Book review

The second part of "Singularity Theory" by V. I. Arnold, V. V. Goryunov, O. V. Lyashko and V. A. Vasiliev, announced as Volume 8 of "Dynamical Systems" and Volume 39 in the series "Encyclopaedia of Mathematical Sciences", is actually quite independent of other volumes and was designed for applications to concrete problems of mathematics, mechanics and theoretical physics. The book is an updated translation of a volume published in 1989 by the USSR Institute of Scientific Information in the framework of the mathematical section of "Progress in Science and Technology".

The subject of the book is the theory of singularities of smooth mappings, which appears helpful in various fields of exact sciences and has been under extensive development mainly for the past 30 years. This branch of functional analysis has been advertized also as the "Catastrophe Theory", since it explains abrupt changes in dynamical systems in response to smooth variations of input conditions. As Arnold wrote in one of his books, "Singularities, bifurcations, catastrophes are different terms for describing the emergence of discrete structures from smooth, continuous ones." The phenomenon itself has been known forever: reflection of light from a smooth surface creates caustics, i.e. surfaces at which the intensity of light has a drastic increase and the energy density is singular in geometrical optics. Investigation of bifurcations of solutions for differential equations depending on parameters was initiated by Poincaré and has been developed in this century, particularly by Moscow scholars in the 30-ties. (The relation to the theory of differential equations explains why this book is a part of the series on dynamical systems.) The modern theory of catastrophes was born after a work by Whitney published in 1955, where it was proven that in any mapping of a surface onto a plane only two types of singularities are stable against small variations: the fold and the cusp. The efforts to extend this result to higher dimensions have been the essence of the new theory. An important step was a work by Arnold in 1972, where he showed that simple degenerate critical points of functions of several variables can be class)fied by series related to symmetries of the root systems for the Lie algebras A_k (for any k >= 1), D_k (for k > 3) and E_k (for k = 6, 7, 8). Since that time, Arnold's group, including his students and collaborators, has obtained a lot of results, some of which are presented in this volume.

The book consists of 5 chapters dealing with different sets of problems and written on different levels of clarity and sophistication. Chapter 1, the longest one, describes classification of singularities of functions and mappings according to the equivalence groups. Functions on manifolds with smooth boundaries are considered in particular, where the critical points related to the Weyl groups for the series Bk and Ck, and the Coxeter groups H_3, H_4 and I_2(p) appear together with those for the series A_k, D_k, E_k. Nonisolated singularities, projections onto the plane and onto line, and critical behavior of tangent vector fields are also considered. A lot of concrete results are presented in numerous tables and figures. Chapter 2 is probably the most elegant; it deals with classification of critical points of functions. Singularities which arise from the Legendre transformations and from a number of geometrical constructions (like evolutes and wavefronts leading to caustics) are described. Section 2.4 presents the theory of bifurcations for dynamical systems described by means of a gradient-type system of differential equations, <formula> (where z stands for a number of functions of t, and F is a function of z), formulating the famous conjecture by Thom and its specification. Section 3 is devoted to classification of singularities of the boundaries of domains for functional spaces, like boundaries of stabilities of solutions of differential equations in the space of their parameters. Chapter 4 treats ramification of oscillating integrals and of solution of hyperbolic equations. It also contains exposition and extension of a statement, called by Arnold "Newton's nonintegrability theorem" (concerning volumes of domains cut out of algebraic bodies by hyperplanes). Application to the Picard-Lefschetz monodromy theory is also considered. Deformations of real singularities and the local Petrovskii lacunas, i.e. domains of the complement to a wave front from the side of which the solution of hyperbolic equation is nonsingular, are the subject of the last short Chapter 5. The volume contains 134 figures and many tables, the extensive bibliography includes more than 250 references.

Actually, the book does not describe applications of the general theory to scientific problems; it is rather a unique collection of strong and exact mathematical results, ready for use by mathematicians and scientists who would take the labor and understand them.