# Bernard Malgrange

### Quick Info

Born
6 July 1928
Paris, France

Summary
Bernard Malgrange is a French mathematician who worked on differential equations and singularity theory.

### Biography

Bernard Malgrange's father was an engineer while his mother's family also contained engineers and men who had studied at the École Polytechnique. Bernard grew up in Paris, living in the family home on the Rue de Rennes near the Luxembourg gardens. He was the eldest of his parents' eight children and he received his early education at a private school. Then he entered the eighth class of the Lycée Montaigne, close to the Luxembourg gardens, before entering the Lycée Louis Le Grand to prepare for university studies. Of course his school years were difficult ones, for he was only eleven years old when World War II began on 1 September 1939, the day when German forces entered Poland. On the following day, Britain, France and several other countries, declared war on Germany but over the following months France was not involved in any fighting, but spent time trying to build defences to protect the country from an invasion by Germany. The war changed dramatically for France on 10 May 1940 when the German army crossed the Dutch and Belgium borders and soon German armies approached Paris. At this stage Malgrange's family moved to the provinces where he continued his education. He did not show particular strengths in mathematics at this provincial college. He said [6]:-
Compared to the class, I was not terribly bright and there was a girl in my class who was very strong competition.
His father was held as a prisoner for a time, but when the family were able to return to Paris and he entered the Lycée there, he found himself to be the best pupil in the class. After taking the baccalaureate examinations, he intended to train as an engineer and follow in his father's footsteps. However, his teacher suggested that he should take the entrance examination for the École Normale Supérieure. This was not the route that his family had envisioned since the École Normale Supérieure was essentially the place where teachers were trained and there was no tradition of teachers in either his father or his mother's families. It had more demanding entrance requirements that the École Polytechnique which would have been a much more natural choice for anyone wishing to train to become an engineer. Malgrange's father certainly would have preferred his son to attend the École Polytechnique.

Even after the decision had been taken that, because of his outstanding performance, he would enter the École Normale Supérieure, he still had to choose between mathematics and physics. Although at first he thought that physics might be the more useful option, he had found that he did not have a great deal of dexterity for physics experiments at school and frequently broke the apparatus he was using. He also realised that the quality of mathematics teaching was superior to that of physics so in the end he opted for mathematics. In his first year at the École Normale Supérieure he was greatly helped by Jean-Pierre Serre who was two years ahead of him. In his second year of study he was taught by Henri Cartan who had arrived back from a visit to Chicago and Harvard in the United States. Malgrange took Cartan's courses on differential geometry and Lie groups in his second year. Following a suggestion by Jean Dieudonné, enthusiastically supported by Henri Cartan, Malgrange and his fellow student André Blanchard spent one semester of their second year of study at the Faculty of Sciences at Nancy. There they were taught by Laurent Schwartz, Jean Dieudonné, Jean Delsarte and Roger Godement. Schwartz wrote [7]:-
It was a pleasure to receive in our university, which at that time had no research students, two young brilliant students from the École Normale Supérieure. It was a breath of fresh air. We had, at Nancy, an active seminar every Saturday in which all the professors and some of our students participated, which studied a variety of different topics. Malgrange and Blanchard were two active participants.
Among the topics at these seminars that Malgrange attended we mention differential topology, particularly de Rham's theorem. In his fourth year of study at the École Normale Supérieure, Malgrange took a course on functions of several complex variables from Henri Cartan and also attended a seminar led by Jean-Pierre Serre on the same topic. In 1951 Malgrange was awarded a grant by the Centre National de la Recherche Scientifique to undertake research at Nancy advised by Laurent Schwartz. At Nancy he was one of three students studying for their doctorates with Schwartz. Jacques-Louis Lions had, like Malgrange, been a student at the École Normale Supérieure, while the other student, Alexander Grothendieck, had been an undergraduate at Montpellier. Paulo Ribenboim arrived from Brazil to work in Nancy at about the same time as Grothendieck. Laurent Schwartz writes [7]:-
So we had a very important group of young men to whom we gave our full attention. I think that, for some years, the Faculty of Science at Nancy was one of the major mathematical centres in the world for everything related to analysis.
In 1952 Schwartz was appointed as a professor in Paris and his students went to Paris with him to complete their doctorates. An interesting story relating to Grothendieck's thesis was told by Malgrange, who was present at the thesis defence on 28 February 1953. He:-
... recalled that after Grothendieck wrote his thesis he asserted that he was no longer interested in topological vector spaces. "He told me, 'There is nothing more to do, the subject is dead'," Malgrange recalled. At that time, students were required to prepare a "second thesis", which did not contain original work but which was intended to demonstrate depth of understanding of another area of mathematics far removed from the thesis topic. Grothendieck's second thesis was on sheaf theory, and this work may have planted the seeds for his interest in algebraic geometry, where he was to do his greatest work. After Grothendieck's thesis defense, which took place in Paris, Malgrange recalled that he, Grothendieck, and Henri Cartan piled into a taxicab to go to lunch at the home of Laurent Schwartz. They took a cab because Malgrange had broken his leg skiing. "In the taxi Cartan explained to Grothendieck some wrong things Grothendieck had said about sheaf theory," Malgrange recalled.
A difficulty arose with Malgrange's thesis which is explained in detail by Schwartz in [7]:-
The idea of finding an elementary solution for all differential operators with constant coefficients might seem a bit far fetched. However, I had made the suggestion of the existence of such a solution using the theory of distributions. But I had no idea if this suggestion was sensible. Malgrange has precisely demonstrated this existence theorem at about the same time as Léon Ehrenpreis in the United States. Ehrenpreis, moreover, put me in an embarrassing situation. Malgrange and Ehrenpreis worked for several years on very similar topics. They had both published notes and were preparing their theses. I worked very closely with Malgrange, but also with Ehrenpreis who regularly sent me material for reflection and told me his plans. But it was hardly possible for me to compare two students preparing their doctoral theses. If I received a letter from Ehrenpreis giving me some suggestions, did I have the right to tell Malgrange? This blurred the whole situation. Sometimes one, sometimes the other, was the more advanced, and I could not keep a balance. Although it is a profoundly anti-scientific principle, I asked Ehrenpreis not to write letters to me about his plans. He could send me notes of papers to be published once they were completely ready, as both of the two sent published notes to each other, he could decide himself whether he wanted to publish or not to publish. He understood this situation which was unpleasant for him and for me, and I'm grateful. He has also done remarkable work on similar topics.
Malgrange was awarded his doctorate in 1955 from the Université Henri Poincaré at Nancy for his thesis Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution . This work was based on ideas in the theory of distributions developed by Laurent Schwartz and also used ideas on analytic functions which had been developed by Henri Cartan. Even before his thesis was published, Malgrange had published a number of papers. These were the two-part paper Équations aux dérivées partielles à coeficients constants (1953, 1954), and the papers Sur quelque propriétés des équations de convolution (1954) and Formes harmonique sur un espace de Riemann à $ds^{2}$ analytique (1955). These were all published in Comptes Rendu of the Academy of Sciences. Malgrange was appointed as an assistant lecturer in the Faculty of Sciences at Strasbourg in 1955 and later promoted to Professeur sans chaire. He was invited to lecture at the Tata Institute in Bombay and his lectures were written up as the book Lectures on the theory of functions of several complex variables (1958). The book was republished in 1984. In 1960 he returned to Paris where he was appointed as an assistant lecturer in the Faculty of Sciences. He was given a personal chair on 1 January 1962.

In 1965 Malgrange became a professor in the Faculty of Science at Orsay. The University of Paris, the École Normale Supérieure and the Collège de France had set up a site at Orsay, in the south west suburbs of Paris, after World War II. It became a faculty in its own right on 1 March 1965 but it was a small department that Malgrange joined. It soon became a large centre and [3]:-
... we spent half of the year discussing recruitment, and the other half solving administrative problems.
In order to have a quieter time where he could devote more time to mathematics, he accepted the position of professor in the Faculty of Science at Grenoble in 1969. In 1971 the Faculty of Science at Grenoble changed its name to become the Université scientifique et médicale de Grenoble. Malgrange remained there until 1973 when he was appointed Director of Research at the Centre national de la recherche scientifique.

We have mentioned his important work on linear partial differential equations above, but he has made numerous other very significant contributions to differential geometry, non-linear differential equations, and singularities of functions and mappings. In particular, he has studied hypoelliptic operators, ideals of differentiable functions, the classification of differential equations with regular singular points, and the algebraic theory of partial differential equations with variable coefficients. Perhaps his best known result is called the Malgrange Preparation Theorem and this appeared in his classic text Ideals of differentiable functions (1966). Michael Atiyah writes:-
This book gives a concise and complete account of a number of central theorems concerning differentiable functions. ... The Weierstrass preparation theorem for differentiable functions, in its various forms, is proved in Chapter V; this result, conjectured by René Thom and proved by the author, is one of the important theorems of the book.
This book was based on a course of lectures given by Malgrange at the Tata Institute of Fundamental Research, Bombay, India in January and February 1964.

In 1991 Malgrange published Équations différentielles à coefficients polynomiaux . Luis Narváez Macarro writes in a review:-
The modern algebraic theory of differential systems, or "theory of D-modules'', brings us new relations between two mathematical areas traditionally far apart: the theory of systems of linear partial differential equations and algebraic geometry. The first benefited from the geometric point of view and the algebraic treatment of singularities. The second has seen it endowed with new structures, which explain some depth relations between discrete and continuous cohomological theories of varieties (Riemann-Hilbert correspondence). In the book under review, the author shows how to fully incorporate some new elements of classical analysis and mathematical physics to the theory of D-modules, especially the Laplace and stationary phase methods. The use of "sophisticated'' tools, with respect to the traditional methods in classical analysis, such as sheaf theory and derived categories, allows us to understand many exceptional phenomena of the classical theory.
Malgrange published Systèmes différentiels involutifs in 2005. It is a 106-page work which contains the first modern presentation of the theory of involutive differential systems. In 2007, in collaboration with Pierre Deligne and Jean-Pierre Ramis, he published Singularités irrégulières . Jan Stevens writes:-
This book documents the development of the theory of linear differential equations with irregular singularities in the interaction between Deligne, Malgrange and Ramis. It contains correspondence and previously unpublished manuscripts, mostly from the period 1976 to 1991, with comments from the authors and references for further reading.
Malgrange has received many honours. On 24 October 1977 he was elected a Corresponding Member of the Academy of Sciences and, on 13 June 1988, he became a full member. The Academy of Sciences has awarded Malgrange a number of prizes: the Prix Carrière in 1961, the Prix Servant in 1970 and the Prix Cognacq-Jay in 1972. He has also been awarded the Prix Peccot-Vimont from the Collège de France in 1962 and delivered the prestigious series of lectures on his research there in that year, the Cours Peccot. He also received the Prix André Vera in 1962 and the Prix du Rayonnement Français in 1987. The University of Geneva awarded him an honorary degree in 1978.

### References (show)

1. Bernard Malgrange (French), C. R. Acad. Sci. Sér. Gén. Vie Sci. 5 (6) (1988), 420-422.
2. L Boutet de Monvel, Les travaux de Bernard Malgrange. II, Ann. Inst. Fourier (Grenoble) 43 (5) (1993), 1211-1222.
3. M D Garay, Une entrevue avec Bernard Malgrange. http://rmages.free.fr/maths/malgrange.pdf
4. Y Laurent, Bernard Malgrange, 80e anniversaire, Ann. Inst. Fourier (Grenoble) 59 (7) (2009), 2591-2592.
5. B Malgrange, Notice sur les travaux scienfifique, Institut Fourier, Université Joseph Fourier (Grenoble I) (1988).
6. J Nimier, Entretien avec le Professeur Bernard Malgrange: Membre de l'Académie des Sciences. http://www.pedagopsy.eu/entretien_malgrange.htm
7. L Schwartz, Les travaux de Bernard Malgrange. I, Ann. Inst. Fourier (Grenoble) 43 (5) (1993), 1199-1209.