'There were lots of young analysts who flocked to Chicago in those
years, but virtually from the start it was clear that Tom had a
special brilliance...Eventually, the mathematical door would open a
crack as Tom discovered a new technique, usually of astonishing
originality. The end would now be in sight, as [he] unleashed his
tremendous technical abilities...Time after time, [Wolff] would
pick a central problem in an area and solve it. After a few more
results, the field would be changed forever...In the mathematical
community, the common and rapid response to these breakthroughs was
that they were seen, not just as watershed events, but as lightning
strikes that permanently altered the landscape' - Peter W. Jones,
Yale University.'Tom Wolff was not only a deep thinker in
mathematics but also a technical master' - Barry Simon, California
Institute of Technology. Thomas H. Wolff was a leading analyst and
winner of the Salem and Bocher Prizes. He made significant
contributions to several areas of harmonic analysis, in particular
to geometrical and measure-theoretic questions related to the
Kakeya needle problem. Wolff attacked the problem with awesome
power and originality, using both geometric and combinatorial
ideas.This book provides an inside look at the techniques used and
developed by Wolff. It is based on a graduate course on Fourier
analysis he taught at Caltech. The selection of the material is
somewhat unconventional in that it leads the reader, in Wolff's
unique and straightforward way, through the basics directly to
current research topics. The book demonstrates how harmonic
analysis can provide penetrating insights into deep aspects of
modern analysis. It is an introduction to the subject as a whole
and an overview of those branches of harmonic analysis that are
relevant to the Kakeya conjecture. The first few chapters cover the
usual background material: the Fourier transform, convolution, the
inversion theorem, the uncertainty principle, and the method of
stationary phase.However, the choice of topics is highly selective,
with emphasis on those frequently used in research inspired by the
problems discussed in later chapters. These include questions
related to the restriction conjecture and the Kakeya conjecture,
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, and they continue to stimulate first-rate work. The
book focuses on laying out a solid foundation for further reading
and research. Technicalities are kept to a minimum, and simpler but
more basic methods are often favored over the most recent methods.
The clear style of the exposition and the quick progression from
fundamentals to advanced topics ensure that both graduate students
and research mathematicians will benefit from the book.
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