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Laurent Schwartz

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5 March 1915

Paris, France

4 July 2002

Laurent Schwartz came from a Jewish background. His father was a surgeon but his family contained other brilliant men such as his uncle, Professor Robert Debre, the founder of Unicef. At school Schwartz excelled at both mathematics and Latin. He entered the École Normale Supérieure in Paris in 1934. He graduated with the Agrégation de Mathématiques in 1937 and studied for his doctorate in the Faculty of Science at Strasbourg which he was awarded in 1943. His political activities at this time are described in :

The intellectual ferment of these years was paralleled by political engagement. Though from a traditionally right-wing background, he was a strong supporter of Leon Blum's Popular Front Government until he became disillusioned by its failure to support the Spanish Republicans. Similarly, his sympathies for communism were soon dampened by Stalin's show trials, though he then spent ten years as a Trotskyite, up to 1947. He claimed never to regret this, even though it almost prevented him travelling to America to receive the Fields Medal .

During the war his political activities and Jewish background put him in all manner of delicate situations.

Schwartz spent the year 1944-45 lecturing at the Faculty of Science at Grenoble before moving to Nancy where he became a professor at the Faculty of Science. It was during this period of his career that he produced his famous work on the theory of distributions described below.

In 1953 Schwartz returned to Paris where he became professor, holding this position until 1959. He taught at the École Polytechnique in Paris from 1959 to 1980. He then spent three years at the University of Paris VII before he retired in 1983. We say a little below about his remarkable mathematical contributions but before we look at these we recount some of the political activity he took part in during his career in Paris.

In 1956 he was one of the leaders of protests in France against the Russian invasion of Hungary. Then in the following year he became involved in an event much closer to him personally, the "Audin Affair" in Algeria :

Audin, a mathematician and communist based in Algiers, was writing his thesis under Schwartz's supervision. But in June 1957 the 25-year-old father of three and opponent of French rule in Algeria was abducted by paratroopers, tortured and killed. Schwartz was tireless in his calls for justice, and organised a presentation of the young man's thesis in his absence.

Vocal in his opposition to the French campaign, he signed the famous "Declaration des 121" in favour of military insubordination. The riposte of Pierre Messmer, the Minister for the French Army (and, by the same token, of the École), was to strip him of his position at the Polytechnique, for reasons of "common sense and honour". To which Schwartz replied that since the Army commanded by Messmer had sanctioned torture and promoted torturers, such remarks were absurd.

After a brief exile in New York, he regained his post two years later ...

The outstanding contribution to mathematics which Schwartz made in the late 1940s was his work in the theory of distributions. The first publication in which he presented these ideas was Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiques which appeared in 1948.

The theory of distribution is a considerable broadening of the differential and integral calculus. Heaviside and Dirac had generalised the calculus with specific applications in mind. These, and other similar methods of formal calculation, were not, however, built on an abstract and rigorous mathematical foundation. Schwartz's development of the theory of distributions put methods of this type onto a sound basis, and greatly extended their range of application, providing powerful tools for applications in numerous areas.

In the article on Analysis in Encyclopaedia Britannica François Treves describes Schwartz's work as follows:

... Schwartz's idea (in 1947) was to give a unified interpretation of all the generalized functions that had infiltrated analysis as (continuous) linear functionals on the space Cç of infinitely differentiable functions vanishing outside compact sets. He provided a systematic and rigorous description, entirely based on abstract functional analysis and on duality. It is noteworthy that such an approach had a precedent, in the presentation by André Weil of the integration of locally compact groups ... Because of the demands of differentiability in distribution theory, the spaces of test-functions and their duals are somewhat more complicated. This has led to extensive studies of topological vector spaces beyond the familiar categories of Hilbert and Banach spaces, studies that, in turn, have provided useful new insights in some areas of analysis proper, such as partial differential equations or functions of several complex variables. Schwartz's ideas can be applied to many other spaces of test-functions beside Cç, as he himself and others have shown ...

Harald Bohr presented a Fields Medal to Schwartz at the International Congress in Harvard on 30 August 1950 for his work on the theory of distributions. Harald Bohr described Schwartz's 1948 paper as one:

... which certainly will stand as one of the classical mathematical papers of our times. ... I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory , potential theory , and indeed the whole theory of linear partial differential equations ...

Schwartz has received a long list of prizes, medals and honours in addition to the Fields Medal. He received prizes from the Paris Academy of Sciences in 1955, 1964 and 1972. In 1972 he was elected a member of the Academy. He has been awarded honorary doctorates from many universities including Humboldt (1960), Brussels (1962), Lund (1981), Tel-Aviv (1981), Montreal (1985) and Athens (1993).

Later work by Schwartz on stochastic differential calculus is described by him in the survey article , see also . Later political campaigns include those against American involvement in Vietnam, the Soviet invasion of Afghanistan, and the Russian war against Chechnya.

With such involvement in mathematics and politics one might imagine that Schwartz would not have had time for a major hobby. This however would be entirely wrong for he was an avid collector of butterflies, with over 20,000 specimens.

Let us end by giving two quotes from Schwartz; the first on politics and the second on mathematics:

I have always thought that morality in politics was something essential, just like feelings and affinities.

To discover something in mathematics is to overcome an inhibition and a tradition. You cannot move forward if you are not subversive.

Source:School of Mathematics and Statistics University of St Andrews, Scotland