The book under review discusses the equivalence problem: when two mathematical objects are the same under a change of variables. The symmetries of a given object are interpreted as the group of self- equivalences, and conditions guaranteeing equivalence are expressed in terms of invariants, whose values are unaffected by the changes of variables.
The author focusses on the continuous counterpart of the theory – differential equations, variational problems, vector fields, differential forms, although algebraic objects, such as polynomials, matrices, and quadratic forms, also play an important role. The book naturally divides into four parts. The first (the algebra-geometric foundation) gives a rapid survey of the basic facts from differential geometry, Lie groups and representation theory. The second provides an in depth study of applications of symmetry methods to differential equations. In the third part the author considers the equivalence problems and the Cartan approach to their solution. The final part surveys the required results from the theory of partial differential equations and differential systems.
The book revolves around themes, arising in a wide variety of mathematical disciplines with significant and substantial applications. It is very well written and provides a very clear exposition of all basic facts, while numerous examples and exercises make it very useful for beginners.