and publications.
- Jean-Pierre Serre at the French Academy of SciencesFrench Academy of SciencesThe French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...
, 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)