## Preface to Equivalence, Invariants, and Symmetry

This volume 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 part, comprising Chapters 4-7, provides an in depth study of applications of symmetry methods to differential equations. We begin with a discussion of jet spaces, on which systems of differential equations are naturally realized as geometrical subsets, and conclude with a discussion of contact transformations. The next three chapters deal with the construction and classification of differential invariants, followed by applications to differential equations and variational problems. My earlier text can be used to supplement the material in this part with many additional applications. In the third part, Chapters 8- 12, the focus shifts to equivalence problems, and the Cartan approach to their solution. The different possible branches that arise in Cartan's equivalence method are discussed in detail, and illustrated by a wide variety of applications. The final three chapters survey the required results from the theory of partial differential equations and differential systems. In particular, we describe the proofs of the fundamental existence theorems of Frobenius, and of Cartan and Kahler, that underlie all the methods.

I should mention that the arrangement of the chapters is slightly unnatural from a developmental standpoint. The most foundational material -- the existence theorems for systems of partial differential equations -- appears in the last three chapters. The geometry of coframes is used in part in the symmetry analysis of differential equations, particularly in the construction of differential invariants, which are used to classify equations admitting prescribed symmetry groups. However, I decided on a more pedagogical arrangement so as not to frighten off the prospective reader with overly technical topics at the outset. Thus, the book begins with the more readily digested methods from Lie group theory, then moves on to the more sophisticated approach of Cartan based on differential forms, and completes the repast with the hard analysis for dessert. I hope the more logically minded reader will not be too inconvenienced by this choice of presentation.

There is a wealth of related material in the mathematical literature, and one of the hardest tasks was choosing which results and applications to include, and which, for reasons of space, to omit. For example, at one time I envisioned a much more substantial section on classical invariant theory; however, this unfortunately had to be considerably shortened for the final version. (I hope, in the near future, to use this additional material as the basis for an introductory text on classical invariant theory and its applications.) Experts in any of the subjects touched on in the book will, no doubt, find many of their favorite theoretical developments or applications omitted or tersely commented upon. In many cases, this was made necessary by the space limitations and the need to choose results that were particularly representative, interesting, and/or interdisciplinary. I hope the omissions will not prevent anyone from enjoying what I did finally decide to include.

For clarity, I have adopted a fairly informal, discursive style to present the methods, with the hope that this will lead to new understanding of their utility and effectiveness. Too often in the recent literature, these powerful and constructive techniques have been obscured by elaborate, theoretical machinery that many have chosen to ``rigorize'' them with. In my opinion, in the practical realm, this only serves to obscure the fundamental issues, thereby interfering with a student's direct understanding. My own presentation relies ultimately on the original sources, particularly Lie and Cartan, which I wholeheartedly recommend to the genuine scholar. As a consequence, results are not always stated in complete generality, and occasionally some minor points of rigor are left for the reader to properly sort out. As long as one exercises the proper amount of caution, such technical details will rarely lead one astray in the practical applications.

Although a substantial fraction of the book covers results that are well known to followers of Lie and Cartan, the modern applicability of these methods is illustrated by a surprising variety of new applications. A large fraction of Chapter 5 on the theory of differential invariants is new, and, apart from being surveyed in conference proceedings, has not appeared in the literature before. Some of the classification results for differential equations and variational problems based on symmetry are also new, and others have only appeared scattered in the literature. The applications of differential invariants to computer vision are the result of a recent collaboration with Allen Tannenbaum and Guillermo Sapiro. 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. The applications of classical invariant theory appear in various contexts, and have been reformulated in a more consistent form; in particular, the connection with the Cartan solution to an equivalence problem from the calculus of variations is based on an earlier paper.

The basic prerequisites for the book are multi-variable calculus -- specifically the implicit and inverse function theorems and the divergence theorem -- basic tensor and exterior algebra, and a smattering of group theory. 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.