These are the lecture notes for a short course entitled "Introduction to Lie groups and
symplectic geometry" which I gave at the 1991 Regional Geometry Institute at Park City,
Utah starting on 24 June and ending on 11 July.
The course really was designed to be an introduction, aimed at an audience of stu-
students who were familiar with basic constructions in differential topology and rudimentary
differential geometry, who wanted to get a feel for Lie groups and symplectic geometry.
My purpose was not to provide an exhaustive treatment of either Lie groups, which would
have been impossible even if I had had an entire year, or of symplectic manifolds, which
has lately undergone something of a revolution. Instead, I tried to provide an introduction
to what I regard as the basic concepts of the two subjects, with an emphasis on examples
which drove the development of the theory.
I deliberately tried to include a few topics which are not part of the mainstream
subject, such as Lie's reduction of order for differential equations and its relation with
the notion of a solvable group on the one hand and integration of ODE by quadrature on
the other. I also tried, in the later lectures to introduce the reader to some of the global
methods which are now becoming so important in symplectic geometry. However, a full
treatment of these topics in the space of nine lectures beginning at the elementary level
was beyond my abilities.