We acknowledge with gratitude the contributions to this book made by
our colleagues, students, and friends over the past forty-five years. It is not
possible to mention by name all those with whom we have had conversations
about this material, but we express appreciation to them here.
We thank Robert Fefferman, Bill Fulton, Denis Hirschfeldt, Carlos Ke-
nig, David Leep, Raghavan Narasimhan, Madhav Nori, Joseph Silverman,
and David Vogan.
We are grateful to Sam Altschul, David Coley, Moon Duchin, Jonny
Gleason, Sam Isaacson, Sean Johnson, Ryan Julian, Tom Koberda, Calvin
Lin, Chris Malon, Mike Miller, Emily Peters, Sam Raskin, Ryan Reich,
Weston Ungemach, Michael Wong, and Alex Zorn for their incisive ideas on
the material in this book, as well as their perceptive reading of portions of
We are in great debt to Mitya Boyarchenko who contributed significant
ideas to the book and to John Boller, Evan Jenkins, and Loren Spice, all of
whom also contributed ideas to the book and carefully read versions of the
My ultimate debt is owed to those who worked with me to produce this
manuscript. The word colleague describes them appropriately. The word
amanuensis could be used as a formal title, but they are much more. We ar-
gued, discussed, rewrote, reaﬃrmed, and readjusted parts of the manuscript
on many occasions. These friends are Chris Jeris, Nick Ramsey, Kaj Gartz,
Nick Ramsey (again), Nick Longo, and finally Nick Ramsey a third time.
Working with Nick Ramsey was an especially rewarding experience, since
he grew from an undergraduate student in my Honors Analysis class to a
graduate student and then to a research mathematician. This book would
never have appeared without his cooperation.
Paul J. Sally, Jr.
July 31, 2012