Tangency quantum cohomology and characteristic numbers

This work establishes a connection between gravitational quantum cohomology and enumerative geometry of rational curves (in a projective homogeneous v ariety) subject to conditions of infinitesimal nature like, for example, ta ngency. The key concept is that of modified psi classes, which are well sui ted for enumerative purposes and substitute the tautological psi classes of 2D gravity. The main results are two systems of differential equations for the generating function of certain top products of such classes. One is to pological recursion while the other is Witten-Dijkgraaf-Verlinde-Verlinde. In both cases, however, the background metric is not the usual Poincare met ric but a certain deformation of it, which surprisingly encodes all the com binatorics of the peculiar way modified psi classes restrict to the boundar y. This machinery is applied to various enumerative problems, among which c haracteristic numbers in any projective homogeneous variety, characteristic numbers for curves with cusp, prescribed triple contact, or double points.