The aim of this book, consisting of two sets of revised and augmented lecture notes prepared for an advanced course given at the Centre de Recerca Matematica, is to “help young mathematicians enter a very active area of research lying on the borderline between dynamical systems, analysis and applications.” The general focus of these lectures is Hilbert’s 16^{th} problem, originally posed in 1900 as a broad problem concerning the topology of algebraic curves and surfaces, but here interpreted as the search for an upper bound for the number of limit cycles in polynomial vector fields of degree *n*.

Colin Christopher’s lectures deal with the *Poincaré center-focus problem*. Given a system of differential equations *x'* = *P*(*x*, *y*), *y'* = *Q*(*x*, *y*) , where *P* and *Q* are polynomials of degree *n*, with a critical point whose linearization yields a center, under what conditions can we conclude that the point is a center for the *nonlinear* system? The author discusses various topics pertaining to this problem and concludes with a 66-item bibliography.

Chengzhi Li’s contribution to this volume addresses a weak form of Hilbert's problem, concerning the maximum number of isolated zeros of Abelian integrals of all polynomial 1-forms of degree *n* over algebraic ovals of degree *m*. Li provides basic concepts and techniques in the study of Abelian integrals and their application to Arnold’s problem. He appends a 192-item bibliography. As is to be expected, there is some overlap in the authors’ reference lists, but their union provides an impressive body of background literature on these research topics.

In summary, these lectures and bibliographies provide valuable introductions by two leaders in the field to some significant open problems in the qualitative theory of differential equations. I recommend this book as suitable for a graduate seminar or for self-study by anyone interested in conducting research on the questions arising from Hilbert’s 16^{th} problem.

Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers College of The City University of New York and serves as Secretary (until May, 2009) and Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations, which has just been translated into Spanish; and he is currently writing a linear algebra text.