Classifying link maps in the four-sphere

This is the first in a series of talks in which we give a careful exposition of a recent ground-breaking paper of Rob Schneiderman and Peter Teichner (arXiv:1708.00358).
A link map is a map of a pair of 2-spheres into the 4-sphere such that the images of the 2-spheres are disjoint, and a link homotopy is a homotopy through link maps. That is, throughout the homotopy each component may self-intersect, but the two components must stay disjoint. Schneiderman and Teichner resolved a long-standing problem by proving that such link maps, modulo link homotopy, are classified by a certain invariant due to Paul Kirk. (This is a higher-dimensional analogue of the classical result in knot theory that the linking number classifies two-component links up to link homotopy.) The goal of these talks is to obtain a complete understanding of the proof of this result.
In this first, introductory talk, we assume no prior knowledge; our goal is to introduce the basic objects at play so as to understand the classification statement. In doing so we will introduce a number of techniques of four-dimensional topology. In particular, we present the basic tools used to study immersions of surfaces in four-manifolds, such as Whitney disks, finger moves, and algebraic intersection numbers.