Introduction to homological geometry: part I

This is an introduction to some of the analytic (or integrable systems) aspects of quantum cohomology which have attracted much attention during the last few years. The small quantum cohomology algebra, regarded as an example of a Frobenius manifold, is described in the original naive manner, without going into the technicalities of a rigorous definition. Then three well known analytic phenomena related to quantum cohomology are reviewed: the Landau-Ginzburg description of cohomology, the phase space of the Toda lattice, and the volume functional. The emphasis is on concrete examples, with the intention of alerting a wider audience to the interesting potential of this area. In part 2, the quantum differential equations will be studied in the same way.