142
IX – Multivariate Differential and Integral Calculus
the “ militant gang ” then in Paris was Henri Cartan, the others being in the
provinces or in the United States. He was the only one to look after students
who were then supposed to follow the classes of the Sorbonne and to prepare
for the advanced degree known as agr´ egation. From this strategic position,
he obvious and inevitably influenced them for about twenty years starting
from 1940, the year when I went to the ´
Ecole. Like everyone, he had his own
conception of mathematics and propagated it, though in a less flamboyant
style than M. Mandelbrot.
Anyhow, before the 1950s, there were almost no other mathematician in
Paris likely to generate enthusiasm among students from the ´
Ecole seriously
attracted to mathematics, and even less to explain to them any thing else
apart from pre-1914 stuff.
7 ´
Elie Cartan being too old did not teach anymore. The Lebesgue integral was sometimes taught by Arnaud Denjoy but
his lectures were incomprehensible. As for Lebesgue, he preferred to teach
elementary geometry at the Coll` ege de France, an institution where, if I am
not mistaken, professors are supposed to present recent subjects likely to be
further developed; so it was Cartan who taught us in a very concise manner
what a (Radon) measure and an integrable function were. Gaston Julia, a
specialist of analytic functions reconverted to Hilbert spaces, was certainly
around; provided one knew German, we could have learnt much more and
much more quickly by reading some sixty pages by von Neumann in Mathematische Annalen of 1928–29 than by following his classes; even Henri Cartan,
a non-specialist, used to teach us almost as much in a few lectures as Julia
whose classes did not lead to any aspects of the subject subsequently developed. Jacques Dixmier experienced this before converting with great success
to von Neumann’s rings of operators,
8 a theory dating from the 1930s and
seemingly unknown to Julia. I also followed lectures by Paul Montel, where
he presented his theory of normal (i.e. compact) families of analytic functions, a 1910 model. We could also learn fluid mechanics from Henri Villat
and Joseph P´ er` es, but the subject did not attract very many students from
´
Ecole Normale.
The only major exception was, shortly after 1945, Jean Leray, a very
temporary member of the initial Bourbaki group which he used to criticize
virulently on a personal plane, as I witnessed during a private conversation
with him in 1950 in Cambridge, Mass, A specialist of partial differential equations in fluid dynamics before the war, he was made a prisoner in June 1940;
detained for almost five years in an officers’ camp in Austria where he had organized some sort of university and not wishing Germans to benefit from his
7 And that is saying a lot since all that the German school had invented, from
Gauss to Hilbert, in algebraico-analytic fields had been forgotten in France since
at least fifty years. See the chapter on “ Jeunes Turcs contre pontifes scl´ eros´ es ”
in the Pour la Science issue on Bourbaki.
8 Dixmier claimed this was due to me. It is true I had discovered them thanks to my
habit, acquired in Le Havre, of opening books randomly, for example the volumes
of Annals of Mathematics. Von Neumann’s papers had a great advantage for
ignorant youngsters like us: they could be read almost without knowing anything.
IX – Multivariate Differential and Integral Calculus
the “ militant gang ” then in Paris was Henri Cartan, the others being in the
provinces or in the United States. He was the only one to look after students
who were then supposed to follow the classes of the Sorbonne and to prepare
for the advanced degree known as agr´ egation. From this strategic position,
he obvious and inevitably influenced them for about twenty years starting
from 1940, the year when I went to the ´
Ecole. Like everyone, he had his own
conception of mathematics and propagated it, though in a less flamboyant
style than M. Mandelbrot.
Anyhow, before the 1950s, there were almost no other mathematician in
Paris likely to generate enthusiasm among students from the ´
Ecole seriously
attracted to mathematics, and even less to explain to them any thing else
apart from pre-1914 stuff.
7 ´
Elie Cartan being too old did not teach anymore. The Lebesgue integral was sometimes taught by Arnaud Denjoy but
his lectures were incomprehensible. As for Lebesgue, he preferred to teach
elementary geometry at the Coll` ege de France, an institution where, if I am
not mistaken, professors are supposed to present recent subjects likely to be
further developed; so it was Cartan who taught us in a very concise manner
what a (Radon) measure and an integrable function were. Gaston Julia, a
specialist of analytic functions reconverted to Hilbert spaces, was certainly
around; provided one knew German, we could have learnt much more and
much more quickly by reading some sixty pages by von Neumann in Mathematische Annalen of 1928–29 than by following his classes; even Henri Cartan,
a non-specialist, used to teach us almost as much in a few lectures as Julia
whose classes did not lead to any aspects of the subject subsequently developed. Jacques Dixmier experienced this before converting with great success
to von Neumann’s rings of operators,
8 a theory dating from the 1930s and
seemingly unknown to Julia. I also followed lectures by Paul Montel, where
he presented his theory of normal (i.e. compact) families of analytic functions, a 1910 model. We could also learn fluid mechanics from Henri Villat
and Joseph P´ er` es, but the subject did not attract very many students from
´
Ecole Normale.
The only major exception was, shortly after 1945, Jean Leray, a very
temporary member of the initial Bourbaki group which he used to criticize
virulently on a personal plane, as I witnessed during a private conversation
with him in 1950 in Cambridge, Mass, A specialist of partial differential equations in fluid dynamics before the war, he was made a prisoner in June 1940;
detained for almost five years in an officers’ camp in Austria where he had organized some sort of university and not wishing Germans to benefit from his
7 And that is saying a lot since all that the German school had invented, from
Gauss to Hilbert, in algebraico-analytic fields had been forgotten in France since
at least fifty years. See the chapter on “ Jeunes Turcs contre pontifes scl´ eros´ es ”
in the Pour la Science issue on Bourbaki.
8 Dixmier claimed this was due to me. It is true I had discovered them thanks to my
habit, acquired in Le Havre, of opening books randomly, for example the volumes
of Annals of Mathematics. Von Neumann’s papers had a great advantage for
ignorant youngsters like us: they could be read almost without knowing anything.