TY - BOOK ID - 14303105 TI - Vitushkin's conjecture for removable sets PY - 2010 SN - 1441967087 9786613076113 1441967095 128307611X PB - New York : Springer, DB - UniCat KW - Analytic functions. KW - Analytic sets. KW - Set theory. KW - Analytic functions KW - Analytic sets KW - Set theory KW - Mathematics KW - Physical Sciences & Mathematics KW - Calculus KW - Functions, Analytic KW - Functions, Monogenic KW - Functions, Regular KW - Regular functions KW - Aggregates KW - Classes (Mathematics) KW - Ensembles (Mathematics) KW - Mathematical sets KW - Sets (Mathematics) KW - Theory of sets KW - Sets, Analytic KW - Mathematics. KW - Functions of complex variables. KW - Functions of a Complex Variable. KW - Several Complex Variables and Analytic Spaces. KW - Complex variables KW - Elliptic functions KW - Functions of real variables KW - Math KW - Science KW - Functions of complex variables KW - Series, Taylor's KW - Logic, Symbolic and mathematical KW - Analytic spaces KW - Differential equations, partial. KW - Partial differential equations UR - https://www.unicat.be/uniCat?func=search&query=sysid:14303105 AB - 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. ER -