Gauge Theory: 4-manifolds with boundary and fundamental group Z

In this talk I will discuss work in progress in which we classify topological 4-manifolds with boundary and fundamental group Z, under some mild assumptions on the boundary. We apply this classification to provide an algebraic classification of surfaces in simply-connected 4-manifolds with 3-sphere boundary, where the fundamental group on the surface complement is Z. We also compare these homeomorphism classifications with the smooth setting, showing for example that every Hermitian form over the ring of integer Laurent polynomials arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group Z. This work is joint with Anthony Conway and Mark Powell.