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An Introduction to Lie Groups and Symplectic Geometry

Introduction 

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 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 xamples 
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. 

After the lectures were over, I contemplated reworking these notes into a comprehensive introduction to modern symplectic geometry and, after some soul-searching, finally 
decided against this. Thus, I have contented myself with making only minor modifications and corrections, with the hope that an interested person could read these notes in a few weeks and get some sense of what the subject was about. 
An essential feature of the course was the exercise sets. Each set begins with elementary material and works up to more involved and delicate problems. My object was to 
provide a path to understanding of the material which could be entered at several different levels and so the exercises vary greatly in difficulty. Many of these exercise sets are obviously too long for any person to do them during the three weeks the course, so I provided extensive hints to aid the student in completing the exercises after the course was over. 
I want to take this opportunity to thank the many people who made helpful suggestions for these notes both during and after the course. Particular thanks goes to Karen Uhlenbeck and Dan Freed, who invited me to give an introductory set of lectures at the RGI, and to my course assistant, Tom Ivey, who provided invaluable help and criticism in the early stages of the notes and tirelessly helped the students with the exercises. While the faults of the presentation are entirely my own, without the help, encouragement, and 
proofreading contributed by these folks and others, neither these notes nor the course would never have come to pass.