§ 1. Classical Differential Calculus
143
expertise, he had converted to algebraic topology. Between 1940 and 1945, he
invented the basic ideas of sheaf theory, which he expounded to Andre Weil
and to Cartan shortly after his release. The theory was immediately adopted
and greatly improved first by the former and then by very young people – my
contemporary Jean-Louis Koszul, Jean-Pierre Serre and the Swiss Armand
Borel, all members of Bourbaki before long. Sorry! –, and was the subject of
a famous Cartan seminar talk in 1950-51. I wrote a book on it some years
later
9 when Grothendieck, a member of the group and converted by Serre,
was starting to revolutionize the subject and algebraic geometry. Elected to
the College de France in 1947, Leray presented in 1947–48 and 1949–50 what
was to become his major article on sheaves in the Journal de Liouville of 1950
and as such certainly contributed to the education of a few people. In 1950,
Leray returned for good to partial differential equations and obviously influenced some of the young people who chose this promising subject, though in
the traditional meaning of the word, he had very few students. In any event,
it is clear that had in 1944 Benoit Mandelbrot opted to come to the ENS
instead of Polytechnique and followed Leray’s lectures from 1947 onwards in
order not to fall into the hands of the militant gang, he would have learnt one
of the most abstract and “ modern ” subjects. Anyhow, no one would have
prevented him from choosing what suited him.
In 1949, Gustave Choquet arrived in Paris . Whilst promoting in his undergraduate lectures, when he had the opportunity to do so from 1954–55 onwards, a version of general topology that most members of the group never
dared to diffuse at this level of abstraction, he was a major expert of fine
structure theory, in line with Lebesgue, Baire and Denjoy and of potential
theory together with Jacques Deny and, briefly, Henri Cartan. The inventor
of fractals would have got on well with him had he not chosen Polytechnique,
an institution where he probably learnt little else than traditional mathematics and the obligation of standing to attention when professors entered
the lecture hall. The only advantage of Polytechnique being that one learnt
other subjects, in particular physics, slightly more modern on some points –
which was not hard – than at the Sorbonne . Besides, Polytechnique offered
its best students much better career prospects and influence networks than
9 It was far removed from the mathematics I was involved in at the time. But a
Bourbaki member is supposed to write down everything in the program of the
groups, and, moreover, I was somewhat upset of constantly hearing discussions
on algebraic topology, which I did not understand at all. When very temporarily,
Bourbaki decided to prepare a book on the subject, I volunteered to write a first
account of sheaf theory. Once the chapter was written and discussed together, it
became clear that Bourbaki could not publish this type of mathematics before a
long time, and I was advised to turn it into a book. It is quite different from my
original report and still sells, Hermann publishers having recently reedited it in
French without consulting me, when the necessity of an English version has been
long obvious (the first one was quickly translated into Russian). . . The moral
of the story and of many others of this type is that the group represented for
its members, in particular its youngest ones, a fantastic opportunity for learning
mathematics.
143
expertise, he had converted to algebraic topology. Between 1940 and 1945, he
invented the basic ideas of sheaf theory, which he expounded to Andre Weil
and to Cartan shortly after his release. The theory was immediately adopted
and greatly improved first by the former and then by very young people – my
contemporary Jean-Louis Koszul, Jean-Pierre Serre and the Swiss Armand
Borel, all members of Bourbaki before long. Sorry! –, and was the subject of
a famous Cartan seminar talk in 1950-51. I wrote a book on it some years
later
9 when Grothendieck, a member of the group and converted by Serre,
was starting to revolutionize the subject and algebraic geometry. Elected to
the College de France in 1947, Leray presented in 1947–48 and 1949–50 what
was to become his major article on sheaves in the Journal de Liouville of 1950
and as such certainly contributed to the education of a few people. In 1950,
Leray returned for good to partial differential equations and obviously influenced some of the young people who chose this promising subject, though in
the traditional meaning of the word, he had very few students. In any event,
it is clear that had in 1944 Benoit Mandelbrot opted to come to the ENS
instead of Polytechnique and followed Leray’s lectures from 1947 onwards in
order not to fall into the hands of the militant gang, he would have learnt one
of the most abstract and “ modern ” subjects. Anyhow, no one would have
prevented him from choosing what suited him.
In 1949, Gustave Choquet arrived in Paris . Whilst promoting in his undergraduate lectures, when he had the opportunity to do so from 1954–55 onwards, a version of general topology that most members of the group never
dared to diffuse at this level of abstraction, he was a major expert of fine
structure theory, in line with Lebesgue, Baire and Denjoy and of potential
theory together with Jacques Deny and, briefly, Henri Cartan. The inventor
of fractals would have got on well with him had he not chosen Polytechnique,
an institution where he probably learnt little else than traditional mathematics and the obligation of standing to attention when professors entered
the lecture hall. The only advantage of Polytechnique being that one learnt
other subjects, in particular physics, slightly more modern on some points –
which was not hard – than at the Sorbonne . Besides, Polytechnique offered
its best students much better career prospects and influence networks than
9 It was far removed from the mathematics I was involved in at the time. But a
Bourbaki member is supposed to write down everything in the program of the
groups, and, moreover, I was somewhat upset of constantly hearing discussions
on algebraic topology, which I did not understand at all. When very temporarily,
Bourbaki decided to prepare a book on the subject, I volunteered to write a first
account of sheaf theory. Once the chapter was written and discussed together, it
became clear that Bourbaki could not publish this type of mathematics before a
long time, and I was advised to turn it into a book. It is quite different from my
original report and still sells, Hermann publishers having recently reedited it in
French without consulting me, when the necessity of an English version has been
long obvious (the first one was quickly translated into Russian). . . The moral
of the story and of many others of this type is that the group represented for
its members, in particular its youngest ones, a fantastic opportunity for learning
mathematics.