Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane focuses on the subject of planar quasiconformal mappings, with an emphasis on building a link between analysis and geometry. The book is a organized by iterating various subjects in PDEs: harmonic analysis, dynamical systems and topology, hold together by the common theme of quasiconformal mappings and their applications. It is remarkably interesting that all the topics can be formulated in the form of second order elliptic PDEs, which has been a well-investigated subject in the context of calculus of variations. Among all the topics, I personally like the chapter on inverse problems, which previously has not been studied from a geometric perspective. The current treatment by this book will definitely bring some new depth to a challenging field.

The book starts with an introduction providing the most important applications of quasiconformal mappings, including holomorphic dynamical systems, singular integral operators, etc. It also provides a motivating example on the technical aspects of the book through a two-dimensional hydrodynamical model. The introduction is followed by two chapters on the background and fundamentals on quasiconformal mappings.

For readers who need some additional background knowledge in analysis, two chapters are devoted to complex analysis and the Beltrami equation. The subject of elliptic PDEs is then discussed in greater detail in Chapters 6, 7, and 8. The existence and uniqueness of the nonlinear Riemann mapping for different domain geometries are established in Chapter 9, and the conformal equivalence of Riemann structures is proved in Chapter 10.

The authors then move to the topic of holomorphic dynamical systems, and they investigate interesting topics such as Hausdorff dimension. After some estimates on Beltrami operators using tools from harmonic analysis, Pucci's conjecture for PDE's not of divergence form is throughly studied and is further studied in Chapter 20. Applications such as inverse problems and calculus of variations are also discussed in detail.

The chapters are clearly written and are very user friendly for readers who may not have a diversified background but are interested in finding how analysis can be useful in revealing geometry, and vice versa. The book could be used by experts who want an updated account of all the recent progress on the subject as well as by graduate students who need an encyclopedic reference. I found the book to be very enjoyable.

Editor's Note: More information, including a free preview of the first chapter, are available on the publisher's website.