# Jean-Pierre Albert Achille Serre

### Quick Info

Bages, Pyrénées Orientales, France

**Jean-Pierre Serre**is a French mathematician who has made important contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was a member of Bourbaki.

### Biography

**Jean-Pierre Serre**'s parents, Jean Serre and Adèle Diet, were both pharmacists. His mother Adèle had been a pharmacy student at the University of Montpellier and she took a calculus course (just for fun, she said, since she liked mathematics). In 1932 Jean-Pierre began his primary school education at the École de Vauvert. It was at the age of seven or eight that he began to enjoy doing mathematics. In 1937 he moved from the École de Vauvert to study at the Lycée Alphonse-Daudet in Nîmes. His mother had kept the calculus books that she had bought when she took the mathematics course at the University of Montpellier and Jean-Pierre began to learn mathematics from these books [9]:-

When I was 14 or 15, I used to look at these books, and study them. This is how I learned about derivatives, integrals, series and such (I did that in a purely formal manner - Euler's style so to speak: I did not like, and did not understand, epsilons and deltas.)At this time mathematics wasn't the only subject he enjoyed. Although he never took much interest in physics, he enjoyed chemistry which was a topic that his parents, being pharmacists, knew a lot about. In particular his father had a lot of chemistry books which Jean-Pierre read when he was fifteen or sixteen years old. In fact he enjoyed one of these books so much that he kept a copy of it - this was the book "Les colloïdes", Gauthier-Villars, 1922 by Jacques Duclaux (1877-1978). However, as he went deeper into chemistry he became less enthusiastic about the subject and became more convinced that mathematics was the topic for him.

He spoke about his time at the Lycée Alphonse-Daudet in Nîmes in [9]:-

In high school I used to do problems for more advanced classes. I was then in a boarding house in Nimes, staying with children older than I was, and they used to bully me. So to pacify them, I used to do their mathematics homework. It was as good a training as any.At the Lycée in Nîmes in the year 1943-44 he had a good mathematics teacher who was nicknamed "Le Barbu" since he had a beard. He was [9]:-

... very clear, and strict; he demanded that every formula and proof be written neatly.He coached Serre for the Concours General in mathematics which he sat in 1944 and was placed first. Also in 1944 he sat the Concours General in physics but since he spent the whole six-hour examination using an incorrect formula, he scored poorly. Serre was awarded his Bachelier ès sciences et ès lettres in 1944 but remained at the Lycée until 1945 preparing to take the entrance examination to enter the École Normale Supérieure in Paris. While at the Lycée [9]:-

I had no idea one could make a living by being a mathematician. It was only later I discovered one could get paid for doing mathematics! What I thought at first was that I would become a high school teacher: this looked natural to me. Then, when I was 19, I took the competition to enter the École Normale Superieure, and I succeeded. Once I was at "l'École", it became clear that it was not a high school teacher I wanted to be, but a research mathematician.From 1945 to 1948 Serre studied at the École Normale Supérieure and was awarded his Agrégé des sciences mathematique in 1948. At this time he became the youngest member of the Bourbaki group of mathematicians. This group included those who had been involved since the mid 1930s such as Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné and André Weil. Around the time that Serre joined the Bourbaki group, others such as Roger Godement, Pierre Samuel and Jacques Dixmier also joined.

Serre married Josiane Heulot (1922-2004) on 10 August 1948; they had one child, a daughter Claudine born on 29 November 1949. Josiane was an organic chemist working at the École Normale Supérieure de jeunes filles at Sèvres.

From 1948 to 1954 Serre held positions at the Centre National de la Recherche Scientifique in Paris, first as attaché and then as chargé de recherches. In 1948-49 he attended Henri Cartan's seminar which was on algebraic topology and sheaf theory. Also attending Henri Cartan's seminar in that year were Claude Chevalley, Jean Delsarte, Jean Dieudonné, Roger Godement, Laurent Schwartz, and André Weil. One of Serre's fellow students was Alexander Grothendieck and he also attended the seminar. Grothendieck and Serre became friends at this time. Serre was advised by Henri Cartan but [18]:-

... Cartan did not suggest research topics to his students: they had to find one themselves; after that he would help them. This is what happened to me. I found that Leray's theory (about fibre spaces and their spectral sequence) could be applied to many more situations than was thought possible and that such an extension could be used to compute homotopy groups.He was awarded his doctorate from the Sorbonne in 1951 for his thesis

*Homologie singulière des espaces fibrés. Applications*Ⓣ. In 1952 he went to Princeton where he lectured on the results in his thesis and also on $C$-theory which was the continuation of this work. While in Princeton he attended the Artin-Tate seminar on class field theory. Returning to Paris, he again attended the Henri Cartan seminar which, in that year, was discussing functions of several complex variables and Stein manifolds. The ideas he met in this seminar motivated the direction of his research. In 1953-54 he was maître de recherches at the Centre National de la Recherche Scientifique.

In 1954 Serre went to the University of Nancy where he worked until 1956. From 1956 he held the chair of Algebra and Geometry in the Collège de France until he retired in 1994 when he became an honorary professor. He describes in [16] his inaugural lecture at the Collège de France:-

I was a young man, about 30, when I arrived at the Collège. The inaugural lecture was almost like an oral examination in front of professors, family, mathematician colleagues, journalists etc. I tried to prepare it, but after a month I only managed to write half a page. When the day of the lecture came, it was quite a tense moment. I started by reading the half page I had prepared and then I improvised.A few months later he was informed that inaugural lectures were published and he was asked to supply a transcript. As the lecture had been improvised he had no transcript but tried to recreate it by giving an impromptu lecture into a tape recorder and then giving the tape to a secretary to type up. However, the secretary said that the recording was inaudible. Serre then gave up and his inaugural lecture was never published. This makes it unique as every other inaugural lecture at the Collège de France that has been published.

He has also spoken about his teaching career at the Collège de France (see for example [16]):-

Teaching at the Collège is both a marvellous and a challenging privilege. Marvellous because of the freedom of choice of subjects and the high level of the audience: Centre National de la Recherche Scientifique researchers, visiting foreign academics, colleagues from Paris and Orsay - many regulars who have been coming for 5, 10 or even 20 years. It is challenging too: new lectures have to be given each year, either on one's own research (which I prefer), or on the research of others. Since a series of lectures for a year's course is about 20 hours, that's quite a lot.His permanent position in the Collège de France allowed Serre to spend quite a lot of time making research visits. In particular he spent time at the Institute for Advanced Study at Princeton (in 1955, 1957, 1959, 1961, 1963, 1967, 1970, 1972, 1978, 1983, 1999) and at Harvard University (in 1957, 1964, 1974, 1976, 1979, 1981, 1985, 1988, 1990, 1992, 1994, 1995, 1996). Here is a list of the universities where Serre delivered courses (in alphabetical order): Algiers (1965, 1966), Bonn (1976), CalTech (1997), Eugene (1998), Geneva (1999), Göttingen (1970), McGill (1967), Mexico (1956), Moscow (1961, 1984), Princeton (1952, 1999), Singapore (1985), U.C.L.A. (2001), Utrect (1974).

Serre's early work was on spectral sequences. A spectral sequence is an algebraic construction like an exact sequence, but more difficult to describe. Serre did not invent spectral sequences, these were invented by the French mathematician Jean Leray. However, in 1951, Serre applied spectral sequences to the study of the relations between the homology groups of fibre, total space and base space in a fibration. This enabled him to discover fundamental connections between the homology groups and homotopy groups of a space and to prove important results on the homotopy groups of spheres.

Serre's work led to topologists realising the importance of spectral sequences. The Serre spectral sequence provided a tool to work effectively with the homology of fiberings. For this work on spectral sequences and his work developing complex variable theory in terms of sheaves, Serre was awarded a Fields Medal at the International Congress of Mathematicians in 1954. Serre's theorem led to rapid progress not only in homotopy theory but in algebraic topology and homological algebra in general. As an example of Serre's approach to attacking a problem, we quote from the interview [9]. Here Serre was answering a question about the importance of inspiration:-

I don't know what "inspiration" really means. Theorems, and theories, come up in funny ways. Sometimes, you are just not satisfied with existing proofs, and you look for better ones, which can be applied in different situations. A typical example for me was when I worked on the Riemann-Roch theorem (circa 1953), which I viewed as an "Euler-Poincaré" formula (I did not know then that Kodaira-Spencer had had the same idea.) My first objective was to prove it for algebraic curves - a case which was known for about a century! But I wanted a proof in a special style; and when I managed to find it, I remember it did not take me more than a minute or two to go from there to the 2-dimensional case (which had just been done by Kodaira). Six months later, the full result was established by Hirzebruch, and published in his well-known Habilitation thesis. Quite often, you don't really try to solve a specific question by a head-on attack. Rather you have some ideas in mind, which you feel should be useful, but you don't know exactly for what they are useful. So, you look around, and try to apply them. It's like having a bunch of keys, and trying them on several doors.Over many years Serre has published many highly influential texts covering a wide range of mathematics. These texts, which show the topics Serre has worked on, are

*Homologie singulière des espaces fibrés*Ⓣ (1951),

*Faisceaux algébriques cohérents*Ⓣ (1955),

*Groupes d'algébriques et corps de classes*Ⓣ (1959),

*Corps locaux*Ⓣ (1962),

*Cohomologie galoisienne*Ⓣ (1964),

*Algèbre Locale. Multiplicités*Ⓣ (1965),

*Lie Algebras and Lie Groups*(1965),

*Algèbres de Lie semi-simples complexes*Ⓣ (1966),

*Abelian l-adic representations and elliptic curves*(1968),

*Représentations linéaires des groupes finis*Ⓣ (1968),

*Cours d'arithmétique*Ⓣ (1970),

*Représentations linéaires des groupes finis*Ⓣ (1971),

*Arbres, amalgames*Ⓣ, $SL_{2}$ (1977),

*Lectures on the Mordell-Weil theorem*(1989),

*Topics in Galois theory*(1992),

*Exposés de Séminaires*Ⓣ 1950-1999 (2001),

*Correspondance Grothendieck-Serre*Ⓣ (2001), (with S Garibaldi and A Merkurjev)

*Cohomological Invariants in Galois Cohomology*(2003), and

*Lectures on*$N_{X}(p)$ (2011).

You can see some extracts from reviews of these books at THIS LINK.

These books are outstanding and led to Serre being honoured. In 1995 he was awarded the Steele Prize for mathematical exposition and the citation for the award reads [2]:-

It is difficult to decide on a single work by a mathematician of Jean-Pierre Serre's stature which is most deserving of the Steele Prize. Any one of Serre's numerous other books might have served as the basis of this award. Each of his books is beautifully written, with a great deal of original material by the author, and everything smoothly polished. It would be hard to make any significant improvement on his expositions; many are the everyday standard references in their areas, both for working mathematicians and graduate students. Serre brings his whole mathematical personality to bear on the material of these books; they are alive with the breadth of real mathematics and are an example to all of how to write for effect, clarity, and impact.The references [8] and [9] provide a fascinating view of Serre's views on some aspects of his career up to 1985:-

Presently, the topic which amuses me most is counting points on algebraic curves over finite fields. It is a kind of applied mathematics: you try to use any tool in algebraic geometry and number theory that you know of, ... and you don't quite succeed!The interview in [8] and [9] also provides a chance to examine Serre's views on mathematics. However, we choose here to quote from [16] on Serre's view on applications of mathematics:-

As for the place of mathematics in relation to other sciences, mathematics can be seen as a big warehouse full of shelves. Mathematicians put things on the shelves and guarantee that they are true. They also explain how to use them and how to reconstruct them. Other sciences come and help themselves from the shelves, mathematicians are not concerned with what they do with what they have taken. this metaphor is rather coarse, but it reflects the situation well enough. (Of course one does not choose to do mathematics just for putting things on shelves; one does mathematics for the fun of it.) Here is a personal example. My wife, Josiane, was a specialist in quantum chemistry. She needed linear representations of certain symmetry groups. The books she was working with were not satisfactory; they were correct, but they used very clumsy notation. I wrote a text that suited her needs, and then published it in book form, as 'Linear Representations of Finite Groups'. I thus did my duty as a mathematician (and as a husband): putting things on shelves.In the interview [18] Serre speaks about his hobbies. He loved rock climbing, skiing and table tennis. I [EFR] can certainly say from personal experience what an excellent table tennis player Serre was, for I've seen him playing at many conferences that we both attended. I always had the (very silly) thought: How can someone who is so good at mathematics be so good at table tennis! Other things that Serre enjoys are chess, reading books and the movies. He said he liked books [9]:-

... of all kinds, from Giono to Böll to Kawabata, including fairy tales and the Harry Potter series.Serre has received numerous awards. In addition to the Fields Medal in 1954 he was elected an honorary member of the London Mathematical Society in 1973, and a Fellow of the Royal Society of London in 1974. He has also been made an Officer Légion d'Honneur and Commander Ordre National du Mérite. He has been elected to many national academies in addition to the Royal Society, in particular the academies of France (1977), the Netherlands (1978), the United States (1979), Sweden (1981), Russia (2003), Norway (2009), Turin (2010) and Taiwan (2010). He was awarded the Prix Peccot-Vimont by the Collège de France (1955), the Prix Francoeur by the Académie des Sciences (1957), the Prix Gaston Julia (1970), the Médaille Émile Picard by the Académie des Sciences (1971), the Prix Balzan (1985), the Gold Medal from the C.N.R.S. (1987), the Steele Prize, described above, from the American Mathematical Society (1995) and the Wolf Prize in 2000. He has been awarded honorary degrees from the University of Cambridge in 1978, the University of Stockholm in 1980, the University of Glasgow in 1983, the University of Athens in 1996, Harvard University in 1998, the University of Durham in 2000, the University of London in 2001, the University of Oslo in 2002, the University of Oxford in 2003, the University of Bucharest in 2004, the University of Barcelona in 2004, the University of Madrid in 2006 and the University of McGill in 2008. In June 2003 he was awarded the first Abel Prize by the Norwegian Academy of Science and Letters [6]:-

... for playing a major role in giving a number of mathematical topics their modern form, notably topology, algebraic geometry and the theory of numbers.The events surrounding this ceremony are described in several articles. The following is extracted from [18]:-

The events started in bright sunshine in Oslo on Sunday, 1 June 2003, with a simple ceremony at the Abel Monument in Slottsparken. After Jens Erik Fenstad, chair of the Abel Board (organizer of the prize events), had given a short speech, the Abel laureate, Jean-Pierre Serre, laid a wreath at the monument. On 2 June the scientific program started in Georg Sverdrup's house, the wonderful new library at the University of Oslo in Blindern. ... Serre's lecture was entitled "Prime numbers, equations and modular forms". He fully lived up to his reputation as a master expositor, lecturing in the old-fashioned way with chalk on the blackboard, and impressed everybody with a very clear presentation without notes. ... Later in the afternoon, Jean-Pierre Serre received several parties of journalists for interviews. On Tuesday morning Serre and representatives of the Abel Committee and the Abel Board met the world press: ten journalists from Norway, England, France, and Germany. On Tuesday afternoon the prize was presented at a ceremony, with due pomp and circumstance, at the University of Oslo. King Harald and Queen Sonja attended, and after some speeches the king presented the prize to Serre.

### References (show)

- Biography in
*Encyclopaedia Britannica*. http://www.britannica.com/biography/Shen-Kua - 1995 Steele Prizes,
*Notices Amer. Math. Soc.***42**(11) (1995), 1288-1292. http://www.ams.org/notices/199511/steele.pdf - R C Alperin, Review: Trees, by Jean-Pierre Serre,
*Bull. Amer. Math. Soc. (N.S.)***8**(2) (1983), 401-405. - R Betti, A 'Nobel' for Jean-Pierre Serre: the Abel Prize (Italian),
*Lett. Mat. Pristem No.***47**(2003), 4. - P Bayer, Jean-Pierre Serre, Fields Medal (Spanish),
*Gac. R. Soc. Mat. Esp.***4**(1) (2001), 211-247. - A Chambert-Loir, Le prix Abel décerné à Jean-Pierre Serre,
*Gaz. Math. No.***99**(2004), 26-32. - C T Chong and Y K Leong, An interview with Jean-Pierre Serre (Czech),
*Pokroky Mat. Fyz. Astronom.***33**(5) (1988), 241-248. - C T Chong and Y K Leong, An interview with Jean-Pierre Serre,
*Math. Intelligencer***8**(4) (1986), 8-13. - C T Chong and Y K Leong, An interview with Jean-Pierre Serre,
*Math. Medley***13**(1) (1985), 11-19. - M Fried, Review: 'Topics in Galois Theory', by J-P Serre,
*Bull. Amer. Math. Soc. (N.S.)***30**(1) (1994), 124-135. - R L Goodstein, Review: Cours D'Arithmétique, by Jean-Pierre Serre,
*The Mathematical Gazette***55**(393) (1971), 342. - W H Gustafson, Review: Linear representations of finite groups, by Jean-Pierre Serre,
*Bull. Amer. Math. Soc.***84**(5) (1978), 939-943. - S Huet, Boss des maths,
*Libération*(28 May 2003). - A Jackson, Serre receives Abel Prize,
*Notices Amer. Math. Soc.***50**(6) (2003), 693. - H Jacquet, Review: A course of arithmetic, by Jean-Pierre Serre,
*American Scientist***62**(1) (1974), 120. - M Kirch, J-P Serre: My first fifty years at the Collège de France, in Helge Holden and Ragni Piene, The Abel Prize: 2003-2007 The first five years (Springer Science and Business Media, 1009), 15-29.
- F Oort, Essay review of Jean-Pierre Serre's Oeuvres - collected papers,
*Nieuw Arch. Wisk.*(4)**9**(1) (1991), 67-72. - M Raussen and C Skau, Interview with Jean-Pierre Serre,
*Notices Amer. Math. Soc.***51**(2) (2004), 210-214. - M Raynaud, Jean-Pierre Serre: une certaine éthique des mathématiques,
*Gaz. Math. No.***98**(2003), 41-43. - K A Ribet, Review: Abelian
*l*-adic representations and elliptic curves, by Jean-Pierre Serre,*Bull. Amer. Math. Soc. (N.S.)***22**(1) (1990), 214-218. - A Travesa, Jean-Pierre Serre, doctor honoris causa at the UB,
*SCM Not.*(2006), Special issue, 38-39. - A Travesa, Professor Jean-Pierre Serre, recipient of an honorary doctorate from the University of Barcelona (Catalan),
*SCM Not. No.***21**(2005), 29. - N Vila, Jean-Pierre Serre, first winner of the Abel Prize (Spanish),
*SCM Not. No.***19**(2003), 22-25. - C T C Wall, Review: Lie Algebras and Lie Groups, by Jean-Pierre Serre,
*The Mathematical Gazette***51**(375) (1967), 76-77.

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Written by J J O'Connor and E F Robertson

Last Update November 2014

Last Update November 2014