# Jean-Pierre Serre

Jean -Pierre Serre (born 15 September 1926 in Bages in the French Department of Pyrénées- Orientales) is one of the leading mathematicians of the 20th century. He is considered a pioneer of modern algebraic geometry, number theory and topology. Serre 's winner of the Fields Medal and the Abel Prize.

## Life

Serres parents were pharmacists. He went to high school in Nîmes ( Lycée de Nîmes ), won the 1944 national Concours général in mathematics and studied from 1945 to 1948 at the École supérieure normal in Paris. He earned his doctorate at the Sorbonne in 1951. During this time he became a member of the mathematician Nicolas Bourbaki circle. From 1948 to 1954 he was at the Centre National de la Recherche Scientifique (CNRS ) in Paris working first as attaché de Recherches and later as Maître de Recherches. 1954-1956 he was Maître de conférences at the University of Nancy and was thereafter since 1956 professor at the Collège de France in Paris (Department of Algebra and Geometry ). Since 1994, he has an honorary professorship there.

He was a visiting professor at Harvard and often at the Institute for Advanced Study ( first from 1955 to 1957 ).

His hobbies include skiing, rock climbing and table tennis.

1983 to 1986 he was with Ludwig Faddejew Vice President of the International Mathematical Union.

He was the chemist Josiane Heulot -Serre (1922-2004) married the former director of the École normale supérieure de jeunes filles in Sèvres. 1949, the daughter Claudine Monteil was born. As a feminist writer and a writer, she wrote under biographies of Simone de Beauvoir and of Charles Chaplin and his wife Oona.

The mathematician Denis Serre is his nephew.

## Works

Even at very young age Serre was one of the most outstanding student of Henri Cartan. He worked in the period around 1950 with algebraic topology and turned Jean Lerays spectral sequences to the fiber bundle spaces from a topological space X as the base and the space of paths in X as the fiber at (loop space method). He was able to find relations between the homology groups in fiber bundle spaces and between homology and homotopy groups. The application in the determination of the homotopy groups of spheres, a notoriously difficult area, then caused a stir (PhD 1951). He proved that the m-th homotopy group of the n-dimensional sphere is finite, except for: n even and m = 2n -1 ( m> n ).

After a stay in Princeton in 1952, where he among other things, the Artin -Tate seminar attended on class field theory, he turned to his return to Paris in the Cartan seminar to there current issues theory of functions of several variables and algebraic geometry, which he with the help of Jean Lerays sheaf theory and the methods of algebraic topology ( cohomology theory ) put on a new foundation. Initially this was done for just achieved results of Cartan and Oka in the theory of functions of several variables. Work on generalizations of the Riemann -Roch theorem ( the same Hirzebruch and Kodaira pushed forward ) 1953 eventually led him in 1954 to algebraic geometry. From the discussions in the Cartan seminar mid-50s then the cornerstone of Alexander Grothendieck's theory of schemes on the Grothendieck and his school was later re- built the algebraic geometry. Two of the most famous products of Serre from this period are FAC ( Faisceaux Algébriques Coherent, on the cohomology of coherent module sheaves ) of 1955 and GAGA ( Géometrie algébrique et Géométrie Analytique ) of 1956. Using " analytical geometry " is meant the theory of functions of several complex variables. Is known of the duality theorem of Serre. During the 1950s until the late 1960s Grothendieck and Serre collaborated intensively.

From 1959 to Serre was mainly interested in number theory, especially with the expansion of Galoiskohomologie for class field theory and the Galois representations in the theory of elliptic curves over the rational numbers. Here he formulated the Serre 's conjecture in the theory of " two-dimensional " representations of "absolute Galois group ". The aim of his work was the formal representation of an absolute Galois group of an arbitrary number field, that is the group of its automorphisms. Therefore, special representations ( sites of action ) are examined in this group, eg in the " n- Torsionspunkten " (rational points of the curve, the n -fold "added" after the Sekanten-/Tangentenmethode Poincaré zero yield ) of elliptic curves. Since these are geometrically ( as doubly periodic) on the shape of a torus, one speaks of " two-dimensional representation ." 1972 proved Serre be open image theorem for elliptic curves ( without complex multiplication ) over algebraic number fields. It states that the representations of the Galois group of field extensions of which were formed by adding the - Torsionspunkte, in the group " are as large as possible ". [A 1]

He also initiated along with Nicholas Katz, the theory of p -adic modular forms.

His book A course in Arithmetic brings a limited space both a discussion of quadratic forms and the theory of modular forms ( with application to grid). He was awarded the Leroy P. Steele Prize.

Serre also made an important contribution in the chain of evidence that led to the proof of Fermat 's conjecture by Gerhard Frey Ken Ribet to Andrew Wiles.

From his friendship with Armand Borel resulted also be interested in Lie groups and their algebras, discrete groups and their geometry and representation theory of groups. It was then only natural that he published the collected works of Ferdinand Georg Frobenius.

Serre is also known for various assumptions. In addition to the above-mentioned conjecture on Galois representations, for example, for a presumption in commutative algebra, which became independent proved by Andrei Alexandrovich Suslin and Daniel Quillen ( that projective modules are free over polynomial ).

## Awards and honors

He is much honorary doctorates: Cambridge (1978 ), Stockholm (1980 ), Glasgow (1983 ), Athens ( 1996), Harvard (1998), Durham (2000 ), London (2001 ), Oslo (2002), Oxford ( 2003), Bucharest ( 2004), Barcelona ( 2004), Madrid ( 2006), McGill University ( 2008).

## Quotes

Precision and Informal brief - that is the ideal in books as well as in lectures (Interview with Leong, Chong 1985)

Some mathematicians have clear and far-reaching " programs " ... I never had such a program, not even a Little

## Writings

- Oeuvres - Collected Works, 1988-1995, 4 vols, Springer Verlag
- Linear representations of finite groups, Springer 1996 ( German vieweg 1972)
- A course in arithmetic, Springer 1996 ( first frz.1970 )
- Galois cohomology, Springer 2002 ( English edition of the College de France course 1962/3 )
- Topics in Galois theory, 1992
- Local fields, Springer, 1979 ( engl.Ausgabe of Corps Locaux, 1962)
- Local Algebra, Springer, 2000 (English edition of Algebre locale- multiplicites 1965)
- Algebraic groups and class fields, Springer 1988 (English edition of Groupes et algebriques corps of classes 1959)
- Lectures on the Mordell -Weil theorem, Vieweg 1997 3.ed.
- Trees, Springer, 1980 ( frz.Original Arbres, amalgames, SL (2 ) 1977)
- Abelian l - adic representations and elliptic curves, Benjamin, New York 1968
- Lie algebras and Lie groups, Springer, 1992 ( first Harvard Lectures 1964)
- Complex semisimple lie algebras, Springer 1987
- Grothendieck -Serre correspondence, 2003, American Mathematical Society ( Colmez ed, the numerous phone calls the two especially during their simultaneous presence in Paris, however, are not recorded)

- Homology singulière of espaces Fibres. Applications. Ann. of Math ( 2) 54, ( 1951). 425-505.
- Groupes d' homotopy classes et de groupes abéliens. Ann. of Math ( 2) 58, ( 1953). 258-294.
- Un theorems de Dualité. Comment. Math Helv 29 ( 1955). 9-26
- Propertys galoisiennes des points d'ordre fini of courbes elliptiques. Invent. Math 15 (1972 ), no 4, 259-331.
- With A. Borel: Corners and arithmetic groups. Avec un appendice: Arrondissement of variétés à coins, par A. Douady et L. Hérault. Comment. Math Helv 48 (1973 ), 436-491.
- Quelques applications de du theorems densité de Chebotarev. Inst Hautes Études Sci. Publ Math No. 54 (1981), 323-401.
- Sur les représentations module aires de degré 2 de. Duke Math J. 54 (1987 ), no 1, 179-230.

- Interview with Leong, Chong in Mathematical Intelligencer, 1986, No.4, online here ( version of July 19, 2011 at the Internet Archive )
- Interview in Notices AMS 2004, Issue 2, PDF