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

1

N. Kamran Lectures on the geometrical study of differential equations, CBMS-NSF Regional

Conf. Ser., AMS, 2001 or 2002

xiii