Abstract: I will describe the ball model and the Siegel space model of the n-dimensional complex hyperbolic space H^n_C. The matrix group SU(n,1) acts on H^n_C. The group of holomorphic isometries of H^n_C is PU(n,1).

Let \Gamma be a discrete subgroup of SU(n,1) which acts freely and properly discontinuously on H^n_C. I will give the definition of an automorphic form for \Gamma. I will talk about constructing automorphic forms associated to certain submanifolds of H^n_C / \Gamma .

Abstract: Levi flat hypersurfaces are characterized by vanishing of the Levi form on them. Their regular part is foliated by complex hypersurfaces, this is called the Levi foliation. It is an open question whether one can extend this foliation to the ambient space. As example of Brunella shows, near singular points the extension may exist in general only as a singular holomorphic web. A smooth holomorphic web is simply the union of several foliations. A singular web is a more general object which loosely can be thought of as a foliation with branching. In this talk I will give a detailed background concerning holomorphic webs, and will discuss some recent progress on extension of the Levi foliation. This is joint work with A. Sukhov.