It is known that many simply connected, smooth topological
4-manifolds admit infinitely many exotic smooth structures. The
smaller the Euler characteristic, the harder it is to construct
exotic smooth structure. In this talk, we construct exotic smooth
structures on small 4-manifolds such as CP^2#k(-CP^2) for k = 2, 3,
4, 5 and 3CP^2#l(-CP^2) for l = 4, 5, 6, 7. We will also discuss the
interesting applications to the geography of minimal symplectic
4-manifolds.

Fundamental groups of 3-manifolds are known to satisfy strong
properties, and in recent years there have been several advances in their
study. In this talk I will discuss how some of these properties can be
exploited to give us insight (and results) in the study of 4-manifolds.

We investigate the transversality of holomorphic mappings between CR submanifolds of complex spaces. In equidimension case, we show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold, provided that the source manifold is of finite type and the map is of generic full rank. In different dimensions, we will show that under certain restrictions on the dimensions and the rank of Levi forms, the mappings whose set of degenerate rank is of codimension at least 2 is transversal to the target. This is a joint work with P. Ebenfelt.