144
IX – Multivariate Differential and Integral Calculus
what rue d’Ulm could propose to its scientists in those days. But no one
would have prevented students in mathematics who so desired to for example take an interest in theoretical physics if only its mathematical aspects
had been taught in Paris. During the war, in one semester of lectures at the
´
Ecole – I have forgotten the year –, Louis de Broglie had not even gone as far
as writing the Schr¨ odinger equation on the blackboard. Physicists themselves
learnt quantum mechanics on their own from German, American, English or
Soviet writers while waiting for Messiah’s classes at the CEA (Centre d’´ etude
atomique).
During the era mentioned by M. Mandelbrot, within a few years, Henri
Cartan had launched his students from the ENS in fields as diverse as potential theory, topology of Lie groups, Lie algebras, homotopy groups, differential topology, functions of several complex variables, etc. Some of his student
from the ´
Ecole normale chose subjects he knew little about, such as noncommutative harmonic analysis in my case (discovered from Andr´ e Weil’s
book and my lectures in the library of the ´
Ecole or of the Henri Poincar´ e
Institute), or nothing about, such as algebraic geometry in the case of Pierre
Samuel, influenced by Chevalley in Princeton, not to mention those who chose
the theory of trigonometric series like Jean-Pierre Kahane, probability theory
like Gerard Debreux, etc. Generally speaking, everyone was perfectly free, on
the understanding that, like everyone, Cartan did not take charge of those
who chose fields he was completely ignorant about or that did not find interesting at all, or that were totally outdated: he directed them to others if
they could be found. If four students graduating from the ENS deserved a
CNRS (Centre nationale de recherches scientifiques) grant and if the CNRS
only offered two, he obviously had to choose those he would support.
By 1955 at the latest, students supervised by Cartan had many opportunities to learn from other “ pure ” or applied mathematicians. Serre’s lectures
and talks at the College de France and those of Laurent Schwartz at the
Henri Poincar´ e Institute, to mention only these two members of the Bourbaki group, met with prodigious success for several decades. The same is true
for those of Jacques-Louis Lions a few years later. As a student of Schwartz,
he could not initially avoid Bourbaki’s influence.
Contrary to what Benoit Mandelbrot and numerous other critics seem to
think, no one in the group was unaware of the existence of other important
fields aside from “ fundamental structures ” or despised them if they were
not outdated or too light; most of the original work of its members goes beyond these. The idea that we held strong prejudices against geometry is odd
from the part of people for whom ´
Elie Cartan was the only French master
and when Andre Weil was giving a solid foundation to the Italians’ algebraic
geometry, with their “ generic points ” ! Bourbaki took no interest in it as
such precisely because they were not part of these structures, which kept it
sufficiently busy: given our program, we would have aroused much mirth if,
instead of starting the El´ ements de Math´ ematique with set theory, algebra
and general topology, we had directly embarked on PDEs, stochastic pro-
IX – Multivariate Differential and Integral Calculus
what rue d’Ulm could propose to its scientists in those days. But no one
would have prevented students in mathematics who so desired to for example take an interest in theoretical physics if only its mathematical aspects
had been taught in Paris. During the war, in one semester of lectures at the
´
Ecole – I have forgotten the year –, Louis de Broglie had not even gone as far
as writing the Schr¨ odinger equation on the blackboard. Physicists themselves
learnt quantum mechanics on their own from German, American, English or
Soviet writers while waiting for Messiah’s classes at the CEA (Centre d’´ etude
atomique).
During the era mentioned by M. Mandelbrot, within a few years, Henri
Cartan had launched his students from the ENS in fields as diverse as potential theory, topology of Lie groups, Lie algebras, homotopy groups, differential topology, functions of several complex variables, etc. Some of his student
from the ´
Ecole normale chose subjects he knew little about, such as noncommutative harmonic analysis in my case (discovered from Andr´ e Weil’s
book and my lectures in the library of the ´
Ecole or of the Henri Poincar´ e
Institute), or nothing about, such as algebraic geometry in the case of Pierre
Samuel, influenced by Chevalley in Princeton, not to mention those who chose
the theory of trigonometric series like Jean-Pierre Kahane, probability theory
like Gerard Debreux, etc. Generally speaking, everyone was perfectly free, on
the understanding that, like everyone, Cartan did not take charge of those
who chose fields he was completely ignorant about or that did not find interesting at all, or that were totally outdated: he directed them to others if
they could be found. If four students graduating from the ENS deserved a
CNRS (Centre nationale de recherches scientifiques) grant and if the CNRS
only offered two, he obviously had to choose those he would support.
By 1955 at the latest, students supervised by Cartan had many opportunities to learn from other “ pure ” or applied mathematicians. Serre’s lectures
and talks at the College de France and those of Laurent Schwartz at the
Henri Poincar´ e Institute, to mention only these two members of the Bourbaki group, met with prodigious success for several decades. The same is true
for those of Jacques-Louis Lions a few years later. As a student of Schwartz,
he could not initially avoid Bourbaki’s influence.
Contrary to what Benoit Mandelbrot and numerous other critics seem to
think, no one in the group was unaware of the existence of other important
fields aside from “ fundamental structures ” or despised them if they were
not outdated or too light; most of the original work of its members goes beyond these. The idea that we held strong prejudices against geometry is odd
from the part of people for whom ´
Elie Cartan was the only French master
and when Andre Weil was giving a solid foundation to the Italians’ algebraic
geometry, with their “ generic points ” ! Bourbaki took no interest in it as
such precisely because they were not part of these structures, which kept it
sufficiently busy: given our program, we would have aroused much mirth if,
instead of starting the El´ ements de Math´ ematique with set theory, algebra
and general topology, we had directly embarked on PDEs, stochastic pro-