Preface xi
a broad range of topics en route—is a good way to help advanced un-
dergraduates integrate what they have learned and prepare for their
careers. Whether or not they pursue graduate studies in the particu-
lar focus areas seems secondary to me, even though I am quite fond
of the subject area of this book.
Turning the lectures into a book introduced another set of chal-
lenges. Undergraduate students reading this book will probably not
have the benefit of lectures at which they can ask questions as they
go along. In addition, the PCMI students were helped by an expert
teaching assistant, Artur Elezi. Since I assume that the reader will
not have direct access to a faculty member, I have added much back-
ground material that was left out of the lectures, while trying to keep
the informal feel throughout. I added material on differential geome-
try which was not needed in the original lectures since this topic was
covered in Ruth Gornet’s lectures. Even so, the reader is warned that
this book is not self-contained. In particular, working through this
book is not a substitute for a more thorough and more pedagogical
treatment of any of the background topics reviewed here. I have cer-
tainly cut out much important foundational material to streamline
the process of getting to the desired results. Even worse, sometimes
for the sake of expediency I have given very nonstandard treatments
of subjects. Some of these have a somewhat clumsy feel to them. This
is especially true of the ad hoc treatment I have given of the beautiful
subject of algebraic geometry. An undergraduate student interested
in algebraic geometry reading this book is urged to consult the refer-
ences provided. A graduate student interested in algebraic geometry
reading this book is urged to provide the standard definitions of all of
the concepts which have been introduced here in an ad hoc fashion.
Each of the chapters of the book corresponds reasonably closely
to one of the lectures, the exception being that the material from a lec-
ture entitled “More on bundles” has been divided up and distributed
through other chapters, so that there are only fourteen chapters re-
placing fifteen lectures. Since material has been added beyond the
content of the original lectures, each chapter contains substantially
more than one lecture’s worth of material.
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