Fractals in Probability and Analysis

"This is a wonderful book, introducing the reader into the modern theory of fractals. It uses tools from analysis and probability very elegantly, and starting from the basics ends with a selection of deep and important results. The authors worked hard to achieve clarity; the book contains many original proofs which are expository gems. The book would serve very well for a graduate course; it is highly recommended both for students and for experts. A notable feature is a wide selection of exercises, some quite challenging, but made more accessible with an appendix containing selected hints."
Boris Solomyak, Bar-Ilan University, Israel

"This is a very valuable contribution to the field of geometric measure theory and its interactions with other branches of mathematical analysis and probability. The notions of Hausdorff measure, Hausdorff dimension and Minkowski dimension are central objects in this text, as in other books on geometric measure theory. What is special in this text, written by two major experts in geometric analysis and probability, is the emphasis on problems lying in the intersection of probability and analysis. In particular, the book studies a variety of questions in connection with self-similar sets, Frostman's theory, Weierstrass functions, Brownian motion and its relationship with the Dirichlet problem for harmonic functions, Besicovitch–Kakeya sets, and Jones' traveling salesman theorem. Many of the problems considered in the book are difficult to find in the literature. Further, very often their proofs are based on new and illuminating arguments. All in all, I think that this is a great book."
Xavier Tolsa, ICREA, Catalan Institution for Research and Advanced Studies, and Universitat Autònoma de Barcelona

'This book, written by two of the best specialists in the world, is centered on the probabilistic aspects of geometric measure theory and fractals, but also contains beautiful pure analysis arguments. The point of view is very concrete, often based on many interesting examples or methods rather than a general theory. The most impressive aspect of the book is the huge collection of exercises of all levels, which will make a serious reading of the book both a pleasure and, if the reader wants to do them all, a performance.' Guy David, Université de Paris Sud

'There are at least two outstanding features of Bishop-Peres’s new textbook that will help it stand with self-assurance … The first feature is the remarkable clarity of exposition. The proofs are beautifully presented, with a stress on communicating ideas and methods (over technicalities). This leads the authors to study the simplest cases of problems/results that already contain the most important ideas. The second feature, which moves this text into its own class among existing graduate texts on the subject, is an exceptional list of 378 exercises.' Tushar Das, MAA Reviews

'This is a technical monograph suited to practioners of geometric measure theory and analysis written by two of the world’s leaders in the field. It would make a serious study for graduate students, containing a large number of helpful examples.' Chris Athorne, Contemporary Physics