The integration of the Chazy equation using prolongation and symmetry appears in joint work with Peter Clarkson. Many of the applications of the Cartan equivalence method are based on results from my long-standing collaboration with Niky Kamran. The emphasis on the classifying manifolds in the solution to the equivalence problem for coframes is new, and offers a substantial advantage over the more traditional classifying function approach. The results on global equivalence and non-associative Lie groups are stated here for the first time. Results from elementary linear algebra and complex analysis, and basic existence theorems for ordinary differential equations are used without comment.
Many standard results are, for lack of space, left unproved, although ample references are supplied.
It is hoped that the book will form the basis of an advanced graduate course in symmetry and equivalence problems. I have included a large number of exercises, which range in difficulty from the fairly obvious to quite substantial. References to the solutions of the more difficult exercises are provided. It is my hope that this book will serve as a catalyst for the further development, both in theory and in applications, of this fascinating and fertile area of mathematics. I am convinced that there are many significant contributions yet to be made, and that the devoted student cannot help but play a role in its accelerating mathematical evolution.