In this talk, I will present recent results on wave localization
in nonlinear random media in the frame work of the stochastic
Gross-Pitaevskii equation (describing Bose-Einstein condensation). In particular, it is shown numerically that the disorder average spatial extension of the stationary density profile decreases with an increasing strength of the disordered potential both for repulsive and attractive interactions.

We present some new results concerning the Dirichlet problem for the prescribed mean curvature equation over a bounded domain in R^n. In the case when the mean curvature is zero this can be posed variationally as the problem of finding a least area representative among functions of bounded variation with prescribed boundary values. We show that there is always a minimizer which is represented by a compact C^{1,alpha} manifold with boundary, with boundary given by the prescribed Dirichlet data, provided this data is C^{1,alpha} and it is of class C^{1,1} if the prescribed data is C^3.

One of the fundamental problems in the classification of complex surfaces is to find a new family of simply connected surfaces with p_g = 0 and K^2 > 0. In this
talk, I will sketch how to construct a new family of simply connected symplectic 4- manifolds using a rational blow-down surgery and how to show that such 4-manifolds
admit a complex structure using a Q-Gorenstein smoothing theory. In particular, I will show explicitly how to construct a simply connected minimal surface of general
type with p_g = 0 and K^2 = 3.

If time allows, I will also sketch how to construct a simply
connected, minimal, symplectic 4-manifold with b_+2 = 1 (equivalently, p_g = 0) and K^2 = 4 using a rational blow-down surgery.