Characteristic Numbers for Cubics and Sparse Quadrics

In the 19th century, Steiner asked: How many smooth conics are tangent to five given conics in the plane? His original answer, $6^5$, turned out to be wrong. The issue of correctly computing numbers like this and rigorously proving them fueled much of the development of modern intersection theory and enumerative geometry. We will see where the problem with Steiner's answer lies and how to solve similar problems that were open until recently, for instance: How many smooth cubic surfaces are tangent to 19 lines in projective 3-space? I will also explain the general notions of characteristic numbers and excess intersection. Time permitting, we might even outline some other typical approaches, like degenerations and compactifications admitting a modular interpretation. This is partly joint work with Mara Belotti, Alessandro Danelon, Jiahe Deng and Claudia Fevola.