Instanton Floer homology groups are groups associated to a 3-manifold. Their construction involves a process analogous to Morse theory where we consider a functional on the space of SU(2) connections on the manifold and study its critical points and gradient flow lines. In these two talks, we'll define the instanton Floer homology and discuss how to use it to understand 3-manifolds, knots, and links.

Given n points on a disk, we will describe how to build an A-infinity category based on the instanton Floer complex of links, and explain why it is finitely generated. This is based on work in progress with Ko Honda.