Quantum Schubert Calculus

String theorists (notably Witten [W]) recently introduced the notion of a “quantum” deformation of the cohomology ring of a smooth projective variety X. This quantum deformation, or quantum cohomology ring, as it is often called, is an algebra over a formal-power-series ring which specializes to the ordinary cohomology ring, and which is defined in terms of intersection data (the Gromov-Witten invariants) on all the spaces of holomorphic maps from pointed curves of genus zero to X.