Summary of the Principal Lectures
The principallectures1 correspond to the spirit of the NSF -CBMS Regional Re-
search Conferences Program by being aimed at both the new and recent entrants
in the area, including post-doctoral fellows and graduate students, and the estab-
lished researchers in the field. They should thus be accessible to a wide audience
of mathematicians interested in the subject. We reproduce here Niky Kamran's
summary of his lectures.
Lecture 1. Differential Equations and their Geometry. Our aim in this
first lecture will be to motivate some of the main ideas and principles underlying
the differential geometrical study of differential equations. These will form the
thread unifying the whole series of lectures. We will first explain how one attaches
a geometric structure and local invariants to a differential equation by viewing
it geometrically as a locus in a jet space. We will then outline the concepts of
symmetry, normal forms, conservation laws, variational principles and geometric
complete integrability. These themes will be explained and illustrated with the help
of a variety of simple yet enlightening examples coming from differential geometry
and mathematical physics.
Lecture 2. External Symmetries of Differential Equations. External
symmetries of differential equations were first discovered and studied by Lie. An
external symmetry is a vector field in the ambient jet space which, when restricted
to the locus defined by the differential equation, is tangent to that locus. The
flow of a symmetry thus maps solutions to solutions. Lie proved that the existence
of non-trivial external symmetries allows one in many cases to construct group-
invariant solutions which are governed by a differential equation involving fewer
independent variables. This principle of reduction by symmetry has thus become a
very powerful and widely used tool for constructing exact solutions. It can be said
to be one of the most successful applications of the geometric study of differential
equations. We will present and illustrate by means of examples the main results
concerning Lie symmetries, and discuss the particularly attractive form that they
take in the Lagrangian and Hamiltonian contexts.
Lecture 3. Internal and Generalized Symmetries There are, besides
the external symmetries studied in the preceding lecture, two additional classes of
symmetries which are of great importance for the geometric study of differential
equations. The first is the class of internal symmetries, which arise when differential
N. Kamran Lectures on the geometrical study of differential equations, CBMS-NSF Regional
Conf. Ser., AMS, 2001 or 2002
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