In 2005, Cambridge University Press began to publish a series of New Mathematical Monographs, "dedicated to books containing an in-depth discussion of a substantial area of mathematics. They will bring the reader to the forefront of research by presenting a synthesis of the key results, whilst also acknowledging the wider mathematical context. As well as being detailed, they will be readable and contain the motivational material necessary for those entering a field; for established researchers they will be a valuable resource."
The book under review is the second in this new series, which already has a total of seven books scheduled for publication and so far promises to be extremely successful.
Harmonic Measure satisfies entirely the goals set forth for the series. It provides, in the first four chapters and several Appendices, a very good introduction to the function theory needed to understand the subject, at a level appropriate for any graduate student who has successfully completed standard graduate courses in real analysis and complex analysis.
The book contains another six chapters providing an in-depth treatment of harmonic measure in the complex plane, including several important results discovered during the last two decades.
Each chapter ends with a section titled Notes (containing further discussions and references to the material presented in the chapter, as well as some open problems) and another section titled Exercises and Further Results (which invites the reader to "try his/her hand" at proving some theorems. Most of the suggested exercises have hints and references to the papers/books they have appeared for the first time.)
The book includes an exhaustive bibliography (very useful for any serious researcher), as well as an Author Index and a Subject Index.
In short: everybody who is interested in function theory and for whom Harmonic Measure sounds somewhat familiar and potentially interesting will find this book extremely useful, wonderfully well written and a joy to read.
Mihaela Poplicher is an assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.
1. Jordan domains; 2. Finitely connected domains; 3. Potential theory; 4. Extremal distance; 5. Applications and reverse inequalities; 6. Simply connected domains, part one; 7. Bloch functions and quasicircles; 8. Simply connected domains, part two; 9. Infinitely connected domains; 10. Rectifiability and quadratic expressions; Appendices.