Mapping Class Groups of 4-Manifolds and Gauge Theory

I will discuss recent progress in the study of the mapping class group—that is, the group of isotopy classes of diffeomorphisms—of 4-dimensional smooth manifolds; in particular, of diffeomorphisms that generalize the classical Dehn twist. This has led to applications and to rich interplay across several areas: algebraic geometry, through the monodromy of complex surface singularities; low-dimensional topology, through exotic diffeomorphisms; and symplectic geometry, via Torelli symplectomorphisms. The techniques I will discuss come from gauge theory, specifically the study of moduli spaces of solutions to the Seiberg–Witten equations on smooth 4-manifold bundles. Based on joint work with H. Konno, J. Lin, and A. Mukherjee.