Preface 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 ix
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