Surfaces in four-manifolds

The genus function (associating the minimal genus representative to a homology class) of a closed four-manifold can measure exotic behaviour in four-dimensional topology.
We know this function only in a handful of cases; in the lecture I will describe some of these.
The notion extends to four-manifolds with boundary by considering slice surfaces for knots in the boundary; the examination of that function leads us to a result concerning our chances of finding exotic four-spheres.