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

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