Thomas Wolff's "Lectures on Harmonic Analysis"
Edited by Izabella Laba and Carol Shubin.
American Mathematical Society, University Lecture Series, vol. 29,
Links to files
For more information on the published version, or to purchase a copy,
go to the AMS bookstore.
This book is based on a graduate course in Fourier analysis taught
by Tom Wolff in the Spring of 2000 at the California Institute of
Technology. Tom wrote up a set of notes which he distributed to
his students and made widely available on the Internet. His
intention was to publish these notes in a book format. After Tom's
untimely death on July 31, 2000, I was asked to complete this work.
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 Prospects in Mathematics (H. Rossi, ed.,
American Mathematical Society, 1999). This article is reprinted here for
the sake of completeness. I have corrected a few misprints and added
footnotes (identified as "editor's notes") indicating further progress on
the problems discussed; no other changes or alterations have been made.
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
Vancouver, March 2003
Links to files
- 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):
- A brief summary of Wolff's recent work, written by Wilhelm Schlag and
Click here for the list of corrections (pdf file)