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The selection of the material is somewhat unconventional in that the book leads us, in Tom's unique and straightforward way, through the basics directly to current research topics. Chapters 1-4 cover standard background material: the Fourier transform, convolution, the inversion theorem, the Hausdorff-Young inequality. Chapters 5 and 6 introduce the uncertainty principle and the stationary phase method. The choice of topics is highly selective, with emphasis on those frequently useful in research inspired by the problems discussed in the remaining chapters. The latter include questions related to the restriction and Kakeya conjectures, distance sets, and Fourier transforms of singular measures. These problems are diverse but often interconnected; they all combine sophisticated Fourier analysis with intriguing links to other areas of mathematics (combinatorics, number theory, partial differential equations); and they continue to stimulate first-rate work. This book focuses on laying out a solid foundation for further reading and, hopefully, research. Technicalities are kept down to the necessary minimum, and simpler but more basic methods are often favoured over the most recent ones.

The book is intended for all mathematical audiences -- a novice and an expert may read it on different levels, but both should be able to find something of interest to them. A background in harmonic analysis is not necessary. Some mathematical maturity, however, will be helpful; the more junior readers should expect to work hard and to be rewarded generously for their efforts.

Tom's original manuscript constitutes Chapters 1-9 of this book. I have edited this part, clarifying a number of points and correcting typos and small errors. Most of the changes are quite minor, with the exception of Chapter 9A which was considerably expanded at the request of many readers of the original version. Chapter 10 is based on Burak Erdogan's notes of Tom's Caltech lectures; I am responsible for its final shape. The last part of Tom's Caltech course covered the material presented in his expository article, "Recent work connected with the Kakeya problem", originally published in

These notes could not have been published in their present form without the help and cooperation of many people. First and foremost, I would like to thank Carol Shubin, Tom's wife and the executor of his estate, for authorizing me to edit his manuscript and for providing additional materials, including Tom's handwritten notes of a series of lectures he gave in Madison in 1996. I am grateful to Burak Erdogan for providing typeset notes which form the core of Chapter 10. Jim Colliander was kind enough to send me his notes of Tom's Madison lectures. In the Spring of 2001 I gave a series of lectures at the University of British Columbia based on Tom's manuscript; I would like to thank all those who participated, including Joel Feldman, John Fournier, Richard Froese, Ed Granirer, and Lon Rosen. Alex Iosevich, Wilhelm Schlag, and Christoph Thiele taught graduate courses based on a preliminary version of this book at the University of Missouri at Columbia, California Institute of Technology, and the University of California at Los Angeles, respectively. I would like to acknowledge the valuable comments I received from them. Michael Christ and Christopher Sogge helped me identify some of the references. I am grateful to Edward Dunne, the AMS Book Program editor, who gave his wholehearted support to this project. Finally, thanks are due to the American Mathematical Society and to the Princeton University Mathematics Department for granting us their permission to reprint Tom's expository article in this book, to Charles Fefferman who kindly provided the foreword, and to the Natural Sciences and Engineering Research Council and the National Science Foundation for their financial support.

Arguments could be made that Tom might have revised significantly the existing manuscript or included other additional topics, had he had a chance to do so. In consultation with Carol Shubin and Edward Dunne, I decided to stay as close to Tom's unfinished original as possible, preserving its character and style, and to modify and complete it only where necessary. Unfinished, perhaps, but very much alive, I hope that this book will become a lasting part of Tom's legacy.

Vancouver, March 2003

**Wolff's lecture notes:****Wolff's review article "Recent work connected with the Kakeya problem"**, originally published in*Prospects in Mathematics*(Hugo Rossi, ed., AMS): .dvi file.**A brief summary of Wolff's recent work, written by Wilhelm Schlag and Carol Shubin:**.dvi file, .ps file, .pdf file.