x Preface
were heavily attended by graduate students. Graduate students re-
peatedly told me that my lectures helped them solidify their knowl-
edge of Gromov-Witten theory, as I presented a more leisurely intro-
duction than they had seen before, made clearer than they had seen
before how Gromov-Witten theory builds upon classical enumerative
geometry, gave many examples, and connected to physics more than
they had seen before. For this reason, while writing this book pri-
marily for advanced undergraduates, I am keeping graduate students
in mind as an important secondary audience.
Designing the course was a bit of a challenge. I could assume
that the students were smart and willing to work hard and that they
had “mathematical maturity”, but I could not assume exposure to any
specific area of mathematics beyond a standard undergraduate course
in linear algebra. For that reason, the lectures contained introductory
material on abstract algebra, geometry, analysis, and topology. Most
of the participating students already knew some of these topics, but
few knew all of them. Thus the lectures served both as a review and
as an introduction to a range of areas in undergraduate mathematics.
The incorporation of physics presented another challenge, as I
could not assume anything more than exposure to a first undergrad-
uate physics course. Here, I did not even pretend to be pedagog-
ical or complete. I cut corners by explaining a range of relevant
ideas of physics via the simplest examples, emphasizing connections
to enumerative geometry throughout. While the physics lectures were
undoubtedly the most difficult part of the course for the students, I
hoped that they would get a firm impression of the myriad connections
between geometry and physics through this very brief introduction.
I have similar hopes for the reader of this book.
It has certainly been gratifying to see a number of “my” PCMI
undergraduate students currently pursuing graduate studies in this re-
search area, near and dear to my own heart. But the PCMI program
has a broader purpose—to give students a research experience that
will benefit them in their chosen careers. I propose that the model
adopted here for undergraduate training—shooting for some reason-
ably advanced ideas from graduate level mathematics while filling in
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