The quantum cohomology ring of a flag manifold encodes information about the enumerative geometry of the rational curves in the manifold. A famous application from the mid 1990’s was Kontsevich’s computation of the number of rational curves in the projective plane of a fixed degree d, which pass through 3d – 1 general points. More recently, the “small” version of this ring was related to Schubert Calculus on infinite dimensional flag manifolds and to integrable systems, especially the Toda lattice.

My goal is to introduce the (small) quantum cohomology ring and some ideas used to perform calculations in it. An important role will be played by the “curve neighborhood” of a variety, i.e., the union of rational curves of a fixed degree intersecting that variety.