Exotic smooth structures on simply connected 4-dimensional manifolds

Two manifolds M and N are called homeomorphic if there exists a bijective map f: M -> N such that both f and f^-1 are continuous. If both f and f^-1 are smooth, then M and N are called diffeomorphic. Note that if M and N are diffeomorphic, then they are also homeomorphic. The converse is false. There are many examples of a pair (M,N) such that M and N are homeomorphic but not diffeomorphic. In this talk we will focus on dimension 4. In particular, given a compact, simply-connected, 4-dimensional smooth manifold (without boundary) X, we can ask whether the underlying topological manifold X has more than one smooth structure. By analyzing the properties of the solution spaces of certain first-order PDE (Seiberg-Witten equations), we can show that many such X possess infinitely many distinct smooth structures.