Links of surface singularities and their fillings

Given an isolated singular point in a complex surface, its link is the intersection of the surface with a small sphere centered at the singular point. The link is a smooth 3-manifold that reflects the topology of the singularity. The link has a natural contact structure, and one can ask questions about its contact and symplectic topology, for example, try to describe symplectic 4-manifolds that the link can bound. A family of symplectic (even Stein) fillings is provided by possible smoothings (Milnor fibers) of the singularity, as well as a deformation of its minimal resolution. We will discuss the relation between these special fillings of algebraic origin and more general Stein fillings of the same contact manifold, and explain different ways to construct and detect Stein fillings that do not come from Milnor fibers. Partly joint with Baykur--Nemethi and with Starkston.