Vitushkin's conjecture, a special case of Painlev '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 1-5 of the 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. 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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