This book uses methods from symplectic and contact geometry to solve non-trivial ordinary and partial differential equations. The methods used to solve the differential equations are in the spirit of those developed by Lie and Cartan at the end of the 19th century to transform solving a differential equation into a problem in geometry. The authors view differential equations as part of symplectic and contact geometry, so that Hodge-de Rham calculus can be applied. These methods are then, in the words of the authors, applied "to introduce the reader to a geometric study of partial differential equations of second order" with a thorough focus on Monge-Ampère equations. In the applications portion of the book, examples from meteorology and non-linear acoustics are also discussed.

The major parts of the book are as follows:

• Part I: Symmetries and Integrals,
• Part II: Symplectic Algebra,
• Part III: Monge-Ampère Equations,
• Part IV: Applications, and
• Part V: Classification of Monge-Ampère Equations.

The first part of this book discusses the geometry of differential equations. In particular, the geometry of distributions, integrability and symmetry are discussed. In this light, the authors examine the various notions of symmetry and their use in solving differential equations. For instance, the main result of the first chapter is the Lie-Bianchi theorem, which gives a condition for integrability in quadratures of a distribution in terms of a Lie algebra of the shuffling symmetries. These results are then applied to linear differential equations by systematically using the notion of symmetries.

The second part of the book is devoted to symplectic algebra. Here the main results associated with the existence of a symplectic structure on a basic vector space are discussed. These results are needed because of the conceptual importance of the symplectic structure for Monge-Ampère partial differential equations.

Part three is specifically devoted to the study of Monge-Ampère equations and operators. In this process, some background information about symplectic and contact manifolds—the odd dimensional analogue to symplectic manifolds—is discussed. The authors' approach in the discussion not only works on Monge-Ampère equations, but also on all linear, and quasilinear second order partial differential equations.

The fifth part of the book discusses specific examples of the applications of the techniques discussed in the book. One such application is to the so-called KZ equation, which models the propagation of 3-dimensional sound beams in a non-linear medium. Symmetries, conservation laws and exact solutions of the KZ equation in 3-dimensions are discussed, and using this, a mathematical explanation to an experimentally verified phenomena of self-diffraction and periodic oscillation of sound beams is given.

Also discussed in the applications portion are the applications of geometric studies of Monge-Ampère-like operators in so-called semi-geostrophic models, which are useful in numerical weather prediction. A short account of the geometric study of balanced rotational models is discussed, which is a very special case of the Navier-Stokes system with the presence of Coriolis-like forces.

The fifth part of the book contains the contact classification results on Monge-Ampère equations, which are obtained by the authors' geometric approach. In particular, the authors address the classical Sophus Lie Problem, which was raised by Lie in his article "Begründung einer Invarianten-Theorie der Berührungs-Transformationen" in Mathematische Annalen (1874).

We find that the text is self-contained and should be accessible to the motivated mathematician who wishes to know more about the geometric study of differential equations by symmetry methods. To aid with the understanding of the material in the book, the authors let the reader know of the importance and development of the mathematical results contained in the book by six illustrations of cats, named: Welcome, Eureka, Mentor, Thinking, Lazy and Terminator. These cats are a nice guide as the reader proceeds through the book.