Preface to the second
Let me thank everyone who over the past decade has provided me with
suggestions and corrections for improving the first edition of this book. I
am extraordinarily grateful. Although I have not always followed these many
pieces of advice and criticism, I have thought carefully about them all. So
many people have helped me out that it is unfortunately no longer feasible
to list all their names. I have also received great help from everyone at the
AMS, especially Sergei Gelfand, Stephen Moye and Arlene O’Sean. The
NSF has generously supported my research during the writing of both the
original edition of the book and this revision. I will continue to maintain
lists of errors on my homepage, accessible through the
When you write a big book on a big subject, the temptation is to include
everything. A critic famously once imagined Tolstoy during the writing of
War and Peace: “The book is long, but even if it were twice as long, if
it were three times as long, there would always be scenes that have been
omitted, and these Tolstoy, waking up in the middle of the night, must have
regretted. There must have been a night when it occurred to him that he
had not included a yacht race...”(G. Moore, Avowals).
This image notwithstanding, I have tried to pack into this second edition
as many fascinating new topics in partial differential equations (PDE) as I
could manage, most notably in the new Chapter 12 on nonlinear wave equa-
tions. There are new sections on Noether’s Theorem and on local minimizers
in the calculus of variations, on the Radon transform, on Turing instabili-
ties for reaction-diffusion systems, etc. I have rewritten and expanded the
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