Every summer, the IAS/Park City Mathematics Institute organizes a graduate summer school, with the main topics varying from year to year. In 2001, the school was devoted to quantum field theory, supersymmetry and enumerative geometry. The courses at the School were centred on supersymmetry and supermanifolds, general relativity, enumerative geometry, and mirror symmetry, and also included introductions to quantum field theory and string theory. The book contains lecture notes on several topics discussed at the school. There are two contributions (by W. Fulton and by A. Bertram) treating enumerative geometry. The first one describes Schubert calculus for Grassmann manifolds, its quantum version, and Gromow-Witten invariants. The second one shows how to compute Gromow-Witten invariants and their generating function by localization techniques.

A classical background needed for discussions between mathematicians and theoretical physicists is covered in the other three contributions. A long contribution (written by D. Freed) is devoted to classical field theory and its supersymmetric version (including basic facts on their quantization) and is complemented by a short lecture (by J. Morgan) on supermanifolds and super Lie groups. The last lecture notes (by V. Johnson) describe in detail the physical principles behind general relativity, along with the associated field equations and variational principles. The book collects together some useful material for anybody hoping to better understand recent important ideas coming to mathematics from theoretical physics. It can be, in particular, strongly recommended to postgraduate students and young mathematicians interested in modern streams of mathematics.