**Pierre Cartier**'s parents were Jean Cartier, who was a medical representative, and Yvonne Suran who was the headmistress of a Lycée. Cartier wrote of his family in [2]:-

Pierre was born in Sedan, a small city in northeastern France near the Belgium border. He said of his background [1]:-The second or third or fourth generation in any family or any social group follows definite sociological patterns. My own family was typical. My grandfather was a self-made man, a very successful businessman. My father and my uncle went into the business, but they were not so devoted to the fight. And people in my generation - well, I suppose I made the right decision not to engage in it. Indeed, people in my generation who did go into our family business did not do so well, because they didn't have anything to fight for.

When Pierre was eight years old the town was destroyed during the German invasion of France in World War II. Food was scarce and life was extremely hard for everyone. The war years, and those immediately after, was an extremely difficult time for the young boy to grow up and receive his education. He attended the Collège Turenne in Sedan, beginning his studies there in 1942. Because of the war the school had few books and struggled to provide an education for the children. He spoke in [2] about his experiences there:-My background, characterised by the contrast between the Alsatian common sense of my grandmother and the somewhat delirious imagination of my father, gave me a huge curiosity for people, countries and books.

He had been taught some algebra by one of his grandmother's brothers and he became fascinated by mathematics, reading on his own a number of mathematics texts. He also leafed through encyclopaedias looking for the articles on mathematical topics. In 1948, at age sixteen, he won first prize in the 'concours général'. One of his parents' friends, delighted at Pierre's achievement, offered to buy him books as a reward. Pierre spent a couple of hours in a bookshop making his choice. He finally decided on André Lichnerowicz's book on tensor calculus and volume 1 of Bourbaki'sI had been a student in a very provincial, very outdated high school. Some of my teachers were very good but of course they were very far away from modern science. The mathematics I was taught was classical geometry, in the uncultivated, synthetic way. I did have the luck to have an imaginative teacher in physics, and so at first I wanted to by a physicist.

*Topology*.

In order to prepare for entry into the École Normale Supérieure in Paris, Cartier left Sedan and went to Paris where he attended the Lycée Saint-Louis [2]:-

He continued to study mathematics on his own and during his time he prepared to begin his university studies by reading Claude Chevalley'sI took private lessons in physics from a very peculiar teacher, Pierre Aigrain.

*Theory of Lie Groups*and works by Hermann Weyl. Cartier entered the École Normale Supérieure in 1950 where Henri Cartan was his professor of mathematics and he also attended courses by Laurent Schwartz. He also learnt much from Samuel Eilenberg who was spending a year in Paris working with Henri Cartan on their book

*Homological algebra*. However Cartier suggested that the education he received at the École Normale Supérieure and the Sorbonne was old fashioned to say the least [2]:-

As well as mathematics and physics courses, he also took philosophy courses and seminars [1]:-Usually a bright student completes the program in two years, but I managed to get through it in one. But both the mathematics and the physics I was taught were totally outmoded at that time, totally. I remember that, in a course called General Physics at the Sorbonne, the professor made a solemn declaration: "Gentlemen in my class what some people call the 'atomic hypothesis' has no place." That was1950, five years after Hiroshima! So I went to Aigrain and said, "What do I do?" and he said, "Well, of course, you have to get your degree, but I will teach you physics properly." This shows what the French university was at the time.

He decided to attend the Bourbaki meeting [1]:-At the end of the year I had to choose: Louis Althusser(1918-1990), the Marxist, advised me that it was better to take the mathematics rather than the philosophy exams, Yves Rocard proposed to me that I help build the French atomic bomb and Henri Cartan invited me to a Bourbaki meeting.

After completing his Agrégé de mathematique in 1951 Cartier married Monique Pissevin on 3 November; they had one daughter Marion. He continued to study at the École Normale for his doctorate and was assigned Roger Godement as an advisor. Godement was in Nancy and Cartier soon decided that he was more interested in the mathematics that Henri Cartan and André Weil were doing so he changed the topic of his research. Weil was by this time on the faculty of the University of Chicago in the United States but he returned to Paris every summer when Cartier could discuss his progress. Cartier began publishing papers and nine had appeared in print while he was still undertaking research for his doctorate:In1950a new generation, whose natural leader was Serre, had taken control[of Bourbaki]. In that period we had huge ambitions; Bourbaki really wanted to write down all of mathematics. For me it was a dazzling experience; I learned so many things during the week I spent with them. According to Bourbaki's method, we studied reports on the topics which were to be treated in the series. Moreover, it was there where I met André Weil, to whom I kept very close during the rest of his life.

*Dualité de Tannaka des groupes et des algèbres de Lie*Ⓣ (1956);

*Démonstration algébrique de la formule de Hausdorff*Ⓣ (1956);

*Théorie différentielle des groupes algébriques*Ⓣ (1957);

*Une nouvelle opération sur les formes différentielles*Ⓣ (1957);

*Calcul différentiel sur les variétés algébriques en caractéristique non nulle*Ⓣ (1957);

*Dualité des variétés abéliennes*Ⓣ (1957); (with J Dixmier)

*Vecteurs analytiques dans les représentations de groupes de Lie*Ⓣ (1958);

*Remarques sur le théorème de Birkhoff-Witt*Ⓣ (1958); and

*Questions de rationalité des diviseurs en géométrie algébrique*Ⓣ (1958).

He defended his thesis *Dérivations et diviseurs en géométrie algébrique* Ⓣ in 1958. He said [1]:-

He had worked for three years, from 1954 to 1957, at the Centre National de la Recherche Scientifique. We have already mentioned that Cartier had been invited to a Bourbaki congress while he was still an undergraduate. This was held at Pelvoux, in the Alps, in June 1951 [2]:-The best result of my thesis solved a problem posed by Weil in his book on Abelian varieties and algebraic groups. One day I had an inspiration: I was aware of what Dieudonné was doing with formal groups, I also had in mind the question posed by Weil and I said to myself: "This is linked". I saw it immediately but it took a very long time to prove because it lies on the crystalline cohomology of schemes(to be developed only after1965).

By 1955 he had become a full member of the Bourbaki group and remained associated with the group until he reached the official retiral age of 50 set by the group in 1983. He made a major contribution to the project [2]:-I remember that we discussed many things, especially a text written by Laurent Schwartz on the foundations of Lie groups; it was one of the first drafts in the well-known series of Bourbaki on Lie groups

He also took over the role of secretary:-I estimate that I contributed about200pages a year during all this time with Bourbaki.

Cartier spent two years from 1957 to 1959 at the Institute for Advanced Study at Princeton in the United States. Returning to France he was appointed Professor in the faculty of Science at Strasbourg in 1961, remaining there until he moved to the Institut des Hautes Études Scientifiques at Bures-sur Yvette ten years later. In addition to this post he was director of research at the Centre National de la Recherche Scientifique from 1974. In 1982 he left he Institut des Hautes Études Scientifiques becoming a professor at the École Polytechnique (1982-88) and at the École Normale Supérieure from 1988.After Dieudonné(and an interlude by Samuel and Dixmier)I was the secretary of Bourbaki, and it was my duty to do most of the proofreading, I think I proofread five to ten thousand pages. I have a good visual memory. I wouldn't compare myself with Dieudonné, but there was a time when I knew most of the printed material in Bourbaki.

Cartier had written papers on a broad range of mathematical topic including algebraic geometry, number theory, group theory, probability, and mathematical physics. In [1] he speaks about his approach to solving mathematical problems:-

The latter part of Cartier's career saw a surprising change of mathematical direction [2]:-My method is my character: I am curious by nature and everything attracts my attention. Since I have acquired a 'savoir faire' in very different directions, when I study a problem I always have several techniques in mind. I am also very interested in questions concerning history and the philosophy of mathematics. André Weil taught me to learn from the mathematicians of the past as if they were our contemporaries. For me the big question is always the same: what guarantees that mathematics tells the truth? How does it tell it? Does it always tell it?

Unlike other members of Bourbaki, Cartier had always been interested in mathematical physics. He had often lectured on applied mathematics topics in the Bourbaki seminar but every time he did so he always felt that others did not approve. There were reasons for this other than the obvious fact that they preferred pure mathematics. Those like Godement and Grothendieck were passionately opposed to the military and mathematical physics was seen by them as "military applications". Long before the 1986 change of direction he speaks of above he had written papers such asI have been personally very happy, because when I reached the time of normal retirement from Bourbaki, I had the very fortunate opportunity to be asked to deliver the lecture on behalf of Vladimir Drinfeld at the International Congress of Mathematicians at Berkeley in1986(Drinfeld was prevented from coming for political reasons). It was a great challenge and a great honour for me; his paper is one of the most important papers in the proceedings. Overnight that changed my mathematical life. I said, "This is what I have to do now." Of course I knew the basic material but the perspective was new. I cannot claim that within the few hours I had to prepare the lecture I could really master it, but I understood enough to explain to the people, "This is new, it is important."

*Quantum mechanical commutation relations and theta functions*(1966). Following his delivery of Drinfeld's lecture in 1986, he delivered the paper

*Sur le développement des mathématiques de 1870 à 1970: quelques exemples d'interaction avec la physique*Ⓣ at the Colloquium on Mathematics and Physics held in Paris from 17 to 19 October 1988. It was published in 1991. Continuing this interest in applications to physics, he did much work on functional integration, collaborating with Cécile DeWitt-Morette. They published jointly authored papers such as

*A new perspective on functional integration, Poisson processes in probability and quantum physics*(1996),

*A rigorous mathematical foundation of functional integration*(1997), and

*Physics on and near caustics*(1997). They produced the excellent survey of their work in this area in

*Functional integration*(2000). In 2006 they published the book

*Functional integration: action and symmetries*. Dada S Fine writes in a review:-

Another interest which Cartier kept throughout his career was in the philosophy of mathematics. For example, he ran a seminar on epistemology at the École Normale Supérieure. He publishedThis monograph presents a wide array of examples of path integration in physics. The list begins with Gaussian integrals, the forced harmonic oscillator, and the semi-classical approximation. It then moves on to more exotic quantum mechanical examples including phase-space path integrals, the spinning top, the Aharonov-Bohm effect and supersymmetric quantum mechanics. Poisson processes, solutions of stochastic differential equations and an introduction to quantum field theory ... The authors write for both mathematicians and physicists. ... Overall, this book provides both mathematicians and physicists with a look at what is out there in the realm of path integral physics. Its extensive coverage and references to recent literature make it potentially very useful.

*Logique, catégories et faisceaux*[

*d'après F Lawvere et Myles Tierney*] Ⓣ (1979) in the

*Séminaire Bourbaki*, and

*The nature and meaning of numbers*(written in 1984 and published in 1989). One of his most recent papers on the philosophy of mathematics, published in 2012, is

*How to take advantage of the blur between the finite and the infinite*. Philip Ehrlich writes in a review of this paper:-

Let us note, however, that in parallel with these topics, he has continued to publish on algebraic geometry. To illustrate the range of topics which interests Cartier we record the fact that he is an editor of a book about art and mathematicsIn this paper, the distinguished author presents a series of philosophical musings based on the distinction between a "true finite" set and a "theoretical finite" set. By a true finite set the author means a set that can be constructed by some feasible human action such as writing numbers on a sheet of paper or on a computer screen, and by a theoretical finite set the author means a set that is set-theoretically finite, i.e. a set for which there is no bijection with a proper subset of itself. Through a series of illustrations the author suggests that the set-theoretic conception is a liberating notion that has important implications for mathematics.

*Les Mathematiques et l'Art*Ⓣ.

Cartier has been very active helping to support mathematics in developing countries. For example, he has helped countries such as Chile, Vietnam, and India to raise their mathematical profile. He was awarded the Ampère Prize by the French Academy of Sciences in 1979.

**Article by:** *J J O'Connor* and *E F Robertson*

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