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Vitushkin´s conjecture, a special case of Painlevé´s problem, states that a compact subset of the complex plane with finite linear Hausdorff measure is removable for bounded analytic functions if and only if it intersects every rectifiable curve in a set of zero arclength measure. Chapters 6-8 of this carefully written text present a major recent accomplishment of modern complex analysis, the affirmative resolution of this conjecture. Four of the five mathematicians whose work solved Vitushkin´s conjecture have won the prestigious Salem Prize in analysis.
Chapters 1-5 of this book provide important background material on removability, analytic capacity, Hausdorff measure, arclength measure, and Garabedian duality that will appeal to many analysts with interests independent of Vitushkin´s conjecture. The fourth chapter contains a proof of Denjoy´s conjecture that employs Melnikov curvature. A brief postscript reports on a deep theorem of Tolsa and its relevance to going beyond Vitushkin´s conjecture. Although standard notation is used throughout, there is a symbol glossary at the back of the book for the reader´s convenience.
This text can be used for a topics course or seminar in complex analysis. To understand it, the reader should have a firm grasp of basic real and complex analysis.

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From the reviews:

"This is a very nice and well-written book that presents a complete proof of the so-called Vitushkin conjecture on removable sets for bounded analytic functions ... . it is accessible to both graduate and undergraduate students." (Xavier Tolsa, Mathematical Reviews, Issue 2011 i)

"The aim of the book is to present a complete proof of the recent affirmative solution to the Vitushkin conjecture, which was preceded by a proof of the Denjoy conjecture. ... The book is a guide for graduate students and a helpful survey for experts." (Dmitri V. Prokhorov, Zentralblatt MATH, Vol. 1205, 2011)

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James J. Dudziak received his Ph.D from Indiana University and is currently a visiting associate professor at Michigan State University at Lyman Briggs College. He published six excellent papers in good journals from 1984 to 1990 when he received tenure at Bucknell University.

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