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4.3
58 reviewshistory, going back to the classical investigations of Sophus Lie, Gaston Darboux
and Elie Cartan. Their ideas are at the source of a number of developments which
currently occupy a central position in several areas of pure and applied mathematics,
including the theory of completely integrable evolution equations, the calculus of
variations and the study of conservation laws.
Our objective in these lectures is to give an overview of a number of significant
ideas and results that have been developed over the past decade in the geometrical
study of differential equations. It is of course impossible in the course of ten lectures
to cover all the important advances that have taken place in such a broad field of
research. This survey is therefore far from complete, and it does not succeed in doing
full justice to all the ideas that it aims to convey. We have chosen to focus our
attention on a number of topics which we have found to be of particular significance,
or in which we have been involved through our own research. We have also tried,
in the spirit of the NSF-CBMS Research Conference Series, to keep a good part of
the exposition at a level sufficiently elementary as to enable the non-expert reader
to gain a reasonable understanding of the main ideas and results of this monograph
by an independent study.
In what follows, we give a brief description of the contents of each chapter.
Chapter 1 serves to motivate some of the main ideas and principles underlying
the geometrical study of differential equations. These form the thread unifying the
whole series of lectures. They include the question of the solvability of the Cauchy
problem for first-order partial differential equations and second order hyperbolic
partial differential equations by ordinary differential equation methods, the con¬
cepts of internal, external and generalized symmetries, and the local, global and
equivariant invers
