Detecting Whitney disks for link maps in dimension four

A link map is a map of spheres into another sphere with pairwise disjoint images, and a link homotopy is a homotopy through link maps. In this talk I will discuss the problem of classifying, up to link homotopy, two-component link maps of two-spheres in the four-sphere. This setting is particularly interesting because, as usual, four-dimensional topology presents unique difficulties. After giving a brief history of the subject, I will describe how invariants of four-dimensional link homotopy arise as obstructions to equipping a link map with Whitney disks, which are the devices for performing the so-called Whitney trick. I will then discuss a result that says an invariant due to Kirk detects a certain nice variety of such Whitney disks.