In this talk I will describe some applications of Seiberg--Witten theory in the study the monodromy diffeomorphisms of Milnor fibrations of isolated hypersurface singularities of complex dimension 2. Namely, we show that in many examples the monodromy diffeomorphism has infinite-order in the smooth mapping class group but has finite order in the topological mapping class group. We deduce this from a more general result about Dehn twist diffeomorphisms along Seifert-fibered 3-manifolds embedded in indefinite symplectic 4-manifolds. Time permitting I will discuss further applications to constructing exotic diffeomorphisms of exotic R4’s and contractible manifolds. 

This is based on joint work with Hokuto Konno, Jianfeng Lin and Anubhav Mukherjee.