This book is based on a series of ﬁfteen advanced undergraduate lec-
tures I gave at the Park City Mathematics Institute (PCMI) during
the summer of 2001. A separate book based on the program of grad-
uate lectures is being published in the Park City Mathematics Series.
My charge was to give students exposure to some of the important
ideas related to modern enumerative geometry as it has evolved in re-
cent years. My goals were to provide some background in and inspira-
tion from classical enumerative geometry, to explain the rudiments of
stable maps and Gromov-Witten theory, and to explain connections
to string theory in physics.
The students were quite good, as I had been told to expect. Stu-
dents were selected from a competitive application process. Further-
more, PCMI runs two series of undergraduate lectures: an “elemen-
tary” lecture series and an “advanced” lecture series. The students
self-select which lecture series they will attend, so my lectures were
populated by the strongest of this already talented group of under-
graduates. The elementary lectures were given by Ruth Gornet on
the subject of diﬀerential geometry.
While my experience with the undergraduate students was close
to my expectations, there was however a big surprise: my lectures