I would like to tell you a story, which – as it seems – can be instructive in various aspects. On the one hand, it is an attempt to solve a very recent problem. On the other hand, it turns out that in the solution one has to use very classical results, which have been obtained during the first 30 years of the twentieth century, and then the interest in them has somehow disappeared. They were forgotten, then reconsidered for completely different reasons, and so on.

This is a story of only a partial success. Starting to work with pleasure, like a hound following a fresh track, you reach a certain place, but the problem wagging its tail disappears somewhere behind a corner. So there is something left to do. I hope very much that those who cannot or do not want to follow all the mathematics still will be involved, interested in other things. And among those wanting to follow the mathematics in detail, somebody will be interested enough to work on the problem itself.

Technically, the problem I am speaking about concerns the attempt to understand the structure of the potential of quantum cohomology of the projective plane. I am just going to write down explicitly what this means. The projective plane, say over the field of complex numbers, yields its three-dimensional cohomology space, which is spanned by the cohomology classes of the whole plane Δ0, of a line Δ1, and of a point Δ2.