||The volume you are about to read will, I hope, prove to be a stimulating, unusual, and provocative blend of mathematical flavors. As its title indicates, the book revolves around three interconnected and particularly fertile themes, each arising in a wide variety of mathematical disciplines, and each having a wealth of significant and substantial applications. Equivalence deals with the determination of when two mathematical objects are the same under a change of variables. The symmetries of a given object can be interpreted as the group of self-equivalences. Conditions guaranteeing equivalence are most effectively expressed in terms of invariants, whose values are unaffected by the changes of variables. Issues of this generality naturally arise in all fields of mathematics, and, particularly in geometry, often lie at the heart of the subject. Although each of these concepts has a discrete counterpart, our primary focus will be on the continuous. The areas of immediate concern are analytical — differential equations, variational problems, vector fields, and differential forms — although algebraic objects, such as polynomials, matrices, and quadratic forms, also play an important role. This book will explore the available methods for systematically and algorithmically solving the problems of symmetry, equivalence, the classification of invariants, and the determination of canonical forms, thereby elucidating the many interconnections, some surprising, between the particular manifestations of these problems in seemingly unrelated situations.
The book naturally divides into four interconnected parts. The first, comprising Chapters 1-3, constitutes the algebro-geometric foundation of our subject. The first chapter provides a rapid survey of the basic facts from differential geometry, including manifolds, vector fields, and differential forms. Chapters 2 and 3 could, with some more fleshing out,, form a basic course on Lie groups and representation theory. The primary omissions are the detailed structure theory of Lie groups and algebras, and the classification theory for irreducible representations, neither of which play a .significant role in our applications. The second