A fully nonlinear time-stepping model for water wave motion over strongly varying topography
in three dimensions is presented. The modl is fully dispersive, fully nonlinear and, and also very rapid. The kinematic and dynamic boundary
condition at the free surface are used to derive the prognostic equations. Conservation of mass yields two integral equations for the normal velocity at the free surface and the wave potential at the sea floor. These are inverted analytically be means of Fourier transform. Various levels of nonlinearity of the equations are derived. A highly efficient computational scheme is obtained by the FFT-part of the formulation. Computations exemplify how a very long tsunami with leading depression running into very shallow water develop very short waves, that in the beginning are linear, developing then into a train of solitary waves of
large amplitude. Numerical examples on the formation of very strong ocean surface waves - rogue waves - are given.

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.

In this talk I will describe the progress that has been made so far
concerning the existence of global strong solutions to the
L^{2}-critical defocusing semilinear Schr\"odinger equation. A long
standing conjecture in the area is the existence
of a unique global strong L^{2}
solution to the equation that in addition scatters to a free solution as
time goes to infinity. I will demonstrate the proofs of partial results
towards an attempt for a final resolution of this conjecture.
I will concentrate on the low dimensions but give the flavor of the results in higher dimensions for general or spherically symmetric initial data in certain Sobolev spaces.
Many authors have contributed to the theory of this equation. I will convey my personal involment to the problem and the results that I have obtained recently. Part of my work is in collaboration with D. De Silva, N. Pavlovic, G.
Staffilani, J. Colliander and M. Grillakis.

In this talk we will survey the development on the Green's functions of the Boltzmann equations. The talk will include the motivation from the field of hyperbolic conservation laws, the connection between the Boltzmann equation
and the hyperbolic conservation laws, and the particle-like and the wave-like duality in the Boltzmann equation. With all these components one can realize a clear layout of the Green's function of the Boltzmann equation. Finally we will present the application of the Green's function the an initial-boundary value problem in the half space domain.